The filter is subjected to a Dolph-. Chebyshev âwindowâ function. ...... is a vector of observations at time tr; n is the total number of time levels on ...... Wang, J., Jenne, Roy, Joseph, Dennis., 1996: The NCEP/NCAR 40-Year Re- analysis Project.
THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES
MESOSCALE DATA ASSIMILATION FOR IMPROVING QUANTITATIVE PRECIPITATION FORECASTS
By
SHIQIU PENG
A dissertation submitted to the Department of Meteorology in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Summer Semester, 2004
The members of the Committee approve the dissertation of Shiqiu Peng defended on 8 July 2004.
Xiaolei Zou Professor Directing Dissertation
I. M. Navon Outside Committee Member
James J. O’Brien Committee Member
Peter S. Ray Committee Member
Albert I. Barcilon Committee Member The Office of Graduate Studies has verified and approved the above named committee members.
ii
ACKNOWLEDGMENT I would like to express my deepest appreciation and gratitude to my advisor, Professor Xiaolei Zou, for her support and guidance, without which this work would not have been accomplished. From her I also “learned the ropes” with regards to publication, successfully producing three journal papers as a result of this research. The encouragement she gave me and the methodologies I learned from her will be invaluable in my future career. I want to thank my committee members, Dr. I. M. Navon, Dr. James J. O’Brien, Dr. Peter S. Ray, Dr. Albert I. Barcilon and Dr. Philip Cunningham for their great help and advice. Many thanks also go to Dr. Jon E. Ahlquist and Dr. Kwang-Yul Kim for their help on filtering and spectral analysis. I would also like to thank Dr. Henry Fuelberg for his helpful discussions with me. I am indebted to many colleagues in Dr. Zou’s lab. Lots of happy memories remind me of the times spent with Dr. Bin Wang, Dr. Q. Xiao, Dr. M. De Pondeca, Dr. H. Liu, Mr. D. Zhang, Mr. K. Park, Ms. H. Shao, Mr. C. Amerault and other graduate students. Thanks go to them for any help they gave me. Thanks are also extended to Dr. Xungang Yin and Dr. Shaoqin Zhang for their academic experience and personal friendship. My heart is filled with deep gratitude towards my mother and my late father for their unselfish love. Thanks to my brothers and sisters for their support and love. No matter where I go or where they are, I will always love them and miss them. This work is partially supported by the Office of Naval Research, under the project grant N00014-99-1-0022.
iii
TABLE OF CONTENTS
LIST OF FIGURES
vi
ABSTRACT
xi
1 Introduction
1
2 Methodology 2.1 4D-Var Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 4D-Var Formulation . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 A Numerical Forecast Model and the 4D-Var System Used in This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Digital Filter and Its Application . . . . . . . . . . . . . . . . . . . . 2.2.1 A regular digital filter . . . . . . . . . . . . . . . . . . . . . . 2.2.2 A Modified Digital Filter for Intensifing Mesoscale Signals . . 2.2.3 Applying Digital Filter in 4D-Var Through a Penalty Term . . 2.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Spectral Analysis on Time Series . . . . . . . . . . . . . . . . 2.3.2 Spectral Analysis within a Limited Area . . . . . . . . . . . . 3 Assimilation of NCEP Multi-Sensor Hourly Rainfall Data for the Prediction of A Squall Line That Occurred in Oklahoma on 5 April 1999 3.1 A Brief Case Description . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 NCEP Multi-Sensor Hourly Rainfall Observations . . . . . . . . . . . 3.3 Design of Numerical Experiments . . . . . . . . . . . . . . . . . . . . 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Impact of Observed No-Rain Information and Data Resolution 3.4.2 Effect of a Digital Filter . . . . . . . . . . . . . . . . . . . . . 3.4.3 Adjustments in Model State Variables Resulting from Rainfall Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
9 9 10 18 20 20 23 24 26 26 27
30 30 31 32 33 33 36 37 40
4 Assimilation of Ground-based GPS ZTD and Raingage Precipitation Observations for a Winter Storm That Occurred in Southern California during 5-6 Dec 1997 58 4.1 A Brief Case Description . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Ground-Based GPS ZTD and Raingage Observation . . . . . . . . . . 59 iv
4.3 4.4
4.5
Design of Numerical Experiments . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 A Mixed Use of Total Zenith Delay and Precipitation Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Analysis of Moisture, Temperature, and Wind Fields . . . . . 4.4.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . .
60 63 63 64 67 69
5 Improving QPFs by Intensifying Mesoscale Gravity Wave Signatures 93 5.1 The synoptic pattern in favor of mesoscale gravity wave occurrence . 93 5.2 Case selection and experimental design . . . . . . . . . . . . . . . . . 94 5.2.1 Case selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.2 Experimental design . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.1 Improvements on QPFs over the entire domain . . . . . . . . . 97 5.3.2 Comparing model forecasts over areas where mesoscale gravity waves propagate . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 100 6 Summary and suggested future work 120 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 122 REFERENCES
123
BIOGRAPHICAL SKETCH
130
v
LIST OF FIGURES 1
2 3 4 5 6 7 8 9
10 11 12 13
(a)surface temperature (contour interval: 1.0 C) and pressure (thin dashed line, contour interval: 4.0 hPa) at 1200 UTC 5 April 1999; (b) Specific humidity (contour interval: 1.0 g/kg) with the surface wind field overlayed (the value of maximum wind vector is 15.9 m/s), also at 1200 UTC 5 April 1999. . . . . . . . . . . . . . . . . . . . . . . . Observed 6-h accumulated precipitation at (a) 0600 UTC and observed 3-h accumulated precipitation at (b) 0900 UTC and (c) 1200 UTC 5 April 1999. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the CTRL forecast. Figure descriptions are indentical to those in figure 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-h accumulated precipitation at 0600 UTC as predicted by (a) EXP1 and (b) EXP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The threat scores for CTRL, EXP1, EXP2 and EXP3 from the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds. . . . . . . (a) The bias and (b) RMS errors for CTRL, EXP1, EXP2 and EXP3 for the accumulated rainfall from 0000 - 0600 UTC, 0600 - 0900 UTC and 0900 - 1200 UTC 5 April 1999. . . . . . . . . . . . . . . . . . . The 3-h accumulated rainfall from 0900 - 1200 UTC over the subdomain for (a) NCEP OBS; (b) CTRL; (c) EXP1; and (d) EXP2. . . . The threat scores for CTRL, EXP1 and EXP2 for the accumulated rainfall from 0900 - 1200 UTC within the subdomain shown in Fig. 7. The composite chart of the differences in the specific humidity at midlevels (around 600 hPa) between EXP2 and CTRL at different times overlayed by the wind field at 600 hPa at 0900 UTC 5 April 1999. Light shading indicates positive values greater than 0.3 g/kg, while dark shading indicates negative values less than -0.3 g/kg. . . . . . . The variation of each term of the cost function with respect to the number of iterations: (a) background term Jb ; (b) observation term Jo ; (c) penalty term Jp ; and (d) total value of the cost function Jtotal . The surface pressure perturbations for (a) CTRL; (b) EXP3; and (c) EXP4 at 0300 UTC 5 April 1999 (contour interval: 1.0 hPa). . . . . The absolute surface pressure tendency at the grid point (27,54) (as indicated in Fig. 11 (a) by the arrow) for CTRL, EXP3, and EXP4 (unit: hPa/3h). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The threat scores for CTRL, EXP3 and EXP4 for the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds. . . . . . . . . . . . vi
42 43 44 45 46 47 48 49
49 50 51 51 52
14
15 16 17 18 19 20
21
22 23 24 25 26
The differences (EXP2 minus CTRL) in (a) the initial 6-h predicted precipitation (contour interval: 2 mm) and in the specific humidity (thick solid and dashed line, contour interval: 0.5 g/kg) on 700 hPa at (b) 0000 UTC and (c) 0600 UTC 5 April 1999 with the specific humidity for CTRL overlayed (thin solid line, contour interval: 1.0 g/kg). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The averaged vertical profiles of the differences (EXP2 minus CTRL) of specific humidity (left panel) and temperature (right panel) for Type I (solid line) and Type II (dashed line) grids at 0000 UTC 5 April 1999. Schematic illustration of the instability in the moist lower layer and dry lower layer. T and T ∗ represent temperature profiles of the environment and the rising air parcel, respectively. . . . . . . . . . . . . Skew-T diagrams at the grid point (43,28) (as indicated in Fig. 14 (a) by the arrow) for CTRL (left panel) and EXP2 (right panel) at 0000 UTC 5 April 1999. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The averaged vertical profiles of the vertical velocity in Type I (thick line) and Type II (thin line) grids for CTRL (solid line) and EXP2 (dashed line), respectively, at 0300 UTC 5 April 1999. . . . . . . . . The averaged vertical profiles of moist static energy of type I grids for CTRL (solid line) and EXP2 (dashed line) at (a) 0000 UTC; (b) 0300 UTC; and (c) 0600 UTC (unit: 1000J/kg). . . . . . . . . . . . . . . The threat scores for CTRL and different sensitivity experiments for the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds. Sea level pressure (heavy solid line), the 850 hPa temperature (thin solid line) and wind fields interpolated from the NCEP global analysis to a 54-km horizontal resolution grid at 0000 UTC 6 December 1997. The contour intervals for SLP and temperature are 4 hPa and 1 K, respectively, and the value of the maximum wind vector is 25.4 ms−1 . Model experiment domain (6-km horizontal resolution) and the locations of GPS receiver sites (cross) and rain gauge stations (circle). . 1-hour accumulated rainfall ending at 0100 UTC from (a) rain gauge observations and (b) the control run of the model (contours: 1.0, 3.0, 5.0, 7.0, 10.0 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-hour accumulated rainfall ending at 0400 UTC from (a) rain gauge observation and (b) the control run of the model (contours: 1.0, 10.0, 20.0, 30.0, 40.0, 50.0 mm). . . . . . . . . . . . . . . . . . . . . . . . The values of different terms of the cost function with respect to the iteration number of the minimization for (a) ERAIN, (b) EZTD, and (c) EBOTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time series of the (a) root mean square error (RMSE) and (b) bias of ZTD between the model-derived and the observed for CTRL, EZTD, and EBOTH (unit: mm). . . . . . . . . . . . . . . . . . . . . . . . .
vii
53 54 54 55 55 56 57
71 72 73 73 74 75
27
28
29
30
31
32 33 34
35 36 37 38
39 40
The rainfall differences (4D-Var experiments minus CTRL) for (a) ERAIN; (b) EZTD; and (c) EBOTH for 1-h accumulated rainfall ending at 0100 UTC (red colour shading represents positive values greater than 0.5 mm; blue colour shading represents negative values smaller than -0.5 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . As in Fig. 27, but for 3-hour accumulated rainfall ending at 0400 UTC. (red colour shading represents positive values greater than 1.0 mm; blue colour shading represents negative values smaller than -1.0 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Threat scores for different thresholds of rainfall amounts for the control forecast and the three 4D-Var experiments for: (a) 1-hour accumulated rainfall ending at 0100 UTC, and (b) 3-hour accumulated rainfall ending at 0400 UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross section along 118.07◦ W of specific humidity differences (4D-Var experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.05 g/kg in (a); 0.2 g/kg in (b) and (c)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross section along 118.07◦ W of temperature differences (4DVar experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.05 K in (a); 0.1 K in (b) and (c)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . As in Fig. 30, except at 0100 UTC (contour interval: 0.2 g/kg in all panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . As in Fig. 31, except at 0100 UTC (contour interval: 0.5 K in all panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross section along 118.07◦ W of cloud water differences (4DVar experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.1 g/kg in (a); 0.2 g/kg in (b) and (c)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross section along 118.07◦ W of rain water differences (4DVar experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.2 g/kg in all panels]. . . . As in Fig. 34, except at 0100 UTC (contour interval: 0.2 g/kg in all panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . As in Fig. 35, except at 0100 UTC [contour interval: 0.2 g/kg in (a); 0.1 g/kg in (b) and (c)]. . . . . . . . . . . . . . . . . . . . . . . . . . Differences at 0100 UTC in the wind vectors overlayed with the differences in wind speed (solid lines) between EZTD and CTRL at: (a) 500 hPa, (b) 700 hPa, and (c) 900 hPa [contour interval: 1.0 ms−1 ; maximum wind vectors in (a), (b), and (c) are 1.9 ms−1 , 4.1 ms−1 , and 4.9 ms−1 , respectively]. . . . . . . . . . . . . . . . . . . . . . . . . . Vertical cross section along 118.07◦ W of vertical velocity at 0100 UTC from: (a) CTRL and (b) EZTD (contour intervals: 0.3 ms−1 ). . . . . Variance spectra of 1-h accumulated observed and model-produced rain data ending at: (a) 0100 UTC, (b) 0200 UTC, and (c) 0300 UTC (unit: mm2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
76
77
78
79
80 81 82
83 84 85 86
87 88 89
41
42 43 44
45
46
47
48 49
50 51
Variance spectra of the increments of 1-h accumulated rainfall between CTRL and OBS, ERAIN, and EZTD ending at: (a) 0100 UTC, (b) 0200 UTC, and (c) 0300 UTC. Values for the heavy solid line and dotted line in (b) are multiplied by a factor of 10, while those in (c) are multiplied by a factor of 100 (unit: mm2 ). . . . . . . . . . . . . . Variance spectra of differences of the v component between CTRL and any of ERAIN and EZTD at: (a) 0000 UTC, (b) 0100 UTC, (c) 0200 UTC, and (d) 0300 UTC (unit: m2 s−2 ). . . . . . . . . . . . . . . . . As in Fig. 42, but for pressure perturbation. The values for the solid line in (a) are multiplied by a factor of 1000 [unit: (Pa)2 ]. . . . . . . Schematic depiction of the conceptual model of Uccellini and Koch (1987). It delineates the 300-hPa height field (solid lines) with a deep trough, an upper-level jet streak (J) approaching the axis of inflection (west boundary of the rectangle) and moving away from the geostrophic wind maximum (Vg ) located in the base of the trough. The surface low, frontal system and the favorable area for mesoscale gravity wave occurrence (rectangular area enclosed by thick dash line) are also indicated (based on Koch and O’Handley 1997). . . . . . . . . . . . . NCEP reanalysis at 18 UTC 11 Mar 2000 of (a) sea level pressure (heavy solid line, contour interval: 1.0 hPa) and 950-hPa temperature (thin solid line, contour interval: 1.0 K) and (b) 300-hPa heights (thin solid line, contour interval: 60 gpm), wind vectors (the value of maximum vector: 62.7 m s−1 ) and isotachs of the geostrophic wind (heavy solid line, contour interval: 10 m s−1 ). . . . . . . . . . . . . . . . . . NCEP multi-sensor rainfall observations for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; (c) 3h-accumulated precipitation ending at 06 UTC; and (d) 3-h accumulated precipitation ending at 09 UTC (unit: mm). . . The response function of the modified digital filter corresponding to an intensified frequency window (0.0028 min−1 , 0.0083 min−1 ) with α1 = 1, α2 = 500 and α3 = 0. The filter is subjected to a DolphChebyshev “window” function. . . . . . . . . . . . . . . . . . . . . . (a) Time series of surface pressure perturbations (unit: hPa) and (b) the corresponding spectra (unit: (hPa)2 ) with and without the modified digital filter applied. . . . . . . . . . . . . . . . . . . . . . . . . The threat scores for CTRL, RDF and MDF over the whole domain for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC. . . . . . . . . . . . . . . . . . . . . 3-h accumulated precipitation ending at 03 UTC 12 March 2000 for (a) CTRL; (b) RDF; (c) MDF; and (d) NCEP multi-sensor observations (unit: mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The threat scores for CTRL, MDF and MDF-A over the whole domain for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC. . . . . . . . . . . . . . . . . . . . .
ix
90 91 92
102
103
104
105 106
107 108
109
52 53
54 55
56 57
58
59
60 61
The threat scores for CTRL, MDF and MDF-A over the whole domain for hourly rainfall within the assimilation window (18-00 UTC) at thresholds of (a) 1 mm; (b) 2 mm; and (c) 3 mm. . . . . . . . . . Filtered surface pressure perturbations on the scale of 50-500 km at 18 UTC 11 March 2000 for MDF (unit: hPa). Line AB and points P1, P2, P3 and P4 associated with other figures are indicated (contour interval: 0.5 hPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtered surface pressure perturbations on the scale of 50-500 km at 05 UTC 12 March 2000 for (a) CTRL; (b) RDF; and (c) MDF (contour interval: 0.4 hPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . The time-cross section along AB for the differences of filtered surface pressure perturbations on the scale of 50-500 km for (a) MDF minus CTRL and (b) RDF minus CTRL (contour intervals are 0.1 hPa with the absolute values of the contours greater than 0.3 hPa). . . . . . . (a) Time series of surface pressure perturbations (unit: hPa) and (b) the corresponding spectra (unit: (hPa)2 ) for CTRL, RDF and MDF at location P1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The threat scores for CTRL, RDF and MDF over the long rectangular area centered along line AB (as indicated in Fig. 53) for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time series of the root-mean-square-error (RMSE) for CTRL, RDF and MDF for 3-h accumulated precipitation from the 15-h forecast starting at 18 UTC 11 March 2000 over the long rectangular area centered along line AB. . . . . . . . . . . . . . . . . . . . . . . . . . Time series of 3-h accumulated precipitation of the 15-h forecast for observations, CTRL, RDF and MDF at different selected slices (170x85 km) centered at location (a) P1; (b) P2; (c) P3; and (d) P4, respectively (unit: mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The time-cross section along AB for 1-h accumulated precipitation from the 15-h forecast for (a) CTRL; (b) RDF; and (c) MDF; and (d) NCEP observations (unit: mm). . . . . . . . . . . . . . . . . . . The vertical-cross section along AB for the vertical velocity (lines) and relative humidity (shading) for (a)-(c) CTRL at 00 UTC, 03 UTC, and 06 UTC respectively; (d)-(f) RDF at 00 UTC, 03 UTC, and 06 UTC respectively; and (g)-(i) MDF at 00 UTC, 03 UTC, and 06 UTC respectively. Contour intervals are 10 mm s−1 , except those for 0600 UTC which are 5 mm s−1 . . . . . . . . . . . . . . . . . . . . . . . .
x
110
111 112
113 114
115
116
117 118
119
ABSTRACT NCEP multi-sensor hourly rainfall data and ground-baed GPS zenith total delay (ZTD) were used for data assimilation and evaluation of quantitative precipitation forecasts (QPFs) through three case studies. Improvements in QPFs were obtained through direct assimilation of these rainfall observations and ZTD data using 4dimensional variational assimilation (4D-Var). Inclusion of the observed no-rain information was shown to be beneficial to QPFs. Although the assimilation of ZTD observations does not produce a rainfall distribution as close to the observations as does the assimilation of rainfall within the assimilation window, the improvement in the QPFs beyond the window from the ZTD experiment is comparable to that from the rainfall experiment. Assimilation of ZTD and rainfall observations modifies the thermodynamic structures of the atmosphere, favoring development of precipitation in the observed rainy areas. The horizontal and vertical wind velocities are also adjusted consistent with the precipitation process. Sensitivity studies indicated that the adjustments in the moisture and temperature fields resulting from precipitation assimilation played a more important role than those of other state variables for improving QPFs. Spectral analysis indicates that rainfall assimilation adjusts the model variables on smaller scales (25 to 50 km) while the ZTD assimilation adjusts the model variables mainly on larger scales (>50 km). A modified digital filter for intensifying mesoscale gravity wave signatures is developed and applied to a real case study of rainfall assimilation. The results show that the rainfall assimilation experiment with the modified digital filter produced further improvements in quantitative precipitation forecasts compared with the rainfall xi
assimilation experiment with a regular digital filter. Spectral analysis confirms that the mesoscale gravity waves are intensified not only within the rainfall assimilation window during which the modified digital filter is applied, but also beyond the assimilation window. The gravity-wave-induced vertical motions along the direction of wave propagation are also intensified, resulting in a more realistic time evolution of the precipitation pattern. It is also found that the assimilation of 6-h accumulated rainfall outperforms the assimilation of hourly rainfall within the same 6-h window.
