GEOPHYSICAL RESEARCH LETTERS, VOL. 33, L01816, doi:10.1029/2005GL024910, 2006
Simulation of sampling error of average rainfall rates in space and time by five satellites using radar-AMeDAS composites Yasuhisa Iida, Ken’ichi Okamoto, and Tomoo Ushio Department of Aerospace Engineering, Osaka Prefecture University, Osaka, Japan
Riko Oki Earth Observation Research and Application Center, Japan Aerospace Exploration Agency, Tokyo, Japan Received 10 October 2005; revised 30 November 2005; accepted 2 December 2005; published 12 January 2006.
[1] Sampling error has long been an important issue in satellite-based rainfall measuring mission planning such as for TRMM and GPM (Global Precipitation Measurement), and in evaluating global maps of space-time average rain rate from multi-satellite rainfall data. Sampling errors due to sparse space-time rainfall observations from five currently operated low-earth-orbiting satellites (Aqua, DMSP-F13, F14, F15, TRMM (Tropical Rainfall Measuring Mission)) were evaluated using 3-year radar-AMeDAS composite data around Japan. We simulate realistic observational patterns for the satellite-borne microwave radiometers. Root-meansquare sampling error is expressed as a fraction of true average rain rate. We formulate sampling error as a function of space scale ranging from 0.1° to 5.0°, time scale from 1 to 30 (day), and true average rain rate from 0.001 to 15 (mm/h) for five different combinations of these five satellites. Citation: Iida, Y., K. Okamoto, T. Ushio, and R. Oki (2006), Simulation of sampling error of average rainfall rates in space and time by five satellites using radar-AMeDAS composites, Geophys. Res. Lett., 33, L01816, doi:10.1029/ 2005GL024910.
1. Introduction [2] Monitoring rainfall on a global scale is crucial for a quantitative understanding of the global hydrologic cycle, climate system (climate change), and flood prediction. Toward this end, the Global Precipitation Measurement (GPM) mission is proposed to observe the global rainfall at a high sampling rate by operating multiple low-earth orbiting (LEO) satellites carrying microwave radiometers (MWRs). The ultimate purpose of GPM is to produce a high-resolution global rain map that combines space-time averaged rainfall data from multiple MWRs, which are obtained about every 3 hours at each global point. [3] Although observations from MWRs on LEO satellites offer global coverage, they cause space-time intermittence of sampling because of low-earth orbits and instrument configuration (e.g., swath width). Intermittent sampling engenders uncertainty in the space-time averaged rainfall data. This sampling-related uncertainty (sampling error) has persisted as an important issue in satellite mission planning [e.g., Simpson et al., 1988; Steiner et al., 2003] and evaluation of the global map of the space-time average
Copyright 2006 by the American Geophysical Union. 0094-8276/06/2005GL024910
rainfall rate in combination with multi-satellite rainfall data [e.g., Bell and Kundu, 1996, 2000]. [4] Numerous global rainfall maps have been created with various space-time scales in combination with rainfall estimates from currently available MWRs and geo-infrared radiometry (IR). To reduce sampling error, many existing operational near-real-time high-resolution precipitation products include data from geo-synchronous satellite observations (e.g., NRL’s algorithm, CMORPH). In such a situation, it is important (1) to evaluate the reliability of maps only from the MWRs and (2) to quantify the reduction of sampling error by including geo-synchronous satellite data, compared with validated results of the products themselves using ground-based rainfall data by the International Precipitation Working Group (IPWG). The resultant information regarding the sampling error estimation is useful for the algorithm developers of IR-MWR combined products. [5] Many attempts have been undertaken to estimate sampling error [e.g., Laughlin, 1981; Shin and North, 1988; Bell et al., 1990; Oki and Sumi, 1994; Steiner, 1996; Bell and Kundu, 1996, 2000; Lin et al., 2002; Steiner et al., 2003]. These previous studies are based on at least one of four assumptions: (1) observations are made at regular time intervals, (2) observations are made with full coverage (flush visits), (3) observations are made by a single satellite like TRMM or two satellites, and (4) sampling error is investigated in a fixed space or time scale. In the actual case, however, satellites observe rainfall in an irregular space-time sampling pattern (satellites sometimes observe only a fraction of a grid box and the sampling time interval is not always uniform). It should be emphasized that a sampling error estimation considering realistic observations patterns of multiple satellites lends additional reliability to the estimated values. [6] Five LEO satellites carrying MWRs are currently operating. Rainfall products are created using combined data from these satellites (e.g., Global Satellite Mapping of Precipitation (GSMaP) [Okamoto et al., 2005] and 3B40RT [Huffman et al., 2003]). Estimating the sampling error for these realistic satellite flight patterns is immediately necessary for the scientific community, which requires a global rain map. These satellites also provide an excellent opportunity to rehearse the GPM. [7] This study evaluates the sampling error for some different sets of these multiple satellites as a function of space-time scales and true average rainfall rate in space and time. We use the 3-year radar-AMeDAS composite data set
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Table 1. Specifications of the Five Satellites Used in This Simulation
Satellite Aqua DMSP-F13 DMSP-F14 DMSP-F15 TRMM
Microwave Swath Altitude, Inclination, Ascending Radiometer Width of km deg Time (MWR) MWR, km 705 830 830 830 350
98.2 98.73 98.73 98.73 35
13:30 21:30 20:21 18:15 non SSO
AMSR-E SSM/I SSM/I SSM/I TMI
1500 1500 1500 1500 700
around Japan as true values of the rainfall rate. This study extends the work of Oki and Sumi [1994] and Steiner et al. [2003] in the following three aspects: (1) considering the simulated flight pattern of microwave radiometers on five currently operating satellites (Aqua, DMSP-F13, F14, F15, TRMM), (2) targeting space-time average rainfall rate in 1 – 30 day integration times and 0.1° – 5.0° space domains, and (3) using a wide range of true average rainfall rate to 15.0 mm/h. [8] This result will be useful for a feasibility study of future missions such as GPM and evaluation of the reliability of existing rainfall products.
2. Methodology and Data [9] This study extends the method of Oki and Sumi [1994] for precipitation radar (PR) and the MWR (TMI) on the Tropical Rainfall Measuring Mission (TRMM) satellite to the MWRs on five operational satellites (Aqua, DMSP-F13, F14, F15, and TRMM). We simulate realistic flight patterns of these five satellites carrying the MWRs in computing the orbital and instrumental characteristics shown in Table 1. Further details are available in work by Shin and North [1988]. [10] We use a 3-year data set (1998 – 2000) of radarAMeDAS composites as the true rainfall rate value. Details about the radar-AMeDAS composites are described by Oki and Sumi [1994]. We chose 11 validation areas (Iriomote, Okinawa, Kagoshima, Fukuoka, Hiroshima, Osaka, Owase, Kanazawa, Nagano, Tokyo, and Niigata) shown in Figure 1 over the radar-AMeDAS composites. Each area has rectangular and space scales (each area’s size is A) with units of the length of one side of the box in degrees. We use the value of 100 km for a 1-degree box. Although Oki and Sumi [1994] use a space scale A of 5°, this study extends to seven space scales A of 0.1°, 0.3°, 0.5°, 0.7°, 1.0°, 2.0°, and 5.0°. Regarding the space scale A of 1.0°, the geo-locations of 11 validation areas are indicated in Figure 1. In each validation area, the geo-location (longitude and latitude) of the center in the area of the other six space scales is the same as those of the space scale of 1.0°. We assume that these 11 validation areas are representative of all areas over Japan for simplicity of calculation. [11] If the swath of satellite instruments passes over the radar-AMeDAS composite, it is assumed that the corresponding-time grid values of radar-AMeDAS rainfall rate within the swath width are observed perfectly without errors. Satellite-based sampling is instantaneous, but we consider the sampled hourly gridded rainfall rate values as the instantaneous sampled rainfall rate. The true average rainfall rate Rt (mm/h) is estimated by averaging fully sampled rainfall rates over a grid box of a given space
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scale A for a fixed averaging period T of 30 time scales ranging 1 –30 (days). On the other hand, satellite-sampled average (satellite estimate) Rs (mm/h) is calculated by dividing the total satellite-sampled rainfall rates by the total satellite-sampled grid points for the corresponding grid box size A and the corresponding averaging period T. [12] The sampling error is defined as the difference Rs Rt (mm/h) between the true average rainfall rate Rt and the corresponding satellite estimate Rs. Many precedent sampling-error studies such as that of Steiner et al. [2003] have found that the sampling error depends on the mean rainfall rate R during period T. Because the rain map is used frequently for flood alerts and El Nino analyses, we attempt to evaluate the sampling error for the true average rainfall rate according to them. It must be emphasized that this study is unique in the following aspects. (1) It classifies the true average rainfall rate Rt into 18 rain rate intensities (mm/h) of 0.001– 15.0 mm/h ([0.001, 0.0075], [0.0075, 0.025], [0.025, 0.075], [0.075, 0.15], [0.15, 0.95 (0.1 interval)], [0.95, 5.5 (1.0 interval)], and [5.5, 15.0]. Here [a, b (h interval)] represents each category ranging from a to b at h intervals in units of mm/h), in which we represent the nominal value R by the mean value of Rt in each of the corresponding 18 rainfall rate categories that are indicated by the range inside parentheses. (2) It evaluates the rootmean-square (RMS) sampling error for each rain rate intensity. (3) It obtains statistics from all seasons and all areas of the 3-year data set and 11 validation areas. Hereafter, we represent the sampling error as a ratio of the RMS sampling error (mm/h) to the true average rainfall rate (i.e., rain rate intensity, mm/h) as follows. S:E: ¼ 100 RMS=R
ð%Þ
ð1Þ
Here, S.E., RMS, and R indicate thersampling error, the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E ðRs Rt Þ2 , and RMS sampling error represented by the true average rainfall rate represented by hRti, respectively. Blanket h i indicates 3-year and 11-area averages for a given intensity.