xii
CHAPTER 1 Introduction Short-range Quantitative Precipitation Forecasts (QPFs) has been a challenge in Numerical Weather Prediction (NWP) throughout history, although large improvements have been obtained for the short-range forecast of other quantities over the last several decades. As indicated by Olson et al. (1995), improvement of QPFs had been very slow, especially during the summer season. The difficulties of short range QPFs come from large uncertainties both in the parameterization of complicated moist physics and in initial conditions. With the development of remote-sensing technology, additional observations are made available from various types of instruments. It is expected that short-range QPFs can be improved by assimilating new observations into NWP techniques. In particular, new satellite and radar observing systems provide a large amount of rainfall/rainrate measurements which are available at asynoptic times. Improvements in the quality of analyses and forecasts produced by NWP centers are expected with increased and improved use of these existing observational data sets. The actual amount of improvement, however, depends on the ability of a data assimilation system to properly and maximally extract useful information from these new observations. Precipitation observations have long been used for NWP. Early studies include Fiorino and Warner (1981), Krishnamurti et al. (1984, 1988), and Donner (1988) who all showed that rainfall observations could be quite valuable in improving moisture analysis and reducing spin-up problems. Rainfall data was used to derive the convective heating rate, divergence, and moisture fields through various empirical 1
and statistical methods. More recently, Treadon (1997) incorporated the GOES Precipitation Index (GPI) and SSM/I-based rain rates into the National Centers for Environmental Prediction (NCEP) global data assimilation system using the nudging and three-dimensional variational (3D-Var) techniques. Falkovich et al. (2000) also used the GPI and SSM/I data to adjust the humidity in the lower atmosphere using nudging techniques. Improvement in the forecast of the precipitation systems was shown. Approaches used in these studies are either diagnostically based or static in time. Given the small-scale spatial variability and the rapid time change of rainfall, four-dimensional variational (4D-Var) assimilation may be a more appropriate approach for the assimilation of rainfall observations. 4D-Var not only allows rainfall observations to be directly assimilated at observation time, but also provides a natural multivariate constraint on model variables, maintaining a dynamical and physical consistency defined by the forecast model itself. Another advantage of 4D-Var (and also 3D-Var using adjoint physics) is that both the convective and non-convective precipitation physics contribute to the adjustments in model variables since separation of precipitation observations into convective and non-convective portions is often difficult. Early efforts have been made in the assimilation of precipitation observations using 4D-Var, including those of Zupanski, D. and Mesinger (1995), Zou and Kuo (1996), and Xiao et al. (2000). It was shown that improvements in QPFs can be expected from rainfall assimilation using 4D-Var. However, one common problem with rainfall assimilation is that, after the rainfall was assimilated, the rainfall forecast improved greatly within the assimilation window but the improvement dropped abruptly beyond the assimilation window. Further studies on the assimilation of rainfall are needed to address issues that remain unresolved. These include the sensitivity of precipitation to observation resolution, the usefulness of the observed no-rain information, the role of gravity waves in precipitation, methods used to suppress the unrealistic rain-assimilation-induced or model-induced gravity-wave oscillations while retaining or intensifying the important 2
mesoscale gravity waves, and the relative importance of various state variables to precipitation assimilation and prediction. The hourly multi-sensor rainfall data are provided routinely by NCEP/Climate Prediction Center (NCEP/CPC). These rainfall observations are available at 4-km resolution and are generated by combining approximately 3000 automated hourly raingage observations available over the contiguous 48 states with radar precipitation estimates from the Next Generation Weather Radar (NEXRAD) network. We first conducted an assimilation study incorporating NCEP/CPC hourly rainfall data to highlight some issues mentioned above, using the hourly multi-sensor rainfall data from NCEP/CPC. Numerical results are presented in chapter 3. It is also well known that water vapor in the atmosphere plays an important role in various atmospheric processes, namely precipitation processes. It was found that accurate model forecasts of precipitation at the mesoscale level are severely impaired by poor knowledge of the water vapor field (Kuo et al. 1996). On the other hand, it is not easy to measure the atmospheric water vapor with a desirable resolution in space and time for various numerical weather prediction and research purposes, mainly because of high small-scale variations of water vapor in both time and space. The traditional way of measuring the atmospheric water vapor is the use of radiosondes. The cost of this operation, however, is relatively high, with an estimated cost of $250 per launch, severely limiting the temporal and spatial resolution of water vapor measurements made with radiosonde networks (which are generally limited to twice a day in time and several hundred kilometers in space). With the rapid development of atmospheric remote sensing techniques, a new way of measuring water vapor has been developed based on the global positioning system (GPS). By making use of the sensitivity of the propagation properties of radio signals to the thermodynamic state of the atmosphere (especially to the water vapor content in the atmosphere), the GPS techniques exploit the delay that radio signals suffer as they propagate through the (electrically) neutral atmosphere. The zenith total delay (ZTD) and/or precipitatble water (PW) values can be derived from GPS 3
raw measurements of radio signal delays. PW is defined as the vertically integrated water vapor content in a column atmosphere. First, ZTD is derived from GPS raw measurements of radio signal delays, while PW is later derived from ZTD through a proportional relationship bewteen ZTD and PW. GPS precipitable water estimates are accurate at the millimeter level and can be superior to those from water vapor radiometers ( e.g., Rocken et al. 1995 and Duan et al. 1996). ZTD and PW are particularly important to NWP because they provide an estimate of available moisture which is needed by the convection and precipitation processes. Kuo et al. (1996) assimilated PW observations from the aforementioned special soundings as well as conventional data into a mesoscale model via the four-dimensional variational (4DVar) approach, and their results showed that the assimilation of PW observations together with other conventional data lead to improvements in the quality of moisture analysis and thus in the short-range precipitation forecast. Xiao et al. (2000) assimilated the Special Sensor Microwave Imager(SSMI)-derived PW into a mesoscale model and obtained improved predictions of cyclone track, cyclone-associated frontal structure, and precipiation along the front. Guo et al. (2000) assimilated GPS-derived PW via a 4D-Var approach and their results showed that the GPS-derived PW had a positive impact on the short-range precipitation forecast. Pondeca and Zou (2001 a and b) conducted both the “observing system simulation experiment” and the real GPS-based ZTD data assimilation experiments to assess the feasibility of directly assimilating the ZTD data into atmospheric models via the 4D-Var approach. They found that the three-dimensional moisture profiles retrieved from the ZTD data were comparable in accuracy to those retrieved from PW data and the assimilation of ZTD data together with profiler-wind and virtual temperature information led to improvements in short-range precipitation prediction. In their study, however, the rainfall observations were not incorporated into the assimilation procedure. It is interesting and worthwhile to assess the differences and improvements in the short-range QPFs when rainfall observations and ground-based GPS ZTD observations are assimilated together. Furthermore, it is also important for us to understand how ZTD data and 4
rainfall observations affect the thermodynamic structures of the atmosphere. Numerical results assimilating ZTD and precipitation individually and jointly are presented in chapter 4. It is well known that mesoscale gravity waves may play a very important role in severe weather development, leading to extraordinary pressure jumps (up to 14 mb in 40 min), wind gusts (up to 68 kt or 35 m/s) and thunderstorm initiation (Bosart and Seimon 1988; Uccellini 1975). They can also affect the precipitation field through modulating individual clouds and organizing them into a cloud complex as well as sparking the development of convective storms within a potentially unstable atmosphere. Generally speaking, mesoscale gravity waves are referred to as gravity waves with horizontal wavelengths of 50 to 500 km, periods of 1 to 5 h, and surface pressure perturbation amplitudes of 0.2 to 7 mb (Fiorino and Correia Jr. 2002; Uccellini and Koch, 1987; Koch and Golus, 1988). It was first demonstrated from a linear theory by Einaudi and Lalas (1975) that a gravity wave generated by wind shear can reach a sufficiently large amplitude in a moist, stratified atmosphere to bring about condensation. Uccellini (1975) showed the capabilities of gravity waves in triggering new severe thunderstorms and enhancing existing ones as the waves passed over the storms through the use of Eom’s (1975) linear model of gravity waves. His results supported the notion that observed gravity waves are capable of lifting air parcels to their level of free convection and thus initiating and modulating thunderstorm activities. Other studies have also suggested that internal gravity waves could provide the mechanical lift neccessary to spark deep convection (Brunk 1949; Matsumoto et al. 1967a, b; Matsumoto and Akiyama 1969; Matsumoto and Tsuneoka 1969; Miller and Sanders 1980). It was also found that convection could be one of the energy sources for the initiation and maintenance of the mesoscale gravity waves, while the mesoscale gravity waves in turn could modulate the convection through wave-CISK interaction (Bosart and Cussen 1973; Lin and Goff 1988; Lindzen and Tung 1976; Miller and Sanders 1980; Stobie et al. 1983; Koch et al. 1988). Mesoscale gravity waves have been studied not only through analyzing data from 5
the intensified observational network, but also through numerical simulation. Powers and Reed (1993) successfully simulated the mesoscale gravity wave event of 15 December 1987 in the central united States using the Penn State/NCAR mesoscale model version 4 (MM4). They showed that convection was the primary wave source. Kaplan et al. (1997) simulated a gravity wave event of 11-12 July 1981 during Cooperative Convective Precipitation Experiment (CCOPE) and studied the role of geostrophic adjustment in mesoscale jetlet formation using the Goddard Mesoscale Atmospheric Simulation System (GMASS) model. Zhang and Koch (2000) conducted numerical simulations of the same case as Kaplan et al.(1997) using the Penn State/NCAR mesoscale model version 5 (MM5) and explored the source of wave generation. These studies demonstrated that numerical models could capture many characteristics of mesoscale gravity waves. However, not all numerical models can simulate the observed mesoscale gravity waves successfully. Poor precipitation forecasts result in cases where the signals of mesoscale gravity waves in the model initial conditions are too weak and are subsequently damped out as the model is integrated forward in time. Precipitation associated with these wave activities is therefore often underestimated. We seek a method which could intensify the signals of mesoscale gravity waves in the model initial condition to improve QPFs. Data assimilation, especially rainfall data assimilation, could introduce high-frequency oscillations due to the imbalance between the observations and the model. Among these oscillations, some are of higher frequency and are either useless or deteriorative to the model forecasts; while some are features of mesoscale gravity waves and may play an important and beneficial role in precipitation forecasts. Therefore, it is desirable to remove the former while keeping the latter in the numerical model. Traditionally, the gravitational noise can be reduced by some appropriate initialization schemes using a geostrophic wind approximation, a balance equation or a quasigeostrophic approximation (Hinkelmann 1951; Charney 1955; Phillips 1960), especailly by the so-called nonlinear normal-mode initialization (NNMI) (Machenhauer 1977; Baer 1977; Baer and Tribbia 1977). It was also shown that the gravity-wave 6
oscillations can be controlled in variational data assimilation with a penalty term constraining time derivatives of the gravity-mode component, the surface pressure and divergence (Courtier and Talagrand 1990; Zou et al. 1993a ). However, these methods are somewhat complicated and can not keep or intensify the important mesoscale gravity waves while removing the unwanted gravitational noise. The digital filter (DF) is one of the techniques for removing the undesired highfrequency oscillations and has been widely applied in model initialization since the work by Lynch and Huang (1992) and Lynch et al. (1997). It was shown that DFs are very effective in removing high-frequency oscillations and are relatively simple both in computation and implementation. Similar to the method of Courtier and Talagrand (1990) and Zou et al. (1993a), DFs can be incorporated in 4D-Var as a penalty term to suppress oscillations with frequencies higher than a cut-off frequency (here after referred to as a regular DF; Polavarapu et al. 2000; Peng and Zou 2002; Wee and Kuo 2004). It is possible that the regular DF applied in 4D-Var may have removed those mesoscale gravity waves which are important to precipitation. As shown in our preliminary results, after applying a regular DF in 4D-Var, the initial fields of model variables became smoother but the impacts on QPFs were mixed. In Chapter 5, we test a modified DF which retains and amplifies the mesoscale gravity wave signatures while eliminating undesirable higher frequency oscillations. Mesoscale rainfall is a localized phenomena, and its direct assimilation often produces a small, passive and short-time positive impact on QPFs. How to extend the information contained in the local rainfall observations both in space and time is probably the key step to further improve the QPFs; This could be done by incorporating rainfall observations from satellite, radar and rain gauges into the NWP models. For example, Bosart and Cussen (1973) as well as Lin and Goff (1988) pointed out that convection could be a source of mesoscale gravity waves. Through wave-CISK interactions, the mesoscale gravity waves can organize the individual clouds into mesoscale cloud complexes which may last longer than normal (Lindzen and Tung 1976; Miller
7
and Sanders 1980; Koch et al. 1988). Hopefully the intensified initial mesoscale gravity waves and their propagation will provide a mechanism to spread out the localized rainfall information to remote areas, resulting in a more realistic QPF.
8
CHAPTER 2 Methodology
2.1
4D-Var Approach
An assimilation process involves various aspects of atmospheric or oceanographic science, statistics, computer science, instrumental physics, and optimal control theory. An ideal assimilation process should be one that can use all available information, including observations, the physical laws governing the flow, and known statistical properties of the flow to produce a complete and consistent description of the flow with respect to both physics and dynamics while maintaining an acceptable computational cost under the current computer framework. Estimation theory provides a general conceptual framework and a number of algorithms for solving this problem. However, because of the nonlinearity and large dimension of the atmospheric data, computational approximations must be developed to efficiently implement these algorithms for atmospheric and oceanic problems. The optimal control theory is one that has made data assimilation research achieve the most significant progress. It offers a deterministic approach of the estimation problem posed by data assimilation. In this approach, variational methods are used to formulate the data assimilation problem as an optimization problem, which can then be solved using classical numerical methods. The introduction of adjoint techniques in the so-called 4D-Var system has dramatically reduced the computational expenses by effectively computing the gradient of any model forecast aspect with respect to the model’s control variables,
9
including model initial conditions, boundary conditions, and model parameters which define the physical and numerical conditions of the integration. 2.1.1
4D-Var Formulation
(a) Fundermental concept The concept of 4D-Var and its application in meteorology was proposed as early as the 1980’s by some scientists (Le Dimet and Talagrand 1986; Derber 1987; Talagrand and Courtier 1987; Courtier and Talagrand 1987) and was further explored afterward by additional studies ( Courtier and Talagrand 1990; Thepaut and Courtier 1991; Navon et al. 1992; Rabier and Courtier 1992; Thepaut et al. 1993a; Zou et al. 1995; Zou 1996 and 1997). Its fundamental concept is to find an optimal state of the atmosphere which minimizes a given measure of the “distance” to the observations while at the same time satisfying an explicit dynamical constraint. It has the great advantage of providing exact consistency between the analysis and dynamics as expressed by the constraint (i.e., the governing equations). Its disadvantage is its high mathematical technicality and computational cost. However, with the development of high-speed computers, 4D-Var is quickly becoming a computationally feasible approach. Let us consider a set of differential equations F (x) = 0, Where x denotes the various meteorological fields which verify (1) over domain
(1) P
with boundary Γ. Suppose some estimate (e.g., observations) y obs of the fields x over P is also available. Thus, to obtain a solution of (1) which is “close” in some sense
to yobs is to find the solution of (1) which minimizes the cost function Z J(x) = P kx − yobs k2 ds,
(2)
where || || denotes a suitable norm. The above 4D-Var constrained minimization problem can be transfered to an unconstrained problem through the so-called Lagrange multiplier method, i.e., by 10
considering the minimization of the Augmented Lagrangian function of J instead of J itself. Let {, } be an inner product defined on the functional space to which F (x) belongs. With λ as an element of that space, we define the Lagrangian function of J(x) as L(x, λ) = J(x) + {λ, F (x)}.
(3)
It can be shown that the problem of determining the stationary points of (2) under constraint (1) is equivalent to the problem of determining the stationary points of (3) with respect to variables x and λ: since ∂L ∂J ∂F (x) = + {λ, } ∂x ∂x ∂x ∂L = {1, F (x)}, ∂λ
(4)
we can have ∂J ∂L ∂L = 0, F (x) = 0 ←→ = 0, = 0. ∂x ∂x ∂λ
(5)
The constrained problem is thus replaced by an unconstrained problem with respect to variables x and λ. The solution to the problem of minimizing J in (2) or L in (3) can be found by introducing an adjoint model and a standard unconstrained minimization algorithm. (b) Optimal control theory and introduction of adjoint operator Let < be a Hilbert space and x ∈ dt t0 Z t1 < R∗ (t, t0 )∇x H(x(t)), δx0 > dt = t0 Z t1 = < R∗ (t, t0 )∇x H(x(t))dt, δx0 >,
(16)
t0
where R∗ (t, t0 ) is the conjugate of R(t, t0 ), and from (16) we have Z t1 ∇x 0 J = R∗ (t, t0 )∇x H(t)dt.
(17)
t0
Let us introduce the adjoint equation of (14): −
dδx0 = A∗ (t)δx0 . dt
(18)
Assuming that the resolvent of (18) is S(t0 , t) between t and t0 , for any two solutions δx(t) and δx(t)0 of the direct and adjoint equations (14) and (18) respectively, it can be proven that their inner product is constant with time: d dδx(t) dδx0 (t) < δx(t), δx0 (t) > = < , δx0 (t) > + < δx(t), > dt dt dt = < A(t)δx(t), δx0 (t) > + < δx(t), −A∗ (t)δx0 (t) > = < A(t)δx(t), δx0 (t) > − < A(t)δx(t), δx0 (t) > = 0.
(19) 14
Let us consider the solutions of (14) and (18) at times t0 and t (t > t0 ): δx(t0 )
Direct Eq.
−→
S(t0 , t)δx0 (t)
R(t, t0 )δx(t0 ),
Adjoint Eq.
←−
δx0 (t).
Thus we have < δx(t0 ), S(t0 , t)δx0 (t) >=< R(t, t0 )δx(t0 ), δx0 (t) > .
(20)
This equality holds for any δx(t0 ) and δx0 (t), which results in S(t0 , t) = R∗ (t, t0 ).
(21)
Therefore, R∗ (t, t0 ) in (17) is just the resolvent of the adjoint equation (18) between t and t0 , allowing (17) to be written as Z t1 S(t0 , t)∇x H(t)dt. ∇x 0 J =
(22)
t0
When ∇x H(t) is added to (18) as a forcing term, we have the following nonhomogeneous adjoint equation −
dδx0 (t) = A∗ (t)δx0 (t) + ∇x H(t), dt
(23)
whose solution determined by the initial condition δx0 (t1 ) = 0 can be expressed as Z t1 0 δx (t) = S(t, τ )∇x H(τ )dτ. (24) t
From (22) and (24), we obtain ∇x0 J = δx0 (t0 ).