3. Results and Discussion 3.1. Estimated Sampling Errors [13] We estimate sampling errors for any combination of seven space scales A ranging from 0.1° to 5.0°, 30 time scales T from 1– 30 (day), and 18 true average rainfall rates
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Figure 1. The 11 validation areas of 1° 1°.
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Figure 2. Examples of the estimated sampling errors against true average rainfall rates in case of five satellites (SSO4 plus TRMM). (rain rate intensities) R of 0.001– 15.0 (mm/h). Here, we target the set of the microwave radiometers on five currently operated satellites (Aqua, DMSP-F13, F14, F15, and TRMM). For example, Figure 2 indicates results for four combinations of space scales A of 0.1°, 1.0°, 2.0°, and 5.0° and the corresponding time scales T of 1-day (daily), 1-day (daily), 7-day (weekly), and 30-day (monthly) integration times. In Figure 2, the sampling error (%) is shown as a function of the true average rainfall rate R (i.e., rain rate intensity). Lines represent the best fit for estimates of the sampling error, as described later. [14] The sampling error largely depends on the area size and integration time as well as the true average rainfall rate (i.e., the rainfall rate intensity), as described in earlier studies [e.g., Oki and Sumi, 1994; Steiner, 1996; Bell and Kundu, 1996, 2000; Steiner et al., 2003]. Figure 2 gives added credence to the scaling of sampling error through the use of these three parameters. 3.2. Formulation of Sampling Error as a Function of Space-Time Scales and the True Average Rainfall Rate [15] Motivated by Figure 2 and previous studies, we evaluate the sampling error as a function of the space scale A (degrees), time scale T (day), and true average rainfall rate R (mm/h). According to Steiner et al. [2003], we assume that the estimate of sampling error can be represented by the empirically guided simple power-law scaling formula. S:E: ¼ C0 AC1 T C2 RC3
ð2Þ
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We determine the coefficient C0 and the power-law exponents C1, C2, and C3 using the least-squares method for any combination of space scale (3 or 4), time scale (30), and true average rainfall rate (18). The result of the best fit for the five satellites is indicated in Table 2 with those in the different sets of the five satellites. Table 2 indicates the two results for the largest three space scales of 1.0° – 5.0° and for the smallest four space scales of 0.1°– 0.7°. The divided space scale of 1.0° was found by trial and error. These scaling formulas can be applied for true average rainfall rates R of 0.001 – 15.0 (mm/h) in the space scale A of 0.1°– 5.0° and time scale T of 1– 30 (days). Best fits by the scaling formula (2) for the four combinations in Figure 2 are indicated in Figure 2 by red, blue, black, and green lines. Figure 2 shows that sampling error well follows the simple power-law scaling of formula (2). [16] We investigate the variation of the coefficient C0, along with the power-law exponents C1, C2, and C3 for different sets of these five satellites. The four sets (TRMM, SSO3 [F13, F14, and F15], SSO3 plus TRMM, SSO4 [SSO3 plus Aqua]) are selected. We extend the above approach to the four sets of satellites. The results are indicated in Table 2, similarly to those for SSO4 plus TRMM. Table 2 shows that the coefficient C0 depends greatly on the number of satellites (N). Satellites in this study each tend to provide the same number of observations (looks) per day (TRMM possibly a little less). We find that the coefficient C0 is approximately proportional to N 0.60 by the power-law scaling of the coefficient C0 in Table 2 with the number of satellites N. That proportionality seems to be consistent with the prediction by Bell and Kundu [1996]. Bell and Kundu [1996] proposed that sampling error might be proportional to S1/2, where S is defined as the effective number of full viewings of an area. Steiner et al. [2003], on the other hand, fit their results to a formula that is almost linear in Dt, the interval between satellites visits. Because the effective number of looks in Steiner et al.’s case would be T/Dt, Steiner et al. [2003] apparently find that the sampling error is proportional to the inverse of the number of looks (instead of the square root). The variation of the power-law exponents C1, C2, and C3 in Table 2 are much smaller than the variation of the coefficient C0 among the five data sets of the satellites. [17] The accuracy of fit (2) to the estimates can be assessed. We evaluate the accuracy using the RMS value (hereafter, the relative RMS difference) of the relative difference between the estimate and the corresponding prediction by formula (2) to the corresponding prediction by formula (2) in any combination of 3 space scales, 30 time scales, and 18 true average rainfall rates. Table 2 indicates the relative RMS differences for different sets of five satellites. The relative RMS difference shown in Table 2 illustrates that the relative RMS