(25)
Therefore, the gradient of the cost-function with respect to the IC x0 at t0 is the solution of the adjoint eq. (18) at t0 , obtained by integrating the adjoint eq. (18) backward from t1 to t0 and starting from ∇x H(t1 ). In summary, for given IC u, we can calculate ∇x0 J through the following steps: (1) Starting from x0 , integrate the forward model (eq. (11)) from t0 to t1 and store the values x(t)(t0 ≤ t ≤ t1 ) which will make up the coefficients of the operator. 15
(2)Starting from δx0 (t1 ) = ∇x H(t1 ), integrate the adjoint model (eq. (18)) backwards in time from t1 to t0 , with the forcing term ∇x H(t) added to the currently computed solution at time t1 ( or at discrete observation times ti ). The final result at time t0 is the gradient ∇x0 J. In the meteorological application of 4D-Var, the following general formulation of cost-function in discretized form is usually adopted: T
−1
J(x0 ) = (x0 − xb ) B (x − xb ) +
n X r=0
obs (Hr (xr ) − yobs )T O−1 ) + Jp (26) r (Hr (xr ) − y
where x0 is the analysis vector on the analysis/forecast grid at time t0 ; x is the forecast background vector; xr is the model forecast at time tr starting from IC x0 ; yrobs is a vector of observations at time tr ; n is the total number of time levels on which observations are available; Or is the observation error covariance matrix of the r th observation time level (assuming uncorrelated observation errors in time) and is usually assumed diagonal (i.e., all observations are independent); B is the background error covariance matrix; Hr is the operator transforming the model variables to the observational quantities; and Jp is a penalty term controlling gravity wave oscillations. General cost-function (26) can be written symbolically as J = Jb + Jo + Jp ,
(27)
where Jb is the background term which measures the misfit between the model initial state and all available information prior to the assimilation period represented by the background field xb and Jo is the observation term which measures the distance of the model state from the observations at appropriate times during the assimilation window. Jo can consist of several individual terms corresponding to various types of observations. (d) Minimization of the cost-function J Once the gradient ∇x0 J is obtained, it is possible to implement a descent algorithm in which the minimum x∗0 is approximated by successive values obtained by varying x0 along its gradient (or along a direction obtained from a combination of the gradients 16
from each successive iteration). One of the major classes of algorithms commonly used for large problems in meteorology is the limited-memory quasi-Newton method (Navon and Legler 1987; Navon et al. 1992; Thepaut and Courtier 1991). From several (m+1) gradient vectors gi = ∇x0 Ji , i = k, k − 1, ..., k − m where m is an integer between 5 and 11 (Zou et al. 1993b), the limited-memory quasi-Newton method forms an approximate to the search direction dk , which is defined as: dk = −Hk gk
(28)
where Hk is the inverse Hessian matrix, i.e., the second derivative of the cost function with respect to the control variables, i.e., the model initial conditions and/or model parameters. The subscript k represents the number of iterations. A step size αk along the descent direction dk is found which satisfies: (k)
(k)
J(x0 + αk dk ) = minα J(x0 + αdk )
(29)
Once the step-size αk is determined, the initial condition obtained at the k th iteration is updated as: (k+1)
x0
(k)
= x 0 + α k dk
(30)
The next iteration continues by finding the next search direction dk+1 according to the new gradient information. This continues until it reaches a convergence, i.e., (k+1)
kgk+1 k ≤ εmax{1, kx0
k}
(31)
Various limited-memory quasi-Newton methods (Shanno and Phua, 1980; Gill and Murray, 1979; Liu and Nocedal, 1989; and Buckley and Lenir, 1983) differ in the selection of m for gi (i = k, k − 1, ..., k − m), the choice of the zeroth iteration Hessian matrix H0 (which is generally taken to be the identity matrix or some other diagonal matrix for preconditioning purposes), the method for computing Hk gk , and the line-search implementation which determines the suboptimal step size αk . For instance, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method 17
of Liu and Nocedal (1989) is used in the MM5 4D-Var system for determining the Hessian Matrix Hk : Hk+1 = Hk + ( (k+1)
where Pk = x0
1 + qTk Hk qk Pk PTk Pk qTk Hk + qk PTk Hk ) − qTk Pk PTk qk qTk Pk
(32)
(k)
− x0 , qk+1 = gk+1 − gk , and H0 is set to be the identity matrix
I. Superscript T represents the transpose of a matrix. 2.1.2
A Numerical Forecast Model and the 4D-Var System Used in This Study
The numerical forecasting model to be used in this study is the Penn State/NCAR nonhydrostatic mesoscale model version 5 (MM5). It is a limited-area, nonhydrostatic primitive equation, finite-difference model with multiple options of physical parameterization schemes (Anthes and Warner 1978; Dudhia, 1993; Grell et al., 1994). The model uses a terrain-following σ-coordinate defined entirely from a reference state (p0 (z), T0 (z), ρ0 (z) ): σ=
p0 − p t ∗ , p = ps − pt , p∗
where ps and pt (pt = 100 hPa) are the surface and top reference pressures of the model, respectively. In terms of the terrain following coordinate (x, y, σ), the basic equations of the nonhydrostatic MM5 model are: σ ∂p∗ ∂p0 ∂m ∂m uw ∂u m ∂p0 + − ∗ −v + Du = −V · ∇u + v f + u − ewcosα − ∂t ρ ∂x p ∂x ∂σ ∂y ∂x rearth ∂v m ∂p0 σ ∂p∗ ∂p0 ∂m ∂m vw + − ∗ −v + Dv = −V · ∇v − u f + u + ewsinα − ∂t ∂x ∂w ρρ0 g∂y p0 ∂σ p0 T 0 gRd p0∂y u2 r+earth v2 ∂p0 p g ∂y + + = −V · ∇w + g − + e(ucosα − vsinα) + + Dw ∂t ρ p∗ ∂σ γp p T0 cp p rearth ! ˙ ∂p0 γp Q T 0 − ρ0 gw + γp∇ · V = −V · ∇p0 + + Dθ , (33) ∂t T cp θ0 where m is the map-scale factor; e = 2Ωcosλ is the other component of Coriolis (usually neglected); α = φ−φc is the rotation angle of the grid; and Du , Dv , Dw , andDθ are the diffusion terms. The representations of other symbols are the same as their 18
common uses in meteorology. The model adopts a second order time-split leapfrog time scheme and a second order centered space scheme. The MM5 provides multiple options for physics (including cumulus parameterization schemes, PBL schemes, explicit moisture schemes, radiation schemes and ground temperature schemes). It also provides options for a one-domain run or a multidomain nested run. The MM5 adjoint-based 4D-Var data assimilation system was developed by Zou et al. (1997). It includes the developments of the MM5 tangent linear model (TLM), adjoint model, and a minimization package which adopted the iterative limited-memory Newtonian method developed by Liu and Nocedal (1989), the former two of which are based on the nonhydrostatic version of MM5. The MM5 TLM was first developed by linearizing the nonlinear MM5 model around the basic state at every time step. The adjoint model was then obtained by transposing the operator of TLM at the coding level. A rigorous check should be conducted to determine the correctness of the TLM, the adjoint model, and the gradient calculation. Some numerical testing and the performance of the system can be found in the works of Zou et al. (1995), Kuo et al. (1995), Zou (1996), Zou and Kuo (1996) and Zou et al. (1997). Adjoints of physical parameterization processes are required in the 4D-VAR system to increase the realism of the numerical model and to be able to assimilate new types of indirect observations which have a strong relation to moist physics and surface processes. In the experiments of this study, the physics selected for both the forecast and the 4D-Var data assimilation are consistent and include the following: the Kuo-type cumulus parameterization scheme (or no cumulus parameterization scheme for those experiments with small grid size), the Blackadar high resolution planetary boundary layer parameterization, the Dudhia simple ice scheme, radiative cooling of the atmosphere, and the five-layer soil model for ground temperature. The model used in this study also includes surface friction, surface heat and moisture fluxes, inflow/outflow boundary conditions, and grid-resolvable precipitation. 19
In this study, the model state vector comprises all gridpoint values of the three components of wind field, temperature, perturbation pressure, specific humidity, cloud water, and rain water. As discussed by Dudhia (1993), nonhydrostatic models have an advantage over hydrostatic models in allowing more localized studies, particularly by resolving topographical effects and deep convection. Since the MM5 is a regional model, it requires an initial condition as well as lateral boundary conditions to run. To produce lateral boundary conditions for a model run, one needs gridded data to cover the entire time period during which the model is integrated. The initial conditions and lateral boundary conditions are generated from the NCEP-NCAR 2.5◦ × 2.5◦ latitude-longitude global reanalysis (Kalnay et al. 1996).
2.2 2.2.1
Digital Filter and Its Application
A regular digital filter
The digital filter, developed by Lynch and Huang (1992), was based on the Convolution theorem: Let Y (f ), X(f ), H(f ) be the Fourier transforms of the functions y(t), x(t), h(t), respectively. According to the convolution theorem, the following conclusions are true: y(t) =
Z
∞ −∞
h(τ )x(t − τ )dτ or
y(t) = h(t)x(t) ←→ Y (f ) =
Z
Z
∞ −∞
x(τ )h(t − τ )dτ ←→ Y (f ) = H(f )X(f ),
∞ 0
−∞
0
0
H(f )X(f − f )df or
Z
(34)
∞ −∞
X(f 0 )H(f − f 0 )df 0 . (35)
These conclusions also hold true for discrete functions. (a) Filtering a continuous function Consider a function of time f (t) with low- and high-frequency components. To filter out the high frequencies one may proceed as follows: 1) calculate the Fourier transform F (ω) of f(t); 2) set the coefficients of the high frequencies to zero; and 3) 20
calculate the inverse transform. Step 2 may be performed by multiplying F (ω) by an appropriate weighting function H(ω). Typically, H(ω) is defined as a step function 1, |ω| ≤ |ω | c H(ω) = , (36) 0, |ω| > |ω | c
where ωc is the cutoff frequency. In other words, we have F DF (ω) = H(ω)F (ω).
(37)
The filtered (high frequency components are removed) time series f DF (t) (i.e., the inverse Fourier transform of F DF (ω)) can be obtained by the convolution theorem: Z ∞ DF h(τ )f (t − τ )dτ, (38) f (t) = −∞
where h(t) is the inverse Fourier transform of H(ω) and is equal to h(t) =
sin(ωc t) . πt
(39)
Therefore, f DF (t) can be obtained by an integration of h(τ )f (t − τ ). The Fourier transform and its inverse transform are not needed. (b) Filtering a discrete function Suppose now that f only has values at discrete moments tn = n∆t, i.e., in the sequence {..., f−2 , f−1 , f0 , f1 , f2 , ...}. The maximum frequency that can be represented with a time step ∆t is ωN = π/∆t, the so-called Nyquist frequency. This time series {fn } may be regarded as the Fourier coefficients of a function F (θ): F (θ) =
∞ X
fn e−inθ ,
(40)
−∞
where θ = ω∆t is the digital frequency and F (θ) is a periodic function such that F (θ) = F (θ + 2π). High-frequency components may be eliminated by multiplying F (θ) by a function H(θ), which is defined as 1, |θ| ≤ |θ | c H(θ) = , 0, |θ| > |θ | c 21
(41)
where the cutoff frequency θc is assumed to fall within the Nyquist range (−π, π), and H(θ) has period 2π and may be expanded as H(θ) =
∞ X
hn e−inθ .
(42)
−∞
Let F DF (θ) = H(θ) · F (θ)
(43)
be the low-frequency part of F (θ). The inverse Fourier transform of F DF (θ), i.e., the time sequence of {fnDF }, is the filtered time series containing only the low frequency part. From the convolution theorem, we have fnDF
=
∞ X
hk fn−k ,
(44)
−∞
where 1 hk = 2π
Z
π
H(θ)eikθ dθ = −π
sin kθc . kπ
(45)
We notice that the filtering is performed directly on the given time series {fn }. In practice, the summation must be truncated at some finite value of k. Thus, an approximation to the low-frequency part of {fn } is given by fnDF =
N X
hk fn−k .
(46)
k=−N
However, truncation of a Fourier series would give rise to the so-called Gibbs oscillations. As demonstrated in Lynch and Huang (1992), the Gibbs oscillations can be greatly reduced by multiplying hk by the Dolph-Chebyshev “window” function (Lynch 1997): M X 1 Wk = [1 + 2r T2M (x0 cos θm /2) cos mθk ], 2M + 1 m=0
(47)
where 1/x0 = cos(π∆t/τs ); 1/r = cosh(2M cosh−1 x0 ); θk = k(2π/M ); T2M is the Chebyshev polynomial of degree 2M; and ∆t is the time interval of the sequence {fn }. We set M = N . As proposed and discussed in Lynch (1997), choosing τs = M ∆t is 22
the most reasonable value. A larger value of τs does not sufficiently filter the Gibbs oscillations while a smaller value will have a damping effect on the frequencies in the pass-band of the filter. 2.2.2
A Modified Digital Filter for Intensifing Mesoscale Signals
The key in designing a new digital filter for intensifying the mesoscale gravity wave signatures while eliminating the high frequency oscillations is to define a suitable window function H(θ). Suppose the frequencies of the mesoscale gravity waves are between θc1 and θc2 (0 < θc1 < θc2 ) while those of the synoptic scale are lower than θc1 and those of random noise are higher α , 1 H(θ) = α2 , 0,
than θc2 . Thus, we define H(θ) as |θ| < θc1
θc1 ≤ |θ| ≤ θc2 ,
(48)
|θ| > θc2
where the cutoff frequencies θc1 and θc2 are assumed to fall within the Nyquist range (−π, π). H(θ) has period 2π and may be expanded as H(θ) =
∞ X
hk e−ikθ ,
−∞
where hk is the inverse Fourier transform of H(θ) and can be derived as follows:
23
(49)
hk
1 = 2π = = = = =
= = 2.2.3
Z
π
H(θ)eikθ dθ −π −θc2
Z π Z θc2 Z θc1 Z −θc1 1 ikθ ikθ ikθ ikθ α2 e dθ + 0eikθ dθ) α1 e dθ + α2 e dθ + ( 0e dθ + 2π −π θc1 θc2 −θc1 −θc2 Z −θc1 Z θc1 Z θc2 α1 α2 eikθ dθ + eikθ dθ + ( eikθ dθ) 2π −θc1 2π −θc2 θc1 Z θc1 Z θc2 Z −θc1 α2 α1 deikθ + deikθ ) deikθ + ( 2πik −θc1 2πik −θc2 θc1 α2 −ikθc1 α1 ikθc1 −ikθc1 (e −e )+ (e − e−ikθc2 + eikθc2 − eikθc1 ) 2πik 2πik α1 α2 [(cos(kθc1 ) + isin(kθc1 )) − (cos(kθc1 ) − isin(kθc1 ))] + [(cos(kθc1 ) − isin(kθc1 )) 2πik 2πik −(cos(kθc2 ) − isin(kθc2 )) + (cos(kθc2 ) + isin(kθc2 )) − (cos(kθc1 ) + isin(kθc1 ))] α1 α2 2isin(kθc1 ) + 2i(sin(kθc2 ) − sin(kθc1 )) 2πik 2πik α1 α2 sin(kθc1 ) + (sin(kθc2 ) − sin(kθc1 )). kπ kπ Z
Applying Digital Filter in 4D-Var Through a Penalty Term
Following the work of Polavarapu et al. (2000) and Peng and Zou (2002), a digital filter (DF) can easily be applied as a penalty term in the cost function during the 4D-Var data assimilation. The penalty term takes the following form DF Jp = α2 < xN − xDF N , xN − xN >,
(50)
where “< , >” is defined as an Euler norm, xN is the vector of model forecast variables at the N th time step (the middle of the assimilation window), and xDF N is the model forecast in the middle of the assimilation window after a DF has been applied, such that: xDF N
=
2N X
hN −k xk .
(51)
k=0
Notice that we have changed the summation over k from 0 to 2N in the above equation instead of from −N to N as in eq. (46). Although the digital filter could be applied at all times within the assimilation window (Gustafsson, 1993), it is found that applying the digital filter only at one 24
time, say, in the middle of the assimilation period, is sufficient (Polavarapu et al. 2000). Minimization of J requires the gradient of Jp . In the following, we derive an expression for calculating the gradient of Jp , ∇Jp . As a first-order approximation, the variation of Jp can be written as DF δJp = 2α2 hxN − xDF N , δxN − δxN i.
(52)
Substituting δxDF N
=
2N X
hN −k δxk ,
(53)
k=0
δxk = Mk δx0
(54)
into (52) we obtain δJp = 2α2 hxN − xDF N , δxN −
2N X
k=0 2N X
= 2α2 hxN − xDF N , (MN − = 2α
2
2N X k=0
hN −k δxk i
k=0
hN −k Mk )δx0 i
gN −k hMTk (xN − xDF N ), δx0 i,
(55) (56) (57)
where Mk is the tangent linear model operator, MTk is the adjoint model operator, and gN −k
−h N −k , k 6= N = 1−h N −k , k = N
(58)
Based on the gradient definition: δJp =< ∇J, δx0 >, we obtain ∇Jp = 2α
2
2N X k=0
gN −k MTk (xN − xDF N ).
(59)
From (59) it is found that the gradient of Jp can be obtained in a straightforward manner during the backward integration of the adjoint model with an additional forcing term 2α2 gN −k (xN − xDF N ) added to the adjoint variable at the kth time step. 25
The implementation of a regular or a modified DF in 4D-Var is the same except for the difference in the calculation of hk . The additional computational cost for the calculation of both Jp and ∇Jp , required for the incorporation of DF in 4D-Var, is very small.
2.3 2.3.1
Spectral Analysis
Spectral Analysis on Time Series
The power spectral density function is defined as ∞ 1 X f (ω) = γ(k)e−ikω 2π k=−∞
(60)
where γ(k) is the covariance function of the time series and is assumed to satisfy P∞ −∞ |γ(k)| < ∞ (i.e., γ(k) is absolutely summable). Since γ(k) = γ(−k), the above equation can be written as
∞ X 1 f (ω) = (γ(0) + 2 γ(k)cos(jkω)). 2π k=1
(61)
From (60), we have γ(k) = Setting k = 0 in (62), we have
Z
∞
f (ω)eikω dω.
−∞ Rπ 2 σ = −π
(62)
f (ω)dω. We see that the total variance of
the process can be “decomposed” into contributions from different frequencies, and f (ω)dω represents the contribution to the total variance of the components in the frequency range (ω, ω + dω). In practice, we usually have discrete sampling time series in limited length. The spectrum estimation of discrete time series can be written as 1 fˆ(ωj ) = 2π
n−1 X
k=−(n−1)
γˆ(k)e
−ikωj
n−1 X 1 = (ˆ γ (0) + γˆ(k)cos(kωj )), 2π k=1
(63)
where ωj = 2πj/n are discrete Fourier frequencies (j=0,1,..,n-1 where n is the length of sample time series), and γˆ(k) is the sample covariance function. 26
In order to reduce the fluctuations in the sample spectrum and get a smoothed spectral density function, a window function is usually applied to the sample spectrum. 2.3.2
Spectral Analysis within a Limited Area
This method was developed by Errico (1985). Consider a limited model domain with grid points NI ×NJ and grid spacing ∆x. The grid points are labeled by coordinates i, j (where i = 1, ..., NI , increasing eastward, and j = 1, ..., NJ , increasing northward). If the domain is small enough (under 5000×5000 km2 ), the variation of the map scale factor is small and can be neglected (i.e., the factor can be set to 1). A spectral analysis of any field over the domain consists of the following two steps: (a) Removal of trend For a chosen domain, there are many global scales (e.g., zonal wave number 1-6) which cannot be resolved. These large unresolved scales may alias into smaller scales if they are not removed from the data, significantly distorting the spectra on the small scales calculated from domain grids if the variance of those large scales is large, a common occurance. Errico’s method to remove the large, unresolved scales from any field ai,j involves removing linear trends along all points of constant i or j . The trends are defined by the boundary values of the field. The procedure includes the following steps: (1) Remove the trend in i-direction: 0
ai,j = ai,j − isj +
NI + 1 sj ; i = 1, ..., NI ; j = 1, ..., NJ , 2
(64)
where sj is the slope in i-direction: sj =
aNI ,j − a1,j ; j = 1, ..., NJ . NI − 1
(2) Remove the trend in j-direction: 0
aN i,j = ai,j − jsi +
NJ + 1 si ; i = 1, ..., NI ; j = 1, ..., NJ , 2 27
(65)
where si is the slope in j-direction: si =
ai,NJ − ai,1 ; i = 1, ..., NI . NJ − 1
aN i,j is the detrended field we want and is independent of the order in which the trends are removed (either i or j ). It can be proven that such a detrended field is periodic, i.e., N aN i,1 = ai,NJ ; i = 1, ..., NI N aN 1,j = aNI ,j ; j = 1, ..., NJ .
(b) Determination of spectra N Suppose aN i,j is the detrended field. The spectral coefficients cp,q of ai,j are deter-
mined by the discrete two-dimensional Fourier transform: cp,q
N J −1 I −1 N X X √ 1 aN = i,j × exp{− −1∆x[p(i − 1) + q(j − 1)]}, (66) (NI − 1)(NJ − 1) j=1 i=1
where NI 2πl 1 ; l = 0, ±1, ..., ± , ∆x NI − 1 2 2πl 1 NJ q= ; l = 0, ±1, ..., ± . ∆x NJ − 1 2 p=
In (66), NI /2 and NJ /2 should be truncated to integers. In practice, only half of the coefficients need to be explicitly calculated since cp,q = c∗−p,−q , where the asterisk denotes complex conjugation. Values of cp,q determine a spectrum in a two-dimensional, discrete vector wavenumber (p, q) space. They can also be transformed to a spectrum in a one-dimensional (k ) space by a summation within discrete annuli in p, q space with a set of k being defined as the central radii of those annuli, i.e., S(k) =
X
cp,q c∗p,q ,
1 1 k − ∆k < (p2 + q 2 )1/2 < k + ∆k. 2 2 28
(67)
In most cases, since NI 6= NJ , the values of ∆k should be determined from the minimum of the fundamental (l = 1 ) values of p and q. For example, if NI > NJ , then p=
1 2π ∆x NI − 1
(68)
is the minimum value which defines ∆k , and the values of k are specified as k = l∆k; l = 0, 1, ...,
29
NI . 2
(69)
CHAPTER 3 Assimilation of NCEP Multi-Sensor Hourly Rainfall Data for the Prediction of A Squall Line That Occurred in Oklahoma on 5 April 1999
3.1
A Brief Case Description
The squall line case that occurred over Oklahoma and Texas from 0000 UTC 5 April 1999 to 0000 UTC 7 April 1999 is chosen for this study. Figure 1 shows the sea-levelpressure (SLP) and temperature at the surface (Fig. 1a) and the specific humidity and wind at the surface (Fig. 1b) at 1200 UTC 5 April. Strong moisture and temperature gradients and a convergence line are seen in southern Kansas, central Oklahoma and Texas. The wind vectors indicate the existence of a convergence line following the maximum low-level moisture gradient. This area of convergence is identified as a squall line (will be called the OK squall line). The observed 6-h accumulated precipitation ending at 0600 UTC, and the 3-h accumulated precipitations ending at 0900 UTC and 1200 UTC are shown in Fig. 2. The initial 6-h precipitation assumed a line-pattern, following the OK squall line. Another heavy precipitation area is observed over northern Louisiana and southern Arkansas, which was induced by the intense moisture convergence from the Gulf of Mexico behind a strong ridge and ahead of a trough. In the following 6 h (0600 - 1200 UTC), the precipitation associated with the OK squall line intensified and moved northeastward. At 1200 UTC, precipitation was observed over northern Oklahoma, central Kansas and southeastern Nebraska, 30
with a 3-h maximum the 3-h rainfall of 50 mm over southern Kansas. The heavy rainfall over Louisiana-Arkansas diminished gradually during this later 6-h period.
3.2
NCEP Multi-Sensor Hourly Rainfall Observations
The rainfall observations used in this case study are the hourly multi-sensor rainfall data from the National Centers for Environmental Prediction (NCEP) Climate Prediction Center (CPC). A prototype, real-time, hourly, multi-sensor National Preciptation Analysis (NPA) has been developed at NCEP in cooperation with the Office of Hydrology (OH). This analysis merges two data sources (i.e., rain gauge and radar) that are currently being collected in real-time by the OH and NCEP. Approximately 3000 automated, hourly rain gauge observations are available over the contiguous 48 states via the GOES Data Collection Platform (DCP) and Automated Surface Observing System (ASOS). In addition, hourly digital precipitation (HDP) radar estimates are obtained as compressed digital files via the Automation of Field Operations and Services (AFOS) network. The HDP estimates are created by the WSR-88D Radar Product Generator on a 131 x 131 4-km grid centered over each radar site. The data analysis routines, including a bias correction of the radar estimates using the gage data, have been adapted by NCEP on a national 4-km grid from algorithms developed by the OH (“Stage II”) and executed regionally at NWS River Forecast Centers (RFC). The analysis schemes used in multi-sensor analyses utilize optimal estimation theory. These were developed by Seo (1998). The schemes optimally estimate rainfall fields using rain gauge and radar data under partial data coverage conditions. This is preferred over previous statistically-based techniques because it takes into account the variability due to fractional coverage of rainfall as well as intrastorm variability. By objectively taking the spatial coverage into account, more accurate estimates of the rain versus no-rain area are obtained. Accurate delineation of this area is as important as the accurate estimation of rainfall within the rain area. One of the underlying assumptions in the radar-gage analysis scheme is that the radar 31
estimates are unbiased. Currently, radar biases are adjusted prior to the multi-sensor analysis by the technique developed by Smith and Krajewski (1991). This method attempts to remove the mean bias but does not attempt to remove range-dependent biases.
3.3
Design of Numerical Experiments
The hourly multi-sensor rainfall data during the 6-h assimilation window from 0000 UTC to 0600 UTC 5 April 1999 was incorporated into the model. Four 4D-VAR experiments (see Table 1 for details) were conducted to study the possible impacts of the resolution of precipitation data, the inclusion of observed no-rain information, and the role of a digital filter (DF). The cost function for these experiments was defined as: T
J(x0 ) = (x0 − xb ) W(x0 − xb ) +
6 X N X
tn =0 i=1
β(H(Rif (tn )) − Riobs (tn ))2 + Jp
(70)
where x represents the vector of model state variables, and the subscripts “0” and “b” denote the initial state and the background field, respectively. H is a unit matrix for EXP1 and EXP2 and represents an interpolation of model-predicted hourly rainfall Rif (tn ) from 30-km to 4-km resolutions for EXP3 and EXP4. Riobs (in units of cm) represents the observed hourly rainfall data with a horizontal resolution of 30 km for EXP1 and EXP2 and 4 km for EXP3 and EXP4, respectively. N denotes the total number of observational points. The first term in (70) is a simple background term measuring the distance between the model initial condition x0 (to be adjusted through an iterative minimization procedure) and the background field xb (where the MM5 standard analysis was used as the first guess). Only approximated variances are included in the background weighting matrix W, which is calculated based on the difference between the 6-h forecast and the initial condition. The weighting parameter for precipitation observations β was set to 100 cm−2 , corresponding to an estimated precipitation observation error of 1 mm. The third term in (70) is a penalty term 32
Table 1: Numerical experiments design
EXP1 EXP2 EXP3 EXP4
Data resolution 30 km 30 km 4 km 4 km
Observed no-rain information No Yes Yes Yes
Digital filter penalty No No No Yes
Assimilation window 6 hours 6 hours 6 hours 6 hours
which applies the digital filter (DF), described in Chapter 2. This term was included only in EXP4. Minimization of J defined in (70) requires the gradient of J, i.e., ∇J, where ∇J = ∇Jb + ∇Jo + ∇Jp
(71)
The derivition of ∇Jp is given in Chapter 2. The lateral boundary conditions are fixed in all of the data assimilation experiments since a relatively large model domain is used in this study.
3.4
Results
In order to see impacts of rainfall data assimilation on QPFs, a comparison among the forecasts from various 4D-Var experiments and the control run (CTRL) is presented in this section. 3.4.1
Impact of Observed No-Rain Information and Data Resolution
We compare the forecasts from each 4D-Var experiment. The 6-h accumulated rainfall from 0000 - 0600 UTC is shown in Fig. 4 for both EXP1 and EXP2. Compared with observations (Fig. 2a) and CTRL (Fig. 3a), the initial 6-h rainfall amount associated with the squall line over northwestern Oklahoma increased to that of the observations.
33
The false precipitation over eastern Colorado and southern Texas in CTRL and EXP1 was removed in EXP2 which incorporated observed no-rain information. The Threat Score (TS) is a typical measure for the verification of QPFs. It is the ratio of the correctly predicted area to the threat area, defined as the envelope of the forecast and observed areas. It can be expressed as: TS =
Nc Nf + N o − N c
(72)
where Nf denotes the number of points on which the forecast rainfall is greater than a given threshold, No the number of points on which the observed rainfall is greater than the threshold, and Nc denotes the number of points on which both the the forecast and observed rainfall are greater than the threshold. A perfect rainfall forecast would have a TS value of one. The threat scores of precipitation from EXP1, EXP2, and EXP3 are shown in Fig. 5. The threat scores for EXP1, EXP2 and EXP3 are shown to be higher than CTRL both within and beyond the assimilation window. The improvements are much more significant during the assimilation window (Fig. 5a) than after the assimilation window (Fig. 5b and 5c). A comparison between EXP1 and EXP2 indicates that the assimilation of observed no-rain information is both important and beneficial to the QPFs. Differences resulting from using different resolution data sets on the rainfall assimilation and QPFs (EXP2 and EXP3) are usually much smaller than the improvements over CTRL. The mean errors and the root mean square (RMS) errors of the rainfall forecast for CTRL, EXP1, EXP2, and EXP3 are shown in Fig. 6. Both the mean and RMS errors from all of the 4D-VAR experiments are smaller than those from CTRL, except for the mean errors from EXP1 from 0000 - 0600 UTC. Results from EXP2 are consistently better than EXP1. Therefore, both the RMS and threat score results show the improvements of forecast skill of the QPFs by assimilation of the NCEP multi-sensor rainfall data and the advantages of including the observed no-rain information. The data resolution impact, however, is relatively small. 34
In order to trace the impacts of including the observed no-rain information in the rainfall assimilation on QPFs, we focus on a subdomain of the southern tail of the squall line. Figure 7 shows the 3-h accumulated rainfall at 1200 UTC in this subdomain from observations, CTRL, EXP1, and EXP2. It is shown that heavy rainfall was observed over the border of Oklahoma and Texas (Fig. 7a), but the rainfall predicted by both CTRL and EXP1 is rather weak. However, in EXP2, where observed no-rain information was included in data assimilation, rainfall amounts over the border of Oklahoma and Texas increased. Figure 8 displays the threat scores for CTRL, EXP1, and EXP2 within this subdomain. It is shown that the threat scores of EXP2 are higher than those of both CTRL and EXP1. The RMS errors of the 3-h accumulated rainfall at 1200 UTC in this subdomain, corresponding to Fig. 7, are 2.7 mm, 2.33 mm, and 2.2 mm for CTRL, EXP1, and EXP2, respectively. Why are the QPFs in EXP2 for the subdomain better than that of EXP1? The evolution of the moisture field provides some answers. Figure 9 shows the positions of the largest differences of the specific humidity between EXP2 and CTRL in the midlevels of the atmosphere at 0600, 0900, and 1200 UTC 5 April 1999 overlayed by the wind field at 600 hPa at 0900 UTC. It is shown that the moisture amounts were reduced over the false-forecasted rain region in southern Texas (also see Figs. 2a and 3a) at 0600 UTC after the assimilation of observed no-rain information. An increase in moisture amounts (positive center) occurred downstream (next to the negative center). Afterward, this positive and negative couplet of moisture adjustment developed and advected northeastward following the wind flow. By 1200 UTC, a strong increment of moisture was seen over southern Oklahoma and northern Texas. This feature of moisture adjustment was not seen in the assimilation without observed no-rain information (EXP1). Therefore, the observed no-rain information had a role not only in correcting the local false forecast of rainfall but also in improving the rainfall forecasts in other regions.
35
3.4.2
Effect of a Digital Filter
Due to observation errors and inconsistency between model output and observations, high frequency oscillations can be introduced by the assimilation of rainfall observations. In order to remove these undesirable high frequency oscillations, a digital filter (DF) (Lynch and Huang 1992) can be applied through a penalty constraint (see Section 3.2). Results from EXP4 illustrate some of the features when applying a DF to rainfall assimilation. In this experiment, the cut-off time period was set to be 6 hours, i.e., the same as the assimilation time window. Figure 10 presents the variation of the cost function with respect to the number of iterations. Adding the DF penalty term (EXP4) improved the performance of the minimization and produced a closer fit to observations before the minimization finds a local minimum. As expected, the value of the background term, which measures the difference between the background field and the initial condition at each iteration and is zero at the 0th iteration, increases with the number of iterations as the observations are fitted closer and closer. The surface pressure perturbation at 0300 UTC 5 April from CTRL, EXP3, and EXP4 are shown in Fig. 11. The absolute surface pressure tendency at a selected grid point (see the dot sign in Fig. 11) is shown in Fig. 12. It is shown that the assimilation of rainfall observations (EXP3) introduces stronger high frequency oscillations, especially in the first 3-h forecast period. Adding the DF penalty constraint (EXP4) effectively reduced these high frequency oscillations. What is the impact of imposing a DF in precipitation assimilation on QPFs? The threat scores given in Fig. 13 indicate that the forecast skill of the QPFs after removing the high frequency oscillations (EXP4) is slightly higher than that without removal (EXP3) in the periods from 0000 - 0600 UTC and 0600 - 0900 UTC. However, the forecast skill of the QPFs is somehow degraded from 0900 - 1200 UTC. This implies that while removing the high frequency gravity wave oscillations results in a better convergence and smoother initial condition, its impact on QPFs is mixed.
36
These high frequency oscillations may contain important, realistic mesoscale gravity waves. As Uccellini and Koch (1987) pointed out, mesoscale gravity waves with wavelengths of 50-500 km and periods of 1-4 h may have important effects on the weather, especially on rainfall in the way of organizing the individual cloud bands into mesoscale cloud clusters. Further studies are required as whether to include a DF for rainfall assimilation and how to maintain the useful gravity waves while removing other undesirable high frequency oscillations in mesoscale data assimilation. 3.4.3
Adjustments in Model State Variables Resulting from Rainfall Assimilation
It is well known that precipitation is a very complicated thermodynamic process during which moisture and temperature interact with each other and are redistributed vertically through the processes of convergence/divergence, updraft/downdraft, and condensation/evaporation. Improvements in QPFs due to precipitation assimilation are produced by making adjustments to model initial conditions. It is natural to think of what adjustments have been made in the model variables and whether these adjustments are reasonable in the sense of physical meanings. In the following, we examine some aspects of initial adjustments generated by 4D-Var rainfall assimilation. Figure 14 shows the differences of the predicted precipitation of the first 6 h (Fig. 14a) and the specific humidity on the 700 hPa level at 0000 UTC (Fig. 14b) and 0600 UTC (Fig. 14c) between EXP2 and CTRL. It is shown that the regions of moisture increase and decrease match the regions of rainfall increase and decrease very well. In order to clearly see the impacts of precipitation assimilation on the moisture, temperature, vertical motion, and divergence, we focus our analysis on two types of grid points. Type I grids include those grid points with observed rainfall greater than zero and Type II grids include those with observed rainfall equal to zero but model-forecasted rainfall greater than zero. Figure 15 shows the vertical distribution
37
of the initial adjustments (EXP2 minus CTRL) of the specific humidity (left panel) and the temperature (right panel) averaged over all Type I grids (solid line) and Type II grids (dashed line). It is shown that, after rainfall assimilation, both moisture and temperature values in Type I grids decreased in the lower levels (below 700 hPa) and increased in the middle levels (around 600 hPa). In Type II grids, moisture decreased in all levels, albeit with smaller magnitude than those in Type I grids. Temperature adjustment in Type II grids, however, is opposite to that in Type I grids. These kinds of adjustments in the moisture and temperature fields are consistent with the work of Augstein (1979), who illustrated two examples of variations of temperature and specific humidity during a passage of precipitating clouds and the gross effect of deep convection on the lower atmosphere for undisturbed and disturbed conditions based on the GARP Altantic Tropical Experiment (GATE). We offer the following explanations: in rain areas, moisture and temperature in the lower levels decreases in favor of convective development. As shown schematically in Fig. 16, when the air in the lower levels is dry and the temperature has a maximum decrease in the subcloud level (for instance, near 900 hPa), the positive area between the temperature profile of the rising air parcel (which we presume to be close to saturation) and that of the environment, which is in proportion to the positive buoyancy or the turbulent kinetic energy of the air parcel (Petterssen 1956), becomes larger. As a result, the instability of the lower atmosphere increases and when moist air parcels rise in the relatively dry lower atmosphere, they obtain more turbulent kinetic energy than in the moist lower atmosphere. Therefore, the development of convection becomes easier. As a part of the compensation for the moisture decrease in the lower levels, an increase of the moisture occurs in the middle levels. On the other hand, because the rainfall assimilation by 4D-Var includes a backward integration of the model (implying that the optimal initial condition carries the information of the precipitation process during the assimilation window), moisture in the lower levels after the assimilation decreases as a result of precipitation. The temperature decrease in the lower levels is likely caused by the evaporative cooling processes in the mesoscale downdraft (Houze and 38
Betts, 1981), while the warming effects of the subsidence are relatively small when the descending air is moist. The temperature increase in the middle levels may be caused by latent heat release due to condensation. In Type II grids, where no rain was observed and thus no condensation or evaporation should occur, the adjustment of temperature is certainly opposite to that of Type I. The decrease of moisture in the middle and lower levels in Type II grids is a result of weak divergence and weak upward motion. If we look at the Skew-T diagrams of one point located in the squall line (Fig. 17) at 0000 UTC, we see that, after EXP2, the separation between temperature and dew point becomes larger below 700 hPa and supersaturation occurs between 700 hPa and 500 hPa. Figure 18 shows vertical profiles of averaged vertical velocity for CTRL and EXP2 over Type I and Type II grids. It is shown that, for Type I grids, the vertical velocity increases in the middle levels for EXP2. In Type II grids, the situation is reversed. Therefore, the convection intensifies in Type I grids and weakens in Type II grids after rainfall assimilation. This is consistent with previous findings, where CTRL under-predicted precipitation over observed rainy regions (Type I grids) and EXP2 was able to suppress the false precipitation in CTRL. Figure 19 displays the distribution of moist static energy in the subcloud layer for CTRL and EXP2 averaged over Type I grids at 0000 UTC, 0300 UTC and 0600 UTC. It is shown that the moist static energy for EXP2 decreases in the lower levels, consistent with the average profile of moist static energy for the disturbed conditions calculated from GATE (Augstein 1979). The decrease of moist static energy in Type I grids is caused by the evaporative cooling effects and drying effects of the downdraft, while changes in moist static energy in Type II grids are very small (figures omitted). In rainfall assimilation, all model variables are adjusted by minimizing the distance between predicted and observed rainfall. The relative importance of these adjustments to the QPFs is examined through a set of sensitivity experiments in which adjustments in only one or two variables are retained. Figure 20 shows the threat scores from these experiments. It can be seen that the TSs for the experiment 39
adjusting only temperature or moisture are higher than those only adjusting wind components, while TSs for the experiment adjusting both temperature and moisture are comparable to those for the experiment adjusting all model variables during 0000 - 0900 UTC. This indicates that the adjustments of temperature and moisture are of the most importance to QPFs, while adjustments in the wind field have less importance to QPFs. However, retaining all model variables gives the best result.
3.5
Summary and Conclusions
A case study of precipitation assimilation is conducted using the NCEP multi-sensor hourly rainfall data. It is shown that: 1. Improvements in QPFs can be obtained through assimilation of multi-sensor rainfall observations over areas where there is observed precipitation. 2. Observed no-rain information included in rainfall assimilation plays an important role in correcting false rainfall and improving QPFs. 3. Assimilation of high-resolution (4 km) observed rainfall produces slightly more improvement in QPFs than the assimilation of low-resolution (30 km) observed rainfall, but the differences are much smaller than the improvements of rainfall assimilation in QPFs. 4. While 4D-Var experiments assimilating precipitation observations adjust all model variables, temperature and moisture adjustments are found to be the primary factors for improving QPFs. 5. Assimilation of rainfall observations can introduce high frequency gravity wave oscillations. These oscillations can be removed by applying a digital filter in 4D-Var. One noticeable feature of rainfall assimilation is the significant drop in the improvement of rainfall assimilation on QPFs beyond the assimilation window. Further 40
improvements of QPFs can be expected if additional observations, such as precipitable water, are available and can be incorporated into rainfall assimilations. This is important to reduce the uncertainty in adjusting 3-dimensional variables of temperature and moisture from 2-dimensional rainfall data.
41
Figure 1: (a)surface temperature (contour interval: 1.0 C) and pressure (thin dashed line, contour interval: 4.0 hPa) at 1200 UTC 5 April 1999; (b) Specific humidity (contour interval: 1.0 g/kg) with the surface wind field overlayed (the value of maximum wind vector is 15.9 m/s), also at 1200 UTC 5 April 1999.
42
Figure 2: Observed 6-h accumulated precipitation at (a) 0600 UTC and observed 3-h accumulated precipitation at (b) 0900 UTC and (c) 1200 UTC 5 April 1999.
43
Figure 3: Results of the CTRL forecast. Figure descriptions are indentical to those in figure 2.
44
Figure 4: 6-h accumulated precipitation at 0600 UTC as predicted by (a) EXP1 and (b) EXP2
45
(a) 0000−0600 UTC
(b) 0600−0900 UTC
0.4
0.4
0.2
0.2
TS
0.6
TS
0.6
0
0
1.0
2.5
5.0
10.0
15.0
0
Threshold (mm)
1.0
2.0
3.0
5.0
Threshold (mm)
(c) 0900−1200 UTC 0.6 CTRL EXP1 EXP2 EXP3
TS
0.4
0.2
0
0
1.0
2.0
3.0
5.0
Threshold (mm) Figure 5: The threat scores for CTRL, EXP1, EXP2 and EXP3 from the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds.