Table 2. Results of Least Squares Fitting to Sampling Error Formulaa 0.1 A < 1.0, deg
1.0 A 5.0, deg
Set of Satellites
C0
C1
C2
C3
Relative RMS Difference
C0
C1
C2
C3
Relative RMS Difference
SSO4 plus TRMM SSO4 (SSO3 plus Aqua) SSO3 plus TRMM SSO3 (F13, F14, F15) TRMM
42.02 44.60 51.49 59.95 104.02
0.26 0.27 0.24 0.24 0.20
0.35 0.35 0.35 0.35 0.35
0.17 0.17 0.17 0.18 0.15
0.26 0.21 0.27 0.21 0.40
44.27 45.00 54.90 60.41 110.37
0.48 0.45 0.49 0.48 0.47
0.39 0.37 0.39 0.37 0.39
0.18 0.18 0.17 0.18 0.16
0.29 0.26 0.27 0.27 0.27
a
The accuracy of that fitting is assessed using the relative RMS difference. See the text for a more detailed definition of the relative RMS difference.
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difference is approximated as 0.3 for any of the five satellite sets and that the sampling error can be well represented by the simple power-law scaling formula (2). [18] We can assess the true sampling error as the range of (1 ± relative RMS difference) times the sampling error estimated by scaling formula (2), using scaling formula (2) and the relative RMS difference in Table 2. For example, the sampling errors in a day over an area with space scales of 1.0° and 0.1° were approximately 33% (±10%) and 58% (±15%), respectively, for heavy daily average rainfall rate of 5.0 mm/h. We can compare our error estimates to those obtained by Oki and Sumi [1994], which used data from the same region as this study. Apparently, the values obtained by Oki and Sumi [1994] were slightly lower than those obtained here for the TRMM sampling case. That difference might be attributable to the different years of data used. We can also approximate the sampling error in other climate regimes using the corresponding averaged rainfall rate. For a realistic flight pattern, for example, Shin and North [1988] and Bell et al. [1990] estimate that the TRMM (TMI) sampling errors are 8% – 12% for the monthly mean rainfall rate of about 0.45 mm/h over a 500 km by 500 km domain during GATE Phase I (summer). For the TRMM sampling error in Table 2, the sampling error in the summer term for the same space-time average rainfall rate is approximately 16% (±4%) and close to the estimates of Shin and North [1988] and Bell et al. [1990]. This fact is considered to prove the validity of the scaling formula (2) itself and indicates that this formula can be used approximately in other climate regions. Although our result of sampling error is slightly larger than those of Shin and North [1988] and Bell et al. [1990], that difference might depend more or less on changes of sampling characteristics by geographical latitude and rainfall characteristics other than the average rainfall rate. [19] The dependence of sampling error on the space scale differs greatly between the two cases depicted in Table 2. The former case for larger space scales is much closer to the square root inverse of the space scale. Note that this study takes the length of one side of the box as the space scale. However, the latter case for the smaller ones is a larger deviation from the square root inverse. The reason for that deviation might be the rainfall data used in this study. It is noteworthy that the radar-AMeDAS data used in this study are hourly averages, not ‘‘instantaneous’’ observations by satellites. For smaller space scales (near the 0.1° range), variability of the hourly averages might not represent the true variability of rain averaged over these smaller areas. Then, the estimate of the sampling error itself for smaller space scales might appear smaller than it really is. Therefore, this dependence on the space scale might differ from the real one. Compared with the result of Steiner et al. [2003], the scaling behavior (C1) of the sampling error with space scale A differs greatly from that of this study. The difference of scaling behavior (C1) of the sampling error with space scale A is dominant compared with the scaling dependence on time scale T and the true average rainfall rate R, approximately the same as that of Steiner et al. [2003]. This difference is considered to be mainly attributable to differences in the following three factors: (1) temporal resolution in the used rainfall data, (2) assumption of observation at even sampling time interval and with full coverage, and (3) rainfall characteristics in used rainfall data. Regarding points 1 and 3, although we use hourly radar-AMeDAS data
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around Japan, Steiner et al. [2003] used 15-min radar data over Central North America. With regard to point 2, we consider simulated realistic observation patterns of satellites, whereas Steiner et al. [2003] did not consider those flight patterns. The influence of these factors demands further investigation.