46
0
Bias (mm)
0.6
0.4
CTRL EXP1 EXP2 EXP3
(a) 0.2
0
0000−0600
0600−0900
0900−1200 UTC
RMS (mm)
4 3 (b)
2 1 0
0000−0600
0600−0900
0900−1200 UTC
Time period Figure 6: (a) The bias and (b) RMS errors for CTRL, EXP1, EXP2 and EXP3 for the accumulated rainfall from 0000 - 0600 UTC, 0600 - 0900 UTC and 0900 - 1200 UTC 5 April 1999.
47
Figure 7: The 3-h accumulated rainfall from 0900 - 1200 UTC over the subdomain for (a) NCEP OBS; (b) CTRL; (c) EXP1; and (d) EXP2.
48
0.4 CTRL EXP1 EXP2
TS
0.3
0.2
0.1
0
1.0
2.0
3.0
Threshold (mm) Figure 8: The threat scores for CTRL, EXP1 and EXP2 for the accumulated rainfall from 0900 - 1200 UTC within the subdomain shown in Fig. 7.
Figure 9: The composite chart of the differences in the specific humidity at mid-levels (around 600 hPa) between EXP2 and CTRL at different times overlayed by the wind field at 600 hPa at 0900 UTC 5 April 1999. Light shading indicates positive values greater than 0.3 g/kg, while dark shading indicates negative values less than -0.3 g/kg.
49
(a)
(b)
10
10000
7
100
Jb
EXP3 (no DF) −7 EXP4 (α=10 ) −6 EXP4 (α=10 ) −5 EXP4 (α=10 )
10
10
6
Jo
1000
1 0
10
20
30
40
50
0
10
20
(c)
30
40
50
10
5
(d) 7
7
10
10
6
10
Jtotal
Jp
6
10 5
10
4
10
5
0
10
20
30
40
50
0
iterations
10
20
30
40
50
10
iterations
Figure 10: The variation of each term of the cost function with respect to the number of iterations: (a) background term Jb ; (b) observation term Jo ; (c) penalty term Jp ; and (d) total value of the cost function Jtotal .
50
Figure 11: The surface pressure perturbations for (a) CTRL; (b) EXP3; and (c) EXP4 at 0300 UTC 5 April 1999 (contour interval: 1.0 hPa).
Pressure Tendency (hPa/3h)
60 CTRL EXP3 (without DF) EXP4 (with DF)
40
20
0
0
3
6
9
12
Time (hour)
Figure 12: The absolute surface pressure tendency at the grid point (27,54) (as indicated in Fig. 11 (a) by the arrow) for CTRL, EXP3, and EXP4 (unit: hPa/3h).
51
(a) 0000−0600 UTC
(b) 0600−0900 UTC
0.4
0.4
0.2
0.2
TS
0.6
TS
0.6
0
1.0
2.5
5.0
10.0
15.0
1.0
Threshold (mm)
2.0
3.0
5.0
0
Threshold (mm)
(c) 0900−1200 UTC 0.6 CTRL EXP3 (without DF) EXP4 (with DF)
TS
0.4
0.2
0
1.0
2.0
3.0
5.0
Threshold (mm) Figure 13: The threat scores for CTRL, EXP3 and EXP4 for the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds.
52
Figure 14: The differences (EXP2 minus CTRL) in (a) the initial 6-h predicted precipitation (contour interval: 2 mm) and in the specific humidity (thick solid and dashed line, contour interval: 0.5 g/kg) on 700 hPa at (b) 0000 UTC and (c) 0600 UTC 5 April 1999 with the specific humidity for CTRL overlayed (thin solid line, contour interval: 1.0 g/kg).
53
0
Pressure (hPa)
200
400
600
800
1000 −0.6
−0.4
−0.2
0
0.2
−0.2
q (g/Kg)
−0.1
0
0.1
0.2
T (K)
Figure 15: The averaged vertical profiles of the differences (EXP2 minus CTRL) of specific humidity (left panel) and temperature (right panel) for Type I (solid line) and Type II (dashed line) grids at 0000 UTC 5 April 1999.
Figure 16: Schematic illustration of the instability in the moist lower layer and dry lower layer. T and T ∗ represent temperature profiles of the environment and the rising air parcel, respectively.
54
Figure 17: Skew-T diagrams at the grid point (43,28) (as indicated in Fig. 14 (a) by the arrow) for CTRL (left panel) and EXP2 (right panel) at 0000 UTC 5 April 1999.
0
Pressure (hPa)
200
400
600
800
1000
CTRL (Type I) EXP2 (Type I) CTRL (Type II) EXP2 (Type II)
0
0.02
0.04
W (m/s) Figure 18: The averaged vertical profiles of the vertical velocity in Type I (thick line) and Type II (thin line) grids for CTRL (solid line) and EXP2 (dashed line), respectively, at 0300 UTC 5 April 1999.
55
(a) 0000 UTC
(b) 0300 UTC 800 CTRL EXP2
900
1000 310
900
312
314
316 312
MSE (1000J/Kg)
314
316
Pressure (hPa)
Pressure (hPa)
800
1000
318
MSE (1000J/Kg)
(c) 0600 UTC
Pressure (hPa)
800
900
1000 314
316
318
320
MSE (1000J/Kg)
Figure 19: The averaged vertical profiles of moist static energy of type I grids for CTRL (solid line) and EXP2 (dashed line) at (a) 0000 UTC; (b) 0300 UTC; and (c) 0600 UTC (unit: 1000J/kg).
56
(a) 0000−0600 UTC
(b) 0600−0900 UTC
0.4
0.4
0.2
0.2
TS
0.6
TS
0.6
0
0
1.0
2.5
5.0
10.0
15.0
0
Threshold (mm)
1.0
2.0
3.0
5.0
0
Threshold (mm)
(c) 0900−1200 UTC 0.6 CTRL adjust all var. only adjust T only adjust q only adjust T & q only adjust U & V
TS
0.4
0.2
0
0
1.0
2.0
3.0
5.0
Threshold (mm)
Figure 20: The threat scores for CTRL and different sensitivity experiments for the accumulated rainfall from (a) 0000 - 0600 UTC; (b) 0600 - 0900 UTC; and (c) 0900 - 1200 UTC 5 April 1999 for different thresholds.
57
CHAPTER 4 Assimilation of Ground-based GPS ZTD and Raingage Precipitation Observations for a Winter Storm That Occurred in Southern California during 5-6 Dec 1997
4.1
A Brief Case Description
The case selected for this study is a California winter storm which occurred during 5-6 December 1997. Intense rainfall was observed over most of southern California. At 0000 UTC 6 December 1997, a strong low pressure center (with a central pressure of 984 hPa) was approaching the coastline of southern California with an occluded front extending south-eastward from approximately 35◦ N, 131◦ W to 34◦ N, 124◦ W (Fig. 21). A very strong southwesterly wind brought large amounts of warm, moist air into the coastal regions. A large area of moderate to intense rain covering the coastline can be seen in radar reflectivity measured at 0100 UTC 6 December 1997 (Fig. 2 of De Pondeca and Zou, 2001b). This rain band moved eastward at roughly 7 ms−1 (De Pondeca and Zou 2001b). Precipitation was observed as early as 1200 UTC 5 December over the area ahead of the warm front, while intense rainfall was observed between 1800 UTC 5 December and 1200 UTC 6 December. We focus our study on a short period from 0000 to 0400 UTC 6 December 1997.
58
4.2
Ground-Based GPS ZTD and Raingage Observation
The observations used in our data assimilation experiments include GPS-derived ZTD and hourly accumulated rainfall from rain gauge measurements. The ZTD observations were obtained from 26 GPS receivers located in the southern California integrated GPS network (Bock and Williams 1997). The average station separation is 20 km. The maximum and minimum differences between the heights of the GPS stations are approximately 1730 m and 1 m, respectively. GPS data processing was done at the Jet Propulsion Laboratory (JPL) using the GIPSY software (Webb and Zumberge 1997). Data from each station is available at 15-min intervals. The mean observational error does not exceed 6 mm. The hourly accumulated rainfall observations are obtained from 51 rain gauge sites over southern California. A constant error of 0.5 mm/h was used in the data assimilation. Figure 22 displays the locations of both types of observations. ZTD consists of two parts: zenith hydrostatic delay (ZHD), and zenith wet delay (ZWD). Estimates of ZTD are readily available from measurements of signal phase from GPS satellites at ground-based GPS receivers. As previously shown (Saastamoinen 1972; Davis et al. 1985; Bevis et al. 1992; Janssen 1993; De Pondeca et al. 2001), ZTD (units: m) can be expressed as Aps ZTD = + 10−6 [ f (φ, H)
Z
e 0 k3 (k + )dz], T 2 T
(73)
where A = (2.2779 ± 0.0024); f (φ, H) = 1 − 0.00266cos2φ − 0.00028H, a correction factor that accounts for variations of the acceleration due to gravity; ps is the pressure (hPa) at the site of the GPS antenna located at latitude φ (rad) and height H (m) above the Earth’s surface; T is the temperature (K); e is the water vapor pressure, 0
a function of the total pressure of the air (p) and specific humidity (q); and k2 = (17 ± 10)K(hPa)−1 and k3 = (3.776 ± 0.004) × 105 K2 (hPa)−1 are two constants. The first term on the right side of (73) is the zenith hydrostatic delay (ZHD) while the second term is the zenith wet delay (ZWD). It is shown that the ZWD is proportional 59
to the precipitable water (PW) (Davis et al. 1985):
ZWD = ΠPW,
(74)
where the non-dimensional constant of proportionality, Π, is a function of a suitably 0
defined column mean temperature, the density of water, and the constants k2 and k3 . ZWD is approximately 6.4 times PW when expressed in the same unit of length (Bevis et al. 1992). Assimilation of ZTD instead of PW is necessary in cases when either pressure or column mean temperature measurements are not available. From (73), we notice that the station pressure ps and the correction factor f (φ, H) are both a function of the station elevation H. An accurate estimate of station elevation H is critical for an accurate estimate of ZHD. For instance, when ZHD is estimated from the model, the difference between the elevation of the GPS antenna and the model terrain must carefully be taken into account (De Pondeca and Zou 2001a). As pointed out by De Pondeca and Zou (2001a), the differences between the 6-km model topography of the MM5 and station elevations over southern California can be as large as 900 m. Near sea level, a 900 m height difference roughly translates into a pressure difference of 90 hPa, corresponding to a significant difference in ZHD values (approximately 204 mm).
4.3
Design of Numerical Experiments
The model domain, as shown in Figure 22, covers southern California, with a horizontal resolution of 6 km (a total of 85×55 horizontal grid points) and 20 vertical levels. Data assimilation experiments are conducted over a one-hour time window from 0000 UTC to 0100 UTC 6 December 1997. Following De Pondeca and Zou (2001b), the initial and boundary conditions at 0000 UTC 6 December are obtained from a 12-h forecast over three nested-grid domains (with horizontal resolutions of 54 km, 18 km and 6 km, respectively) initialized at 1200 UTC 5 December 1997. De Pondeca and Zou (2001b) provided details of this forecast. The use of a 12-h forecast before the 60
start of the data assimilation is intended to remove unrealistic gravity waves that are usually excited at the beginning of model integration and allows the model to generate mesoscale features in its initial conditions. The cost function for the 4D-Var experiment is defined as follows: J(x0 ) = (x0 − xb )T B−1 (x0 − xb ) N1 X P f cst T −1 [HPi (Pf cst ) − Pobs ) − Pobs + i ] RP [Hi (P i ] i=1
+
N2 M X X r=0 i=1
−1 obs obs ZT D D [HZT r,i (x) − ZTDr,i ]RZT D [Hr,i (x) − ZTDr,i ],
(75)
where x represents the vector of model state variables and the subscripts “0”and “b” denote the initial state and the background field, respectively. N 1 and N 2 are the numbers of rain gauge stations and ground-based GPS sites, respectively, and M denotes the number of time levels at which ZTD observations are assimilated. HP and HZT D represent the operators of precipitation and ZTD, respectively. B , RP , and RZT D are three diagonal error covariance matrices for the background vector, rainfall observations, and GPS-derived ZTD, respectively. Pf cst and Pobs represent the 1-h accumulated rainfall from the model forecast and rain gauge observations, respectively, while ZTDobs is the GPS-derived ZTD observations. The first term in (75) is a simple background term measuring the distance between the model’s initial condition x0 (to be adjusted through an iterative minimization procedure) and the background field xb , which is a 12-h forecast. The error covariance matrix B is obtained based on the differences between the 1-h model forecast and the initial state. The second term in (75) is the observation term for rainfall. One hour accumulated rainfall observations ending at 0100 UTC are assimilated. The operator HP represents the interpolation of model-produced rainfall from model grids to rain gauge stations. The error variances for the rainfall observations are set to be constant, corresponding to an estimated precipitation observation error of 0.5 mm.
61
The third term in (75) is the observation term for ZTD, which is assimilated at 15min intervals during the assimilation window. For the ZTD observational operator, the input variables are pressure, specific humidity, and temperature. The station elevation enters the calculation of ZTD [eq. (refeq.ztd)] in three ways: the pressure ps , which depends strongly on the station elevation; the correction factor f (φ, H); and the lower boundary of the vertical integrals of the second term of (73) (i.e., the ZWD term). As discussed by De Pondeca and Zou (2001a), three distinct cases occur when one compares the station elevation with the model topography: where the station elevation is higher, lower, or equal to the model topography. In all three cases, a 16 point bi-quadratic interpolation (Guo and Chen 1994) is adopted to carry out the horizontal interpolation of the model fields to the GPS sites. The error variances for ZTD are time-dependent and derived along with ZTD itself from signal-phase measurements at the GPS receivers. Errors in the ZTD observations generally vary from 0.1 mm to several millimeters. We conduct three experiments: experiment ERAIN assimilates only rainfall, EZTD includes only ZTD observations, while EBOTH incorporates both rainfall and ZTD observations. Each experiment is performed at a 6-km horizontal resolution over a 1-h period between 0000 UTC and 0100 UTC 6 December. The minimization procedure adopts the iterative limited-memory Newtonian method developed by Liu and Nocedal (1989). The values of the cost-function (J) are calculated after the forward model integration. The values of the gradient of J (∇J) are obtained by integrating the adjoint model backwards in time through the assimilation window from the ending time to the initial time. Calculation of both J and ∇J is required at each iteration. In this study, the maximum number of iterations for minimization was set at 30. Forecasts initialized with initial conditions obtained by ERAIN, EZTD, and EBOTH are compared with each other and with a control experiment in which no data assimilation was performed (CTRL).
62
4.4 4.4.1
Results
A Mixed Use of Total Zenith Delay and Precipitation Observations
Figures 23 and 24 show the observed and predicted first 1-h and subsequent 3-h accumulated rainfall, respectively. From 0000 UTC to 0100 UTC, there are three observed heavy rain locations with rain rates exceeding 10 mm/hr along the coastline (Fig. 23a, hereafter referred to as R1, R2 and R3 from north to south) accompanied by two inland light rain locations (hereafter referred to as R4 and R5 from north to south). For the subsequent 3-h accumulated rainfall (Fig. 24a), a large precipitation area is seen along the coastline with small heavy rain areas embedded. Compared with observations, we find that the control forecast (Figs. 23b and 24b) captures the overall observed precipitation area, but the locations of the heavy rainfall from the model do not match well with those from the observations. Furthermore, the control forecast also underestimates the heavy rainfall along the coastline and produces small amounts of unobserved rainfall over Santa Catalina Island and near Point Conception. We next consider the impact of assimilating both ZTD and hourly rainfall observations. We show how well the model can be fitted to the observations and then analyze the forecast differences with and without data assimilation. The values of the different terms in the cost-function with respect to the number of iterations of the minimization for ERAIN, EZTD, and EBOTH are shown in Fig. 25. The total cost-function values decrease by more than one order of magnitude for each 4D-Var experiment, with most of the reduction occurring during the first ten iterations. Figure 26 shows the time evolution of the root mean square error (RMSE) and bias error of model-derived ZTD with respect to GPS-observed ZTD within the assimilation window. It is shown that the RMSE and the ZTD bias for the control run within the assimilation window are as large as approximately 12 mm and -9 mm, respectively. After ZTD assimilation, dry bias in the control forecast was reduced from 9 mm to less than 1 mm and the RMSE was also significantly reduced. 63
Figures 27 and 28 show differences in predicted rainfall between the 4D-Var experiments (ERAIN, EZTD, and EBOTH) and the control run for 1-h accumulated rainfall ending at 0100 UTC and 3-h accumulated rainfall ending at 0400 UTC, respectively. By comparing Fig. 27 with Fig. 23a, it is shown that, after rainfall assimilation, the positions of positive difference (increased rainfall after data assimilation) areas match those of observed rainfall locations R1, R2, R3, and R5 very well. The positions of negative difference (decreased rainfall after data assimilation) centers also match well with those of observed light-rain or no-rain regions between the heavy rain bands. In EZTD, which does not assimilate rainfall observations, smaller changes are found near R3 and R5 during the first hour. This is due to the fact that most GPS ground sites are located to the south of the rain gauge-measured precipitation areas. For the 3-h accumulated rainfall prediction beyond the assimilation window (Fig. 28b), we find that the impact of ZTD extends further to the north. The threat scores (Peng and Zou 2002) of the rainfall forecasts from CTRL, ERAIN, and EZTD for the 1-h accumulated rainfall ending at 0100 UTC and 3-h accumulated rainfall ending at 0400 UTC are displayed in Fig. 29. Within the assimilation window, assimilation of ZTD alone (EZTD) produces minor improvements in the QPFs at 1, 2, and 5 mm thresholds. The other two experiments in which observed rainfall data are assimilated (ERAIN and EBOTH) result in significant improvements in the QPFs for all threshold values. However, beyond the assimilation window (Fig. 29b), improvements in QPFs for ERAIN and EBOTH drop quickly to near the level of EZTD. It is encouraging that the assimilation of ZTD produced a positive impact on the QPFs. Given more GPS stations, a greater positive impact should be expected from assimilating these observations. 4.4.2
Analysis of Moisture, Temperature, and Wind Fields
As shown in Chapter 3 (also in Peng and Zou 2002), among all model variables, changes made in the water vapor and temperature fields by rainfall assimilation have
64
the greatest impact on QPFs. Hence, in this section, changes in the water vapor and temperature fields will be analyzed. Since a microphysics scheme is included in the model, changes in rain water and cloud water will also be examined, followed by an analysis of the wind field. Analysis increments of specific humidity and temperature from the three 4DVar experiments are plotted in vertical cross section along longitude 118.07◦ W. This longitude was selected because it goes through a region where both GPS sites and rain gauges are located and where the maximum adjustments of precipitation and model state variables occur. At 0000 UTC (the initial time), water vapor changes in ERAIN are mainly in the middle troposphere (between 500-700 hPa, Fig. 30a), while those in EZTD are found from the surface all the way to 300 hPa (Fig. 30b) with a magnitude much larger than those in ERAIN. The water vapor increments in EBOTH appear to represent the sum of the previous two increments (Fig. 30c). Similarly, the temperature adjustments in EZTD are larger than those in ERAIN (Fig. 31). At 0100 UTC (the end of the assimilation window), water vapor and temperature adjustments in all experiments shifted to the north of the initial adjustments (Fig. 32-33). The order of magnitude of the water vapor changes in ERAIN is similar to that in EZTD. At 0100 UTC, water vapor changes in all experiments display similar structures, i.e., a strong positive center in the middle to upper levels with a negative center in the lower levels. South of these largest increments is a weak negative center in the middle to lower levels with a positive center near the surface. The vertical structure of temperature changes (Fig. 33) shows a complex structure with several regions of cooling and warming. Because the feedback between all of the processes associated with precipitation is highly nonlinear and complex, adjustments in the water vapor and temperature fields are complex on both horizontal and vertical scales. Figures 34-37 show the differences in microphysics variables (cloud water qc and rain water qr ) between the 4D-Var experiments and CTRL at both 0000 UTC and 0100 UTC. At the initial time of 0000 UTC, adjustments in both qc and qr from EZTD are much larger than those from ERAIN (Figs. 34-35). At 0100 UTC, however, 65
adjustments in qc from ERAIN increased to a magnitude comparable to those from EZTD (Fig. 36). The adjustments in qr from ERAIN are much stronger than those from EZTD, with its maximum at the surface (Fig. 37). Such an adjustment in qr is consistent with the fact that the microphysics scheme used in this study calculates precipitation as a surface flux of rain water. Therefore, compared to the assimilation of observed rainfall alone, assimilating ZTD can have a similar or even stronger impact on the fields of moisture, temperature, cloud water and rain water. This implies that ZTD could be a useful data source for improving precipitation forecasts, especially for those regions where observed rainfall data are not available. The changes in the thermal structure of the atmosphere have a strong impact on the winds fields through pressure forces, and in turn, changes in the wind fields of the atmosphere affect the thermal fields through convergence (divergence) and vertical motion. Thus, the question of how changes in moisture and temperature due to the assimilation of observed rainfall or ZTD lead to changes in the dynamical fields arises. This is discussed with respect to the results from EZTD. Differences in the horizontal wind field between EZTD and CTRL at 500 hPa, 700 hPa, and 900 hPa at 0100 UTC are shown in Fig. 38. Adjustments in the horizontal wind field occurred mainly in the regions where adjustments of moisture and temperature occurred, with a magnitude of approximately 5 ms−1 observed at 900 hPa (Fig. 38c). These changes in the horizontal wind field are associated with significant changes in horizontal divergence and vertical motion. Figure 39 shows the vertical velocity along 118.07◦ W for both CTRL and EZTD at 0100 UTC. Assimilation of ZTD is associated with a somewhat more complex and intense pattern of updrafts and downdrafts. The way data assimilation adjusts the model variables is different for the rainfall and GPS-derived ZTD observations. Since the microphysics scheme calculates precipitation as the surface flux of rain water, the rainfall data assimilation first affects
66
the rain water and its tendency at the surface. These changes then affect the water vapor, temperature and pressure at the surface, later affecting these variables at higher levels. The adjustments of the temperature, water vapor and pressure fields affect the wind fields, the microphysical variables and the precipitation amounts. The ZTD assimilation, however, has a direct impact on the temperature, water vapor and pressure fields at all levels through the observation operator of ZTD and then on the wind fields, the microphysical variables and the precipitation amounts. 4.4.3
Spectral Analysis
As previously mentioned, one major problem with the rainfall experiment is that the precipitation forecast skill drops quickly after the assimilation window. In contrast, although the ZTD experiment does not fit the observed rain as well as the rainfall experiment within the assimilation window, it provides comparable results after the window. In order to help understand these results, we perform a spectral analysis of selected variables. The methodology of computing the spectrum over a limited area was developed by Errico (1985) and is briefly described in Chapter 2. We first perform a spectral analysis on 1-h accumulated observed and modelproduced (CTRL) rainfall as well as on differences in 1-h accumulated rainfall between CTRL and OBS, ERAIN and EZTD (Figs. 40 and 41). Note that the domain length is approximately 500 km, where wave number 1 corresponds to a wavelength of 500 km, wave number 10 to a wavelength of 50 km, wave number 20 to a wavelength of 25 km, and so on. From Figure 40, we find that, at 0100 UTC and 0200 UTC, the observed rainfall has a major peak at wave number 3 and a secondary peak at wave number 10. At 0100 UTC, the model simulates the rainfall along larger scales (wave number 3) very well but not as well at smaller scales (wave number 10). At 0200 UTC, the model-produced rainfall peak at wave number 3 is weaker than the observations, and there is still a significant difference between the observations and the model-produced rainfall amounts along smaller wave scales. At 0300 UTC, the
67
observed rainfall has a single peak along larger scales, while the control model forecast is rather weak along all scales. The spectrum distribution of the 1-h accumulated rainfall differences between observations and CTRL shown in Fig. 41 indicates the differences between the observations and the model forecast, i.e., on what scales the model precipitation forecast needs to be improved. From Fig. 41, we find that improvement of the precipitation forecast is mostly needed for wave numbers near 10 for the first two hours (Figs. 41ab), and wave numbers near 3 at hour 3 (Fig. 41c). The increments of precipitation by rainfall assimilation (ERAIN-CTRL) are mainly found along smaller scales (wave numbers 8 to 20) (Fig. 41a). Beyond the assimilation window, the increments of precipitation by rainfall assimilation (ERAIN-CTRL) remain in smaller scales while the major differences between the observations and the control forecast move from smaller scales to larger scales (Figs. 41b-c). Although ZTD assimilation does not produce a fit to the rainfall observations as close as ERAIN with the assimilation window (Fig. 41a), the peaks in the increments (EZTD-CTRL) move from smaller scales to larger scales from 0200 UTC to 0300 UTC (Figs. 41b-c). This may be one of the reasons why the forecast skill of precipitation from EZTD is comparable to that from ERAIN beyond the assimilation window. Next, we perform a spectral analysis on the increments (4D-Var experiments minus CTRL) of the horizontal wind field and pressure perturbations at the surface for ERAIN and EZTD. Figures 42 and Figure 43 show the variance spectra of the differences of the meridional component (v) and pressure perturbation (p0 ), respectively. It is shown that, at the initial time (0000 UTC), increments of the v-component from ERAIN are mainly along smaller scales (wave numbers 8 to 25), while the increments from EZTD are mainly along larger scales (wave numbers 3 to 10). At the end of the assimilation window (Fig. 42b), large increments from ERAIN are found near wave number 10, while those from EZTD are along wave numbers 3 to 15. However, beyond the assimilation window, the increments from ERAIN decrease and remain at smaller scales, while those from EZTD increase at larger scales (Figs. 42c-d). Increments of 68
pressure perturbations are similar to those of the v-component, except that: 1) increments from EZTD are found along all scales and are much larger than those from ERAIN at the initial time (see Fig. 43a, noting that the values for ERAIN-CTRL [solid line] are multiplied by a factor of 1000); and 2) at the end of the window (Fig. 43b), increments from ERAIN have two peaks at wave numbers 3 and 10. Therefore, scale analysis of the forecast differences indicates that the adjustments in model variables by rainfall assimilation are mainly at small scales and remain at these small scales during the subsequent forecast. Conversely, ZTD assimilation adjusts model variables along both large and small scales, which is more consistent with the scales of rainfall observations beyond the assimilation window.
4.5
Summary and Discussions
The impacts of assimilating GPS-derived ZTD and rain gauge rainfall rate observations on short-range QPFs are assessed through a case study of a winter storm occurring on 5-6 December 1997. Three 4D-Var experiments were conducted: one where only rainfall observations was assimilated, one where only ZTD data were assimilated, and one where both rainfall observations and ZTD data were assimilated. Numerical results reveal that: 1. Assimilation of hourly observed rainfall produces a close fit to observed rainfall within the assimilation window, but the improvement drops sharply beyond the window. 2. Although the ZTD data alone do not produce a rainfall distribution as accurate as does the use of rainfall data within the assimilation window, improvements in the QPFs beyond the window from the ZTD data are comparable to those from the rainfall data. 3. Assimilation of both rainfall observations and ZTD combines the effects of the two complementary data sources and thus produces the greatest improvement 69
in short-range QPFs. 4. Assimilation of ZTD or rainfall observations modifies the thermodynamic structures of the atmosphere. In rainy areas, water vapor, temperatures and winds are adjusted in favor of precipitation, consistent with the model’s precipitation processes. 5. A spectral analysis of observed and simulated hourly rainfall and forecast differences between the 4D-Var experiments and the control forecast indicates that the assimilation of rainfall observations adjusts model variables mainly at small scales, while the assimilation of ZTD adjusts model variables at both large and small scales. The latter is more consistent with the scales of observed rainfall beyond the assimilation window. This is likely one of the reasons why the increase in forecast skill of QPFs from ZTD observations is comparable to that from rainfall observations beyond the assimilation window. It should be noted that in this study observations from 26 GPS sites and 51 rain gauge stations were used over a relatively small model domain and a short assimilation period. Additional GPS sites and rain gauge stations over a larger area and being assimilated over a longer time period would likely produce a more significant impact on short-range QPFs.
70
Figure 21: Sea level pressure (heavy solid line), the 850 hPa temperature (thin solid line) and wind fields interpolated from the NCEP global analysis to a 54-km horizontal resolution grid at 0000 UTC 6 December 1997. The contour intervals for SLP and temperature are 4 hPa and 1 K, respectively, and the value of the maximum wind vector is 25.4 ms−1 .
71
Figure 22: Model experiment domain (6-km horizontal resolution) and the locations of GPS receiver sites (cross) and rain gauge stations (circle).
72
Figure 23: 1-hour accumulated rainfall ending at 0100 UTC from (a) rain gauge observations and (b) the control run of the model (contours: 1.0, 3.0, 5.0, 7.0, 10.0 mm).
Figure 24: 3-hour accumulated rainfall ending at 0400 UTC from (a) rain gauge observation and (b) the control run of the model (contours: 1.0, 10.0, 20.0, 30.0, 40.0, 50.0 mm).
73
Cost−function J
10000 (a)
1000 100 10 1 0
10
20
30
Cost−function J
10000 (b)
1000 100 10 1 0
10
20
30
Cost−function J
10000 (c)
1000 100 background term ZTD term rain term total
10 1 0
10
20
30
iterations
Figure 25: The values of different terms of the cost function with respect to the iteration number of the minimization for (a) ERAIN, (b) EZTD, and (c) EBOTH.
74
RMSE (mm)
20 (a) 15 10 5 0
0
15
30
45
60
10
BIAS (mm)
(b) 0 −10 CTRL EZTD EBOTH
−20 −30
0
15
30
45
60
Time (minute) Figure 26: Time series of the (a) root mean square error (RMSE) and (b) bias of ZTD between the model-derived and the observed for CTRL, EZTD, and EBOTH (unit: mm).
75
Figure 27: The rainfall differences (4D-Var experiments minus CTRL) for (a) ERAIN; (b) EZTD; and (c) EBOTH for 1-h accumulated rainfall ending at 0100 UTC (red colour shading represents positive values greater than 0.5 mm; blue colour shading represents negative values smaller than -0.5 mm).
76
Figure 28: As in Fig. 27, but for 3-hour accumulated rainfall ending at 0400 UTC. (red colour shading represents positive values greater than 1.0 mm; blue colour shading represents negative values smaller than -1.0 mm).
77
(a) 0000−0100 UTC 1
TS
0.8 0.6 0.4 0.2 0
1.0
2.0
3.0
4.0
5.0
(b) 0100−0400 UTC 1 CTRL EZTD ERAIN EBOTH
TS
0.8 0.6 0.4 0.2 0
1.0
2.0
3.0
4.0
5.0
Thresholds (mm) Figure 29: Threat scores for different thresholds of rainfall amounts for the control forecast and the three 4D-Var experiments for: (a) 1-hour accumulated rainfall ending at 0100 UTC, and (b) 3-hour accumulated rainfall ending at 0400 UTC.
78
Figure 30: Vertical cross section along 118.07◦ W of specific humidity differences (4DVar experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.05 g/kg in (a); 0.2 g/kg in (b) and (c)].
79
Figure 31: Vertical cross section along 118.07◦ W of temperature differences (4D-Var experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.05 K in (a); 0.1 K in (b) and (c)].
80
Figure 32: As in Fig. 30, except at 0100 UTC (contour interval: 0.2 g/kg in all panels).
81
Figure 33: As in Fig. 31, except at 0100 UTC (contour interval: 0.5 K in all panels).
82
Figure 34: Vertical cross section along 118.07◦ W of cloud water differences (4D-Var experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.1 g/kg in (a); 0.2 g/kg in (b) and (c)].
83
Figure 35: Vertical cross section along 118.07◦ W of rain water differences (4D-Var experiments minus CTRL) for: (a) ERAIN, (b) EZTD, and (c) EBOTH at 0000 UTC [contour interval: 0.2 g/kg in all panels].
84
Figure 36: As in Fig. 34, except at 0100 UTC (contour interval: 0.2 g/kg in all panels).
85
Figure 37: As in Fig. 35, except at 0100 UTC [contour interval: 0.2 g/kg in (a); 0.1 g/kg in (b) and (c)].
86
(a)
(b)
(c)
Figure 38: Differences at 0100 UTC in the wind vectors overlayed with the differences in wind speed (solid lines) between EZTD and CTRL at: (a) 500 hPa, (b) 700 hPa, and (c) 900 hPa [contour interval: 1.0 ms−1 ; maximum wind vectors in (a), (b), and (c) are 1.9 ms−1 , 4.1 ms−1 , and 4.9 ms−1 , respectively].
87
Figure 39: Vertical cross section along 118.07◦ W of vertical velocity at 0100 UTC from: (a) CTRL and (b) EZTD (contour intervals: 0.3 ms−1 ).
88
(b)
(a) 0.4
2
Spectrum (mm )
0.4 OBS CTRL
0.3
0.3
0.2
0.2
0.1
0.1
0
0
10
20
30
40
50
Wave number
0
0
10
20
30
40
Wave number
(c) 2
Spectrum (mm )
4 3 2 1 0
0
10
20
30
40
50
Wave number Figure 40: Variance spectra of 1-h accumulated observed and model-produced rain data ending at: (a) 0100 UTC, (b) 0200 UTC, and (c) 0300 UTC (unit: mm2 ).
89
50
(b)
(a) 0.2
2
Spectrum (mm )
0.2 OBS−CTRL ERAIN−CTRL EZTD−CTRL
0.15
0.15
0.1
0.1
0.05
0.05
0
0
10
20
30
40
50
Wave number
0
0
10
20
30
40
Wave number
(c) 2
Spectrum (mm )
3
2
1
0
0
10
20
30
40
50
Wave number Figure 41: Variance spectra of the increments of 1-h accumulated rainfall between CTRL and OBS, ERAIN, and EZTD ending at: (a) 0100 UTC, (b) 0200 UTC, and (c) 0300 UTC. Values for the heavy solid line and dotted line in (b) are multiplied by a factor of 10, while those in (c) are multiplied by a factor of 100 (unit: mm2 ).
90
50
(a)
(b)
2 −2
Spectrum (m s )
0.00015
0.03 ERAIN−CTRL EZTD−CTRL
0.0001
5e−05
0
0.01
0
10
20
30
40
50
Spectrum (m s )
0
0
10
20
30
40
Wave number
Wave number
(c)
(d)
50
0.05
0.04 2 −2
0.02
0.04
0.03
0.03 0.02 0.02 0.01 0
0.01 0
10
20
30
40
Wave number
50
0
0
10
20
30
40
Wave number
Figure 42: Variance spectra of differences of the v component between CTRL and any of ERAIN and EZTD at: (a) 0000 UTC, (b) 0100 UTC, (c) 0200 UTC, and (d) 0300 UTC (unit: m2 s−2 ).
91
50
(a)
(b)
2
Spectrum ((Pa) )
0.05
15
0.04 10
0.03 0.02
5
0.01 0
0
10
20
30
40
50
0
0
10
20
30
40
Wave number
Wave number
(c)
(d)
50
20
15 2
Spectrum ((Pa) )
ERAIN−CTRL EZTD−CTRL
15
10
10 5
0
5
0
10
20
30
40
50
Wave number
0
0
10
20
30
40
50
Wave number
Figure 43: As in Fig. 42, but for pressure perturbation. The values for the solid line in (a) are multiplied by a factor of 1000 [unit: (Pa)2 ].
92
CHAPTER 5 Improving QPFs by Intensifying Mesoscale Gravity Wave Signatures
5.1
The synoptic pattern in favor of mesoscale gravity wave occurrence
According to the conceptual model proposed by Uccellini and Koch (1987), a common synoptic pattern in favor of gravity wave occurrence is characterized by: (i) a jet streak propagating away from the upper-level trough and toward the ridge axis; (ii) a diffluent trough in the exit region of the maximum geostrophic wind; and (iii) a stationary or warm front to the south or southeast of the supposed “wave” region. Figure 44 schematically shows such a synoptic pattern in favor of gravity wave occurrence. According to the study of Koppel et al. (2000), the mesoscale gravity waves most likely occur in central-eastern U.S., especially along the narrow area of Arkansas, Missouri, Iowa, Minnesota, Wisconsin, Illinois and Michigan, and the likelihood of the mesoscale gravity wave occurrence in Western U.S. is nearly zero. The seasons in which mesoscale gravity waves most likely occur are winter and spring. This is mainly because in winter and spring, the synoptic environment favorable for mesoscale gravity wave occurrence is more easily satisfied.
93
5.2 5.2.1
Case selection and experimental design
Case selection
The precipitation event we chose to study is associated with a low pressure system over the mid-Atlantic region of the U.S. which occurred during March 11-12, 2000. Figure 45a shows the sea level pressure and the temperature on 950 hPa at 18 UTC 11 March 2000 from the NCEP reanalysis. A low pressure center with a minimum pressure of 1007 hPa was located over southeastern Kentucky. A strong temperature gradient is found over the southwestern and the northeastern regions of the low pressure center, indicating a cold front extending southward and a stationary warm front extending eastward. Figure 45b shows the geopotential height, the wind and the geostrophic wind fields at 300 hPa. There is a highly diffluent trough, a jet downstream of the axis of inflection, and a geostrophic wind maximum at the base of the trough, providing a favorable environment for gravity wave occurrence over Kentucky, Virginia, West Virginia, Pennsylvania, Ohio and Indiana. The rainfall observations used in this study are the hourly multi-sensor rainfall data from the NCEP/Climate Prediction Center (NCEP/CPC). These rainfall observations are available at a 4-km resolution and are generated by combining approximately 3000 automated hourly rain gauge observations available over the contiguous 48 states with radar precipitation estimates from the Next Generation Weather Radar (NEXRAD) network. Figure 46 shows the observed 6-h and 3-h accumulated precipitation data during the period from 18 UTC 11 March to 09 UTC 12 March. It can be seen that the precipitation pattern is characterized by several narrow rain bands oriented in the southwest-northeast direction with localized heavy rain patches embedded. The rain bands over the northeastern part of the U.S. favor the occurrence of mesoscale gravity waves and are of particular interest for assessing the performance of the modified DF.
94
5.2.2
Experimental design
The Penn State/NCAR nonhydrostatic Mesoscale Model 5 (MM5) version 2 and its adjoint model are used in this study. The physical options chosen for the forecast model in this study include the Blackadar high resolution planetary boundary layer parameterization, grid-resolvable precipitation, the Grell cumulus parameterization (Grell 1993) and the Dudhia microphysics scheme (Dudhia 1989). The MM5 adjoint modeling system is described in Zou et al. (1997). The model domain is shown in Fig. 45. The horizontal resolution is 30-km with total horizontal grid points of 80x60 and 27 vertical σ-levels. Four data assimilation and forecast experiments are carried out to examine the effect of the modified DF for intensifying the signals of mesoscale gravity waves in rainfall assimilation as well as the benefits of assimilating 6-h accumulated rainfall data rather than hourly rainfall data. The first experiment is a control run using the NCEP reanalysis as the initial conditions at 18 UTC 11 March 2000 (CTRL). The second experiment assimilates 6-h accumulated rainfall and incorporates a regular DF with a cut-off period of 6-h (RDF). The third experiment is the same as RDF except the modified DF is incorporated (MDF). The fourth experiment is the same as MDF except the hourly rainfall data are assimilated (MDF-A). Starting at 18 UTC 11 March 2000 (referred to as 0 hour), a 6-h assimilation window was used in all 4D-Var experiments and a 15-h forecast was made from the NCEP reanalysis (CTRL) and the three “optimal” initial conditions from the RDF, MDF and MDF-A experiments. The cost function for 4D-Var experiments is defined as: J(x0 ) = (x0 − xb )T B−1 (x0 − xb ) +
M X N X
tn =1 i=1
β(Rif (tn )) − Riobs (tn ))2 + Jp ,
(76)
where x represents the vector of model state variables, the subscripts “0” and “b” denote the initial state and the background field, respectively, M denotes the number of time levels of the observations, N is the total number of observational points, and Riobs (in units of cm) represents the observed 6-h accumulated rainfall data (M = 1) 95
or hourly rainfall data (M = 6). The first term in (76) is a simple background term measuring the distance between the model initial condition x0 (to be adjusted through an iterative minimization procedure) and the background field xb (the MM5 standard analysis was used as the first guess). Only approximated variances were included in the background weighting matrix B−1 , which is calculated based on the difference between the 6-h forecast and the initial condition. The weighting parameter for precipitation observations β was set to 100 cm−2 , corresponding to an estimated precipitation observation error of 1 mm. The third term in (76) is the penalty term to which a DF is applied, as discussed in section 2. The lateral boundary conditions are fixed in all the data assimilation experiments since a relatively large model domain is used in this study. The description of a regular DF is given in Chapter 2. A modified DF for intensifying mesoscale gravity wave signatures and its application in 4D-Var are also described in Chapter 2. In order to intensify the signals of mesoscale gravity waves and keep the synoptic-scale waves, we can set α1 equal to 1 and α2 greater than 1. It should be noticed that, since the filter is subjected to a “window” function for removing the Gibbs oscillations caused by the truncation of a Fourier series, α2 is not exactly the amplifying factor of the mesoscale gravity waves. It is usually set to a large value in order to get a proper filter response, which is defined as the function by which a pure sinusoidal oscillation is multiplied when subjected to the filter. In this study, we set α2 = 500 for the time period between 2 h and 6 h. Figure 47 plots the response function of such a filter after a Dolph-Chebyshev “window” function (Lynch and Huang 1992; Lynch 1997) is applied. Waves with the frequency of 0.0055 min−1 (i.e., the period of 182 min) have the maximum amplification. Before the modified DF is applied in rainfall assimilation using 4D-Var, it is tested by simply applying it in a time series. Figure 48 shows the time evolutions of the surface pressure perturbation at a single grid point (point P1 in Fig. 53) with and without applying the modified DF and its corresponding spectra. The length of the time series is 15 h with a 1.5-min interval. It can be seen that after applying the 96
modified DF, the mesoscale gravity waves are intensified significantly.