4. Conclusion [20] The salient conclusions of this study are the following two points. [21] First, the sampling error for realistic flight patterns can be confirmed to follow the simple scaling law as a function of the space scale, the time scale, and the true average rainfall rate. [22] Second, the scaling behavior of the sampling error with the area size differs greatly from those not considering the satellite flight pattern, as seen in earlier work, such as that of Steiner et al. [2003]. In addition, exponents C2 and C3 in formula (2) are similar to those used by Steiner et al. [2003]. [23] Acknowledgments. The authors thank the anonymous reviewers for their valuable comments. This study was supported by the fund of Japan Science and Technology Corporation – Core Research for Evolutional Science and Technology.
References Bell, T. L., and P. K. Kundu (1996), A study of the sampling error in satellite rainfall estimates using optimal averaging of data and a stochastic model, J. Clim., 9, 1251 – 1268. Bell, T. L., and P. K. Kundu (2000), Dependence of satellite sampling error on monthly averaged rain rates: Comparison of simple models and recent studies, J. Clim., 13, 449 – 462. Bell, T. L., A. Abdullah, R. L. Martin, and G. R. North (1990), Sampling errors for satellite-derived tropical rainfall: Monte Carlo study using a space-time stochastic model, J. Geophys. Res., 95, 2195 – 2205. Huffman, G. J., R. F. Adler, E. F. Stocker, D. T. Bolvin, and E. J. Nelkin (2003), Analysis of TRMM 3-hourly multi-satellite precipitation estimates computed in both real and post-real time, paper presented at 12th Conference on Satellite Meteorology and Oceanography, Am. Meteorol. Soc., Long Beach, Calif., 9 – 13 Feb. Laughlin, C. R. (1981), On the effect of temporal sampling on the observation of mean rainfall, workshop on Precipitation Measurements from Space, Rep. D59 – D66, NASA Goddard Space Flight Cent., Greenbelt, Md. Lin, X., L. D. Fowler, and D. A. Randall (2002), Flying the TRMM satellite in a general circulation model, J. Geophys. Res., 107(D16), 4281, doi:10.1029/2001JD000619. Okamoto, K., T. Ushio, T. Iguchi, N. Takahashi, and K. Iwanami (2005), The Global Satellite Mapping of Precipitation (GSMaP) project, paper presented at 25th International Geoscience and Remote Sensing Symposium, Inst. of Electr. and Electron. Eng., Seoul, Korea, 25 – 29 July. Oki, R., and A. Sumi (1994), Sampling simulation of TRMM rainfall estimation using radar-AMeDAS composites, J. Appl. Meteorol., 33, 1597 – 1608. Shin, K. S., and G. R. North (1988), Sampling error study for rainfall estimate by satellite using a stochastic model, J. Appl. Meteorol., 27, 1218 – 1231. Simpson, J., R. F. Adler, and G. R. North (1988), Proposed tropical rainfall measuring mission (TRMM) satellite, Bull. Am. Meteorol. Soc., 69, 278 – 295. Steiner, M. (1996), Uncertainty of estimates of monthly areal rainfall for temporally sparse remote observations, Water Resour. Res., 32, 373 – 388. Steiner, M., T. L. Bell, Z. Zhang, and E. F. Wood (2003), Comparison of two methods for estimating the sampling-related uncertainty of satellite rainfall averages based on a large radar dataset, J. Clim., 16, 3759 – 3778.
Y. Iida, K. Okamoto, T. Ushio, Department of Aerospace Engineering, Osaka Prefecture University, Gakuencho 1-1, Sakai, Osaka 599-8531, Japan. (
[email protected]) R. Oki, Earth Observation Research and Application Center, Japan Aerospace Exploration Agency, Harumi Island Triton Square Office Tower X 23F 1-8-10 Harumi, Chuo-ku, Tokyo 104-6023, Japan.
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