5.3 5.3.1
Numerical results
Improvements on QPFs over the entire domain
Figure 49 shows the threat scores of CTRL, RDF and MDF for the rainfall forecasts. Within the assimilation window (Fig. 49a), significant improvements of QPFs can be seen for both RDF and MDF. Beyond the assimilation window (Fig. 49b-c), QPFs from MDF compare more favorably to observations than those from RDF, especially for the first 3-h accumulated rainfall beyond the assimilation window (Fig. 49b). The horizontal distribution of the observed and simulated 3-h accumulated rainfall from 6 to 9 h by CTRL, RDF and MDF is shown in Fig. 50. The precipitation pattern of MDF is closer to the observations than those of CTRL and RDF. The main improvements in MDF are seen in the eastern part of West Virginia where both CTRL and RDF underpredicted the precipitation amount; north central Massachusetts and southeastern New Hampshire where CTRL underpredicted the precipitation amount; and east central New York and Vermont where CTRL overpredicted the precipitation amount. As mentioned in section 3, these regions favor mesoscale gravity wave occurrence; a more detailed diagnosis is conducted in the following subsection. Although the rainfall observations used in this study are available at an hourly interval, it seems more robust for the model to fit the 6-h observed rainfall than hourly rainfall. Figure 51 shows the threat scores of the model forecasts without rainfall assimilation (CTRL) and with hourly rainfall (MDF-A) and 6-h rainfall (MDF). One can see that within the assimilation window (Fig. 51a), the improvements of QPFs from MDF-A are much smaller than those from MDF, implying that it is more difficult for the model to fit the hourly rainfall than the 6-h accumulated rainfall. Beyond the assimilation window (Fig. 51b-c), the improvements from the assimilation of hourly rainfall within a 6-h window are also not as significant as those from the assimilation of 6-h rainfall covering the same 6-h time period. Fig. 52 shows the threat scores 97
of CTRL, MDF and MDF-A for the hourly rainfall forecast within the assimilation window for the three thresholds: 1mm, 2mm and 3mm. Although 6-h accumulated rainfall observations are assimilated in MDF, the resulting hourly rainfall compares more favorably with observations than MDF-A (which assimilated hourly rainfall observations), especially after 3 h. 5.3.2
Comparing model forecasts over areas where mesoscale gravity waves propagate
In order to determine the area where the mesoscale gravity waves likely occurred, a bandpass filtering using the limited-area spectrum analysis technique (Errico 1985) is performed on the surface pressure perturbations over the whole domain. The wavelengths for the pass bands are set to be between 50 and 500 km, which are the typical wavelengths of mesoscale gravity waves. Figure 53 shows the filtered surface pressure perturbations for MDF at 18 UTC 11 March. It can be seen that the pressure perturbations on the mesoscale gravity wave scales are mainly located along the border between Tennessee and North Carolina and the border between West Virginia and Virginia, oriented from southwest to northeast. These perturbations propagate slowly with time from southwest to northeast and weaken gradually. Fig. 54 shows the filtered pressure perturbations for CTRL, RDF and MDF at 05 UTC 12 March. The surface pressure perturbations for MDF (Fig. 54c) are intensified over southern West Virginia and northern Pennsylvania and reduced over Vermont where unrealistic gravity waves may be produced by the model. Figure 55 is the time-cross section along the line AB in Fig. 53 for the differences of the filtered surface pressure perturbations between MDF and CTRL and those between RDF and CTRL (Fig. 55b). It can be seen that the waves propagate from the southwest to the northeast along the line AB and that the wave signals in MDF (Fig. 55a) are stronger than those in RDF (Fig. 55b). Figure 56 shows the time series of surface pressure perturbations from 0 to 15
98
hours (Fig. 56a) and the corresponding spectra (Fig. 56b) at the point P1. A 2-6 h bandpass filter is applied to all three time series of the surface pressure perturbations. The mesoscale gravity wave oscillations are intensified effectively in MDF (Fig. 56a). The peak value of the spectrum is increased from 1.7 (hPa)2 in RDF to 4.5 (hPa)2 in MDF. The maximum amplitude occurred at a frequency of about 0.004 min−1 for both RDF and MDF. In order to obtain a closer look at the performance of MDF on QPFs related to gravity wave activities, the threat scores of rainfall forecasts were calculated over a rectangular area (about 1700x170 km2 ) centered along the line AB (Fig. 57). The improvements on QPFs by MDF over the rectangular area are much more significant than over the entire domain (see Fig. 49), especially between 9-15 h (Fig. 57c). The root-mean-square-error (RMSE) of the 3-h accumulated rainfall forecasts from MDF over the same rectangular area is the smallest (Fig. 58). In order to see how the precipitation maxima moved along this long rectangular area and how well the 4D-Var experiments fit the observations, the 3-h accumulated precipitation amount averaged over four different areas (about 170x85 km2 ) centered at the points P1, P2, P3 and P4 are shown in Fig. 59. The four small rectangular areas are perpendicular to the line AB. It can be seen that the observed rainfall maxima moved from P1 to P4 with time and decreased greatly by the time the precipitation maxima reached P4. The model underpredicted these observed rainfall maxima at P1, P2 and P3 and overpredicted the observed rainfall maxima at P4. The time change of the precipitation amount from the MDF and RDF experiments is more in phase with the observations than CTRL. The improvements in QPFs by the RDF experiment are not as significant as those in the MDF experiment. Figure 60 shows the cross-section of the hourly precipitation along the line AB shown in Fig. 53. The precipitation produced by the MDF experiment compares much more favorably to the observations than does that from both CTRL and RDF. CTRL underpredicted the rainfall in the first few hours near the point A and overpredicted the rainfall near the point B, while the RDF experiment demonstrates some 99
improvements in the QPFs over the CTRL experiment, but not as large as those in MDF. The propagation of precipitation in both observations and the MDF forecast is similar to the propagation of the mesoscale gravity waves shown in Fig. 55b for the surface pressure perturbations of the MDF experiment. As reviewed by Uccellini and Koch (1987), case studies by many researchers have indicated that mesoscale gravity waves can induce strong rising and sinking motions which lead to the formation of heavy precipitation bands. This is also confirmed in our experiments. Figure 61 is the vertical-cross sections for the vertical velocity and relative humidity along the line AB for CTRL, RDF and MDF at 00 UTC, 03 UTC and 06 UTC, respectively. At 00 UTC over West Virginia, strong upward motion is seen in MDF (Fig. 61g), a weaker upward motion is seen in RDF (Fig. 61d), and no upward motion is found at this place in CTRL (Fig. 61a). The strong upward motion moves northeastern along the line AB with time, while the upward motion in CTRL which produced unrealistic precipitation over eastern Pennsylvania and Vermont is suppressed in both RDF and MDF. The relative humidity also increases over regions where there is strong upward motion.
5.4
Summary and conclusions
A modified DF was developed for intensifying the signals of mesoscale gravity waves in 4D-Var rainfall assimilation, aiming at improving QPFs. Numerical results from this experiment are compared with results from the control forecast and the forecast from the rainfall assimilation with a regular DF. Differences between the assimilation of 6-h accumulated rainfall and the hourly rainfall are also examined. It was found that mesoscale gravity wave oscillations can be intensified by the modified DF. By applying the modified DF in rainfall assimilation, the precipitation associated with mesoscale gravity waves can be better simulated, resulting in more improvements in QPFs with the same rainfall observations. The assimilation of 6-h accumulated rainfall outperformed assimilating hourly rainfall in the same 6-h assimilation window. 100
It should be pointed out that the usefulness of the modified DF for intensifying the signals of mesoscale gravity waves for data assimilation and QPFs needs to be tested on many more cases. Sensivitities on the selection of the values of parameter α1 and α2 , the choice of the type of window function for suppressing the Gibbs oscillations, and the determination of the intensified frequency window must be studied. Furthermore, the resolution of the model grids is also an important factor which may affect the effectiveness of the modified DF in intensifying the gravity wave signatures. For instance, if the model cannot capture any signal of the existing gravity waves, the modified DF may fail to intensify the signal of the gravity waves.
101
Figure 44: Schematic depiction of the conceptual model of Uccellini and Koch (1987). It delineates the 300-hPa height field (solid lines) with a deep trough, an upper-level jet streak (J) approaching the axis of inflection (west boundary of the rectangle) and moving away from the geostrophic wind maximum (Vg ) located in the base of the trough. The surface low, frontal system and the favorable area for mesoscale gravity wave occurrence (rectangular area enclosed by thick dash line) are also indicated (based on Koch and O’Handley 1997).
102
(a)
(b) Figure 45: NCEP reanalysis at 18 UTC 11 Mar 2000 of (a) sea level pressure (heavy solid line, contour interval: 1.0 hPa) and 950-hPa temperature (thin solid line, contour interval: 1.0 K) and (b) 300-hPa heights (thin solid line, contour interval: 60 gpm), wind vectors (the value of maximum vector: 62.7 m s−1 ) and isotachs of the geostrophic wind (heavy solid line, contour interval: 10 m s−1 ).
103
Figure 46: NCEP multi-sensor rainfall observations for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; (c) 3h-accumulated precipitation ending at 06 UTC; and (d) 3-h accumulated precipitation ending at 09 UTC (unit: mm).
104
6 Response function
5 4 3 2 1 0 −1
0
0.01 −1 Frequency (min )
0.02
Figure 47: The response function of the modified digital filter corresponding to an intensified frequency window (0.0028 min−1 , 0.0083 min−1 ) with α1 = 1, α2 = 500 and α3 = 0. The filter is subjected to a Dolph-Chebyshev “window” function.
105
(a)
(b) 20
1
15 2
Spectrum ((hPa) )
Surf. Pres. Pert. (hPa)
before DF after DF
0
10
5
−1
0
3
6 9 Time (hour)
12
0
15
0
0.01 0.02 −1 Frequency (min )
0.03
Figure 48: (a) Time series of surface pressure perturbations (unit: hPa) and (b) the corresponding spectra (unit: (hPa)2 ) with and without the modified digital filter applied.
106
(b) 0.4
0.6
0.3 TS
TS
(a) 0.8
0.4 0.2 0
CTRL RDF MDF
0.2 0.1
1
2
3 4 5 7 Threshold (mm)
0
10
1
2
3 4 5 7 Threshold (mm)
10
(c) 0.4
TS
0.3 0.2 0.1 0
1
2
3 4 5 7 Threshold (mm)
10
Figure 49: The threat scores for CTRL, RDF and MDF over the whole domain for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC.
107
Figure 50: 3-h accumulated precipitation ending at 03 UTC 12 March 2000 for (a) CTRL; (b) RDF; (c) MDF; and (d) NCEP multi-sensor observations (unit: mm).
108
(b) 0.4
0.6
0.3 TS
TS
(a) 0.8
0.4 0.2 0
CTRL MDF−A MDF
0.2 0.1
1
2
3 4 5 7 Threshold (mm)
0
10
1
2
3 4 5 7 Threshold (mm)
10
(c) 0.4
TS
0.3 0.2 0.1 0
1
2
3 4 5 7 Threshold (mm)
10
Figure 51: The threat scores for CTRL, MDF and MDF-A over the whole domain for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC.
109
(b) 0.4
0.3
0.3 TS
TS
(a) 0.4
0.2 0.1 0
CTRL MDF−A MDF
0.2 0.1
1
2
3 4 Time (hour)
5
6
5
6
0
1
2
3 4 Time (hour)
5
6
(c) 0.2
TS
0.15 0.1 0.05 0
1
2
3 4 Time (hour)
Figure 52: The threat scores for CTRL, MDF and MDF-A over the whole domain for hourly rainfall within the assimilation window (18-00 UTC) at thresholds of (a) 1 mm; (b) 2 mm; and (c) 3 mm.
110
Figure 53: Filtered surface pressure perturbations on the scale of 50-500 km at 18 UTC 11 March 2000 for MDF (unit: hPa). Line AB and points P1, P2, P3 and P4 associated with other figures are indicated (contour interval: 0.5 hPa).
111
Figure 54: Filtered surface pressure perturbations on the scale of 50-500 km at 05 UTC 12 March 2000 for (a) CTRL; (b) RDF; and (c) MDF (contour interval: 0.4 hPa).
112
Figure 55: The time-cross section along AB for the differences of filtered surface pressure perturbations on the scale of 50-500 km for (a) MDF minus CTRL and (b) RDF minus CTRL (contour intervals are 0.1 hPa with the absolute values of the contours greater than 0.3 hPa).
113
(a)
(b)
5
0.3
CTRL RDF MDF
4 2
Spectrum (hPa )
Surf. Pres. Pert. (hPa)
0.5
0.1 −0.1 −0.3 −0.5
3 2 1
0
3
6
9
12 15
0
0
0.01
0.02 −1
Time (hour)
Frequency (min )
Figure 56: (a) Time series of surface pressure perturbations (unit: hPa) and (b) the corresponding spectra (unit: (hPa)2 ) for CTRL, RDF and MDF at location P1.
114
(a)
(b)
1
0.6 CTRL RDF MDF
0.8 0.4 TS
TS
0.6 0.4
0.2
0.2 0
1
2
3 4 5 7 Threshold (mm)
0
10
1
2
3 4 5 7 Threshold (mm)
10
(c) 0.6
TS
0.4
0.2
0
1
2
3 4 5 7 Threshold (mm)
10
Figure 57: The threat scores for CTRL, RDF and MDF over the long rectangular area centered along line AB (as indicated in Fig. 53) for (a) 6-h accumulated precipitation ending at 00 UTC; (b) 3-h accumulated precipitation ending at 03 UTC; and (c) 3-h accumulated precipitation ending at 06 UTC.
115
6
RMSE (mm)
5
4
3 CTRL RDF MDF
2
1
3
6
9 Time (hour)
12
15
Figure 58: Time series of the root-mean-square-error (RMSE) for CTRL, RDF and MDF for 3-h accumulated precipitation from the 15-h forecast starting at 18 UTC 11 March 2000 over the long rectangular area centered along line AB.
116
(a)
(b) 5
OBS CTRL RDF MDF
5 4
3−h rain (mm)
3−h rain (mm)
6
3 2 1 0
3
6
9 12 Time (hour)
4 3 2 1 0
15
3
6
(c) 10
3−h rain (mm)
3−h rain (mm)
15
(d)
6
4
2
0
9 12 Time (hour)
3
6
9
12
8 6 4 2 0
15
Time (hour)
3
6
9
12
15
Time (hour)
Figure 59: Time series of 3-h accumulated precipitation of the 15-h forecast for observations, CTRL, RDF and MDF at different selected slices (170x85 km) centered at location (a) P1; (b) P2; (c) P3; and (d) P4, respectively (unit: mm).
117
Figure 60: The time-cross section along AB for 1-h accumulated precipitation from the 15-h forecast for (a) CTRL; (b) RDF; and (c) MDF; and (d) NCEP observations (unit: mm).
118
Figure 61: The vertical-cross section along AB for the vertical velocity (lines) and relative humidity (shading) for (a)-(c) CTRL at 00 UTC, 03 UTC, and 06 UTC respectively; (d)-(f) RDF at 00 UTC, 03 UTC, and 06 UTC respectively; and (g)-(i) MDF at 00 UTC, 03 UTC, and 06 UTC respectively. Contour intervals are 10 mm s−1 , except those for 0600 UTC which are 5 mm s−1 . 119
CHAPTER 6 Summary and suggested future work
6.1
Summary
In this study, NCEP multi-sensor rainfall observations and ground-based GPS ZTD data were incorporated into a numerical model using the 4D-Var technique and their impacts on QPFs were assessed. Through three real case studies, it was found that improvements on QPFs can be obtained through assimilation of multi-sensor rainfall observations over areas where there is observed precipitation. The observed no-rain information included in rainfall is also very helpful for reducing the overpredicted precipitation by the models. Among all the adjustments on the model variables by the rainfall assimilation, the adjustments on temperature and moisture were found to be the primary factors for improving QPFs. ZTD data was found to be beneficial in improving QPFs. Although the ZTD data alone did not produce a rainfall distribution as accurately as did the use of rainfall data within the assimilation window, improvements in the QPFs beyond the window from the ZTD data are comparable to those from the rainfall data. Assimilation of both rainfall observations and ZTD data produced the greatest improvements in shortrange QPFs, indicating that rainfall observations and ZTD can be complementary to each other. The numerical results also show that, in rainy areas, assimilation of ZTD or rainfall observations adjusted water vapor, temperatures and winds in favor of precipitation. Moreover, spectral analysis indicates that the assimilation of rainfall 120
observations adjusted model variables mainly at small scales, while the assimilation of ZTD adjusted model variables at both large and small scales. The latter is more consistent with the scales of observed rainfall beyond the assimilation window. The results also indicate that, in the scope of the model resolution, the resolution of the rainfall observations is not a primary factor in rainfall assimilation. Using highresolution (4 km) observed rainfall produced slightly more improvement in QPFs than using low-resolution (30 km) observed rainfall, but the differences are much smaller than the improvements of rainfall assimilation in QPFs. However, the frequency in which the rainfall observations are assimilated may have an important impact on QPFs; numerical experiments show that the assimilation of 6-h accumulated rainfall outperformed assimilating hourly rainfall in the same 6-h assimilation window. In all the improvements on QPFs by rainfall assimilation, one noticeable feature is that there is a significant drop of precipitation forecast skill beyond the assimilation window. This may be caused by the fact that the mesoscale rainfall is a localized phenomenon, and its direct assimilation often produces a small, passive and shorttime positive impact on QPFs. Thus, how to extend the information contained in the local rainfall observations both in space and time is probably the key step to further improve the QPFs. This could be done by incorporating a penalty term in the rainfall assimilation which is associated with the inherent mechanism of precipitation formation, such as the wave-CISK interactions. Mesoscale gravity waves may play a very important role in the prediction of various mesoscale weather systems, especially in the prediction of mesoscale precipitation. The failure of capturing the mesoscale gravity wave signature in the model may lead to poor precipitation forecasts. In some cases, the signals of mesoscale gravity waves in the model initial conditions are too weak and are subsequently damped out as the model is integrated forward in time. On the other hand, rainfall data assimilation could introduce high-frequency oscillations because of the imbalance between the observations and the model. Among these oscillations, some are of higher frequency and are either useless or deteriorative to the model forecasts, while some are of the 121
features of mesoscale gravity waves and may play an important and beneficial role in precipitation forecasts. Therefore, it is desirable to remove the former while keeping the latter in the numerical model. We seek a method which could intensify the signals of mesoscale gravity waves and suppress the undesired high frequency oscillations in the model initial condition to improve QPFs. While the regular digital filter was able to remove high frequency oscillations, it may remove all the waves within the cut-off frequency which may contain the important mesoscale gravity waves, and thus, its impacts on QPFs were mixed. Therefore, in this work, a modified DF was developed to accomplish our purpose. Numerical results from the rainfall assimilation experiment with the modified DF show that mesoscale gravity wave oscillations can be intensified by applying the modified DF. The precipitation associated with mesoscale gravity waves can be better simulated, resulting in more improvements in QPFs with the same rainfall observations.
6.2
Suggested Future Work
Further improvements of QPFs can be expected if additional observations, such as precipitable water, intensified wind and temperature observations are available and can be incorporated into the rainfall assimilation, which may reduce the uncertainty in adjusting 3-dimensional variables of temperature and moisture from 2-dimensional rainfall data. More experiments and more case studies need to be carried out to determine the optimal parameters in the modified DF which still have much uncertainty in this study. Finally, the rainfall assimilation should be carried out over an ensemble of spring and summer precipitation cases, and the average impacts on QPFs over the whole period should be assessed before it can be implemented in the operational (or quasioperational) precipitation forecasts.
122
REFERENCES
Anthes, R.A., and T.T. Warner, 1978: Development of hydrodynamic models suitable for air pollution and other mesometeorological studies. Mon. Wea. Rev., 106, 1045-1078. Augstein, E., 1979: The atmospheric boundary layer over the tropical oceans. Meteorology Over the Tropical Oceans, Royal Meteorol. Soc., Bracknell, pp. 73-103. Baer, F., 1977: Adjustments of initial conditions required to suppress gravity oscillations in nonlinear flows. Contrib. Atmos. Phys., 50, 350-366. Baer, F., and J. Tribbia, 1977: On comlete filtering of gravity waves through nonlinear initialization. Mon. Wea. Rev., 105, 1536-1539. Barnes, S. L., F. Caracena, and A. Marroquin, 1996: Extracting synoptic-scale diagnostic information from mesoscale models: the Eta model, gravity waves, and quasigeostrophic diagnostics. Bull. Amer. Meteor. Soc., 77, 519-528. Bevis, M., S. Businger, T. A. Herring, C. Rocken, R. A. Anthes, and R. H. Ware, 1992: GPS meteorology: Remote sensing of atmospheric water vapor using the Global Positioning System. J. Geophys, Res., 97, 15787-15801. Bock, Y., and S. Williams, 1997: Integrated satellite interferometry in southern California. EOS, Trans. Amer. Geophys. Union, 78, 293 and 299-300. Bosart, L. F., and A. Seimon, 1988: A case study of an unusually intense atmospheric gravity wave. Mon. Wea. Rev., 116, 1857-1886. Basart, L. F., and J. P. Cussen, Jr., 1973: Gravity wave phenomena accompanying east coast cyclogenesis. Mon. Wea. Rev., 101, 446-454. Brunk, I. W., 1949: The pressure pulsation of 11 April 1944. J. Meteor., 6, 181-187. Buckley, A., and A. LeNir, 1983: BBVSCG –A variable storage algorithm for function minimization. ACM Transactions on Mathematical Software, 11, 103-119. Charney, J. G., 1955: The use of the primitive equations of motion in numerical prediction. Tellus, 7, 22-26. Chimonas, G., F. Einaudi, and D. P. Lalas, 1980: A wave theory for the onset and initial growth of condensation in Atmosphere. J. Atmos. Sci., 37, 827-845.
123
Courtier, P., and O. Talagrand, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. Part II: Numerical results. Quart. J. Roy. Meteor. Soc., 113, 1329-1368. Courtier, P., and O. Talagrand, 1990: Variational assimilation of meteorological observations with the direct and adjoint shallow-water equation. Tellus, 42A, 531-549. Davis, J. L., T. A. Herring, I. I. Shapiro, A. E. Rogers, and G. Elgered, 1985: Geodesy by radio interferometry: Effects of atmospheric modeling errors on estimates of baseline length. Radio Sci., 20, 1593-1607. De Pondeca, M. S. F. V., and X. Zou, 2001a: Moisture retrievals from simulated zenith delay “observations” and their impact on short-range precipitation forecasts. Tellus, 53, 192-214. De Pondeca, M. S. F. V., and X. Zou, 2001b: A case study of the variational assimilation of GPS zenith delay observations into a mesoscale model. Journal of Applied Meteorology, 40, 1559-1576. Derber, John C. 1987: Variational Four-dimensional Analysis Using Quasi-Geostrophic Constraints. Mon. Wea. Rev., 115, 998-1008. Donner, L.J., 1988: An initialization for cumulus convection in numerical weather prediction models. Mon. Wea. Rev., 116, 377-385. Dudhia, 1989: Numerical study of convection observed during the winter monsoon experiment using a mesoscale two-dimensional model. J. Atmos. Sci., 46, 30773107. Dudhia, J., 1993: A nonhydrostatic version of the Penn State-NCAR mesoscale model: validation tests and simulation of an Atlantic cyclone and cold front. Mon. Wea. Rev., 121, 1493-1513. Duan, J., and Coauthors, 1996: GPS meteorology: Direct estimation of the absolute value of precipitable water. Journal of Applied Meteorology, 35, 830-838. Einaudi, F., and D. P. Lalas, 1973: On the growth rate of an unstable disturbance in a gravitationally stratified shear flow. J. Atmos. Sci., 30, 1707-1710. Eom, J. K., 1975: Analysis of the internal gravity wave occurrence of 19 April 1970 in the Midwest. Mon. Wea. Rev., 103, 217-226. Errico, R. M., 1985: Spectra computed from a limited area grid. Mon. Wea. Rev., 113, 1554-1562. Falkovich, A., E. Kalnay, S. Lord and M. B. Mathur, 2000: A new method of observed rainfall assimilation in forecast models. Journal of Applied Meteorology, 39,12821298. Fiorino, M., T. T. Warner, 1981: Incorporating Surface-Winds and Rainfall Rates into the Initialization of a Mesoscale Hurricane Model. Mon. Wea. Rev., 109, 1914-1929. Fiorino, S.T., and James Correira Jr., 2002: Anlysis of a mesoscale gravity wave event using empirical orthogonal functions. earth Interaction, 6, 1-19. 124
Gauthier, Pierre, and J.-N. Thepaut, 2001:Impact of the digital filter as a weak constraint in the preoperational 4DVAR assimilation system of Meteo-France. Mon. Wea. Rev., 129, 2089-2102. Gill, P. E., and W. Murray, 1979: Conjugate-gradient methods for large-scale nonlinear optimization. Tech. Report SOL 79-15, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, CA. Grell, G. A., J. Dudhia and D. R. Stauffer, 1994: A description of the fifth-generation Penn State/NCAR Mesoscale Model (MM5). NCAR Tech. Note NCAR/TN398+ STR, 138 pp. Guo, Y.-R., and S. Chen, 1994: Terrain and land use for the Fifth-Generation Penn State/NCAR Mesoscale Modeling System (MM5). NCAR Tech. Note NCAR/TN-397+IA, National Center for Atmospheric Research, Boulder, CO, 114 pp. Guo, Y.-R., Y.-H. Kuo, J. Dudhia, D. B. Parsons,and C. Rocken, 2000: Fourdimensional variational data assimilation of heterogeneous mesoscale observations for a strong convective case. Mon. Wea. Rev., 128, 619-643. Gustafsson, N., 1993: Use of a digital filter as weak constraint in variational data assimilation. Proc. Workshop on Variational Assimilation, with Special Emphasis on Three-Dimensional Aspects, Reading, United Kingdom, ECMWF, 327-338. Hinkelmann, K., 1951: Der mechanismus des meteorologischen larmes. Tellus, 3, 285296. [Available in English as NCAR Tech. Note TN-203+STR from the National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.] Houze, R. A., Jr. and A. K. Betts, 1981: Convection in GATE. Reviews of Geophysics and Space Physics, 19, 541-576. Janssen, M. A., 1993: Atmospheric Remote Sensing by Microwave Radiometry. John Wiley and Sons, 592 pp. Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Leetmaa, A., Reynolds, B., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, K.C., Ropelewski, C., Wang, J., Jenne, Roy, Joseph, Dennis., 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bulletin of the American Meteorological Society, 77, 437-472. Kaplan, M. L., S. E. Koch, Y.-L. Lin, and R. Weglarz 1997: Numerical simulations of a gravity wave event over CCOPE. Part I: The role of geostrophic adjustment in mesoscale jetlet formation. Mon. Wea. Rev., 125, 1185-1211. Koch, S. E., and R. E. Golus, 1988: A mesoscale gravity wave event observed during CCOPE. Part I: Multiscale statistical analyses of wave characteristics. Mon. Wea. Rev., 116, 2527-2544. Koch, S. E., R. E. Golus, and P. B. Dorian, 1988: A mesoscale gravity wave event observed over CCOPE. Part II: Interactions between mesoscale convective systems and the antecedent waves. Mon. Wea. Rev., 116, 2545-2569.
125
Koch, S. E., and P. B. Dorian, 1988: A mesoscale gravity wave event observed over CCOPE. Part III: Wave environment and probable source mechanisms. Mon. Wea. Rev., 116, 2570-2592. Koch, S. E., and C. O’Handley, 1997: Operational forecasting and detection of mesoscale gravity waves. Wea. and Forecasting, 12, 253-281. Koch, S. E., F. Einaudi, P. B. Dorian, S. Lang, and G. M. Heymsfield, 1993: A mesoscale gravity wave event observed during CCOPE. Part IV: Stability analysis and doppler-derived wave vertical structure. Mon. Wea. Rev., 121, 2483-2510. Koch S.E. and Leanne M. Siedlarz 1999: Mesoscale Gravity Waves and Their Environment in the Central United States during STORM-FESTM. it Mon. Wea. Rev., 127, 2854-2879. Koppel, L. L., F. Bosart, and D. Keyser, 2000: A 25-yr climatology of large-amplitude hourly surface pressure changes over the conterminous United States. Mon. Wea. Rev., 128, 51-68. Krishnamurti, T. N., K. Ingles, S. Cocke, T. Kitade, and R. Pasch, 1984: Details of low latitude medium range numerical weather prediction using a global spectral model. Part II. Effects of orography and physical initialization. J. Meteor. soc. Japan, 62, 613-649. Krishnamurti, T. N., H.S. Bedi, W. Heckley, and K. Ingles, 1988: Reduction of the spinup time for evaporation and precipitation in a spectral model. Mon. Wea. Rev., 116, 907-920. Kuo, Y.-H., Gyakum, John R., Guo, Zitian. 1995: A Case of Rapid Continental Mesoscale Cyclogenesis. Part I: Model Sensitivity Experiments. Mon. Wea. Rev., 123, 970-997. Kuo, Y.-H., X. Zou, and Y.-R. Guo, 1996: variational assimilation of precipitable water using a nonhydrostatic mesoscale adjoint model. Part I: Moisture retrieval and sensitivity experiments. Mon. Wea. Rev., 124, 122-147. Kuo, Y.-H., J. Dudhia, D. B. Parsons, and C. Rocken, 2000: Four-dimensional variational data assimilation of heterogeneous mesoscale observations for a strong convective case. Mon. Wea. Rev., 128, 619-643. LeDimet, F. X., and O. Talagrand, 1986:Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus, 38A, 97-110. Lindzen, R. S., and K. K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, 1602-1617. Lin, Y.-L., and R. C. Goff, 1988: A study of a mesoscale solitary wave in the atmosphere originating near a region of deep convection. J. Atmos. Sci., 45, 194-205. Lions, J. L., 1971: Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin, 369 pp. Liu, D. C., and J. Nocedal, 1989: On the limited memory BFGS method for large scale optimization. Math. program., 45, 503-528. 126
Lynch, P., and X.-Y. Huang, 1992: Initialization of the HIRLAM model using a digital filter. Mon. Wea. Rev., 120, 1019-1034. Lynch, P., D. Giard, and V. Ivanovici, 1997: Improving the efficiency of a digital filtering scheme for diabatic initialization. Mon. Wea. Rev., 125, 1976-1982. Machenhauer, B., 1977: On the dynamics of gravity oscillations in a shallow water model, with applications to normal mode initialization. Conrib. Atmos. phys., 50, 253-271. Matsumoto, S., K. Ninomiya and T. Akiyama, 1967: A synoptical and dynamic study on the three-dimensional structure of mesoscale disturbances observed in the vicinity of a cold vortex center. J. Meteor. Soc. Japan, 45, 292-304. Matsumoto, S., and T. Akiyama, 1969: Some characteristic features of the heavy rainfalls observed over wester Japan on July 9, 1967. I: Mesoscale structures and short period pulsation. J. Meteor. Soc. Japan, 47, 255-266. Matsumoto, S., and Y. Tsuneoka, 1969: Some characteristic features of the heavy rainfalls observed over wester Japan on July 9, 1967. II: Displacement and life cycle of mesoscale rainfall cells. J. Meteor. Soc. Japan, 47, 267-278. Miller, D. A., and F. Sanders, 1980: Mesoscale conditions for the severe convection of 3 April 1974 in the east-central United States. J. Atmos. Sci., 37, 1041-1055. Navon, I. M. and D. M. Legler, 1987: Conjugate-gradient methods for large-scale minimization in meteorology. Mon. Wea. Rev., 115, 1479-1502. Navon, I. M., X. Zou, J. Derber, J. Sela, 1992: Variational data assimilation with an adiabatic version of the NMC spectral model. Mon. Wea. Rev., 120, 1433-1446. Olson, D. A., N. W. Junker, and B. Korty, 1995: Evaluation of 33 years of quantitative precipitation forecasting at the NMC. Weather and Forecasting, 10, 498-511. Peng, S.-Q. , and X. Zou, 2002: Assimilation of NCEP multi-sensor hourly rainfall data using 4D-Var approach: A case study of the squall line on 5 April 1999. J. Met. Atm. Phy., 81, 237-255. Petterssen, S., 1956: Weather Analysis and Forecasting. Vol. II: Weather and Weather Systems. New York, pp266. Phillips, N.A., 1960: On the problem of initial data for the primitive equations. Tellus, 12, 121-126. Pokrandt, P. J., G. J. Tripoli, and D. D. Houghton, 1996: Processes leading to the formation of mesoscale waves in the midwest cyclone of 15 December 1987.Mon. Wea. Rev., 124, 2726-2752. Polavarapu, Saroja, Monique Tanguay, Luc Fillion, 2000: Four-dimensional variational data assimilation with digital filter initialization. Mon. Wea. Rev., 128, 2491-2510. Powers, J. G., 1997: Numerical model simulation of a mesoscale gravity-wave event: Sensitivity tests and spectral analyses. Mon. Wea. Rev., 125, 1838-1869. 127
Powers, J. G., and R. J. Reed, 1993: Numerical simulation of the large-amplitude mesoscale gravity wave event of 15 December 1987 in the central United States. Mon . Wea. Rev., 121, 2285-2308. Rabier, F., and P. Courtier, 1992: Four-dimensional assimilation in the presence of baroclinic instability. Quart. J. Roy. Meteor. Soc., 118, 649-672. Ramamurthy M. K., R. M. Rauber, B. P. Collins, and N. K. Malhotra 1993: A comparative study of large-amplitude gravity-wave events. Mon. Wea. Rev., 121, 2951-2974. Raymond, D. J., 1984: A wave-CISK model of squall lines. J. Atmos. Sci., 41, 1946-1958. Rocken, C., T. Van Hove, J. Johnson, F. Solheim, R. Ware, M. Bevis, S. Chiswell, and S. Businger, 1995: GPS/STORM–GPS sensing of atmospheric water vapor for meteorology. J. Atmos. Oceanic Technol., 12, 468-478. Rottman, J. W., F. Einaudi, S. E. Koch, and W. L. Clark, 1992: A case study of penetrative convection and gravity waves over the PROFS mesonetwork on 23 July 1983. Meteorol. Atmos. Phys., 47, 205-227. Saastamoinen, J., 1972: Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites. The Use of Artificial Satellites for Geodesy, Geophys. Monogr., Ser. 15. Amer. Geophys. Union, 247-251. Seo, D.J., 1998: Optimal estimation of rainfall fields using radar rainfall and rain gauge data. submitted to Journal of Hydrology. Shanno, D. F., and K. H. Phua, 1980: Minimization of unrestrained multivariate functions. ACM Trans. Math. Soft., 6, 618-622. Smith, J.A., and W.F. Krajewski, 1991: Estimation of mean field bias of radar rainfall estimates. J. Appl. Meteor., 30, 397-412. Stobie, J. G., F. Einaudi, and L. W. Uccellini, 1983: A case study of gravity wavesconvective interaction: 9 May 1979. J. Atmos. Sci., 40, 2804-2830. Talagrand, O., and P. Courtier, 1987: Variational assimilation of meteorological observations with the adjoint vorticity equation. Part I: Theory. Quart. J. Roy. Meteor. Soc., 113, 1311-1328. Thepaut, J.-N. and P. Courtier, 1991: Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model. Quart. J. Roy. Meteor. Soc., 117,1225-1254. Thepaut, J.-N. and P. Courtier, 1993a: Variational assimilation of coventional meteorological observations with a multilevel primitive equation model. Quart. J. Roy. Meteor. Soc., 119,153-186. Treadon, R. E., 1997: Assimilation of satellite derived precipitation estimates with the NCEP GDAS. Ph. D. dissertation, Depart. of Met., Florida State Univ.. Trexler C. M., and S. E. Koch 2000: The life cycle of a mesoscale gravity wave as 128
observed by a network of Doppler wind profilers. Mon. Wea. Rev., 128, 24232446. Uccellini, L. W., 1975: A case study of apparent gravity wave initiation of severe convective storms. Mon. Wea. Rev., 103, 497-513. Uccellini, L. W., and S. E. Koch, 1987: The synoptic setting and possible energy sources for mesoscale wave disturbances. Mon. Wea. Rev., 115, 721-729. Webb, F. H. and J. F. Zumberge, 1997: An introduction to GIPSY/OASIS II. Jet Propulsion Laboratory Rep. JPL D-11088. Wee, Tae-Kwon and Y.-H. Kuo, 2004: Impact of a digital filter as a weak constraint in MM5 4DVAR: An observing system simulation experiment. Mon. Wea. Rev., 132, 543-559. Xiao, Q., X. Zou, and Y.-H. Kuo, 2000: Incorporating the SSM/I-derived precipitable water and rainfall rate into a numerical model: A case study for the ERICA IOP-4 cyclone. Mon. Wea. Rev., 128, 87-108. Zhang F. Q., and S. E. Koch 2000: Numerical simulations of a gravity wave event over CCOPE. Part II: Waves generated by an orographic density current. Mon. Wea. Rev., 128, 2777-2796. Zou, X., I. M. Navon, J. Sela, 1993a: Control of gravitational oscillations in variational data assimilation. Mon. Wea. Rev., 121, 272-289. Zou, X., I. M. Navon, M. Berger, P. K. H. Phua, T. Schlick, and F. X. Le Dimet, 1993b: Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM J. on optimization 3, 582-608. Zou, X., Kuo, Y.-H., Guo, Y.-R.. 1995: Assimilation of Atmospheric Radio Refractivity Using a Nonhydrostatic Adjoint Model. Mon. Wea. Rev., 123, 2229-2250. Zou, X. and Y.-H. Kuo, 1996: Rainfall assimilation through an optimal control of initial and boundary conditions in a limited-area mesoscale model. Mon. Wea. Rev., 124, 2859-2882. Zou, X., F. Vandenberghe, M. De Pondeca, and Y.-H. Kuo, 1997: Introduction to adjoint techniques and the mm5 adjoint modeling system. NCAR Technical Note, NCAR/TN-435-STR, pp122. Zupanski, D. and Fedor Mesinger, 1995: Four-dimensional assimilation of precipitation data. Mon. Wea. Rev., 123, 1112-1126.
129
BIOGRAPHICAL SKETCH Mr. Shiqiu Peng was born in 1968 in Qinzhou, Guangxi Province, China. He obtained his pre-college education in his hometown. He then entered Ocean University of Qingdao (Now called Ocean University of China) in Qingdao in 1987 and received his B.S. degree in 1991 in the field of atmospheric and marine science. He then continued his study in the field of atmospheric science at the Chinese Academy of Meteorological Sciences and the Nanjing Institute of Meteorology and graduated in 1994 with an M.S. degree. He worked as a researcher at the Chinese Academy of Meteorological Sciences between 1994-1998. In the fall of 1998, he entered the Department of Meteorology at the Florida State University as a Ph.D. student.
130
