Variational Retrieval of Temperature and Humidity Profiles from TRMM ...

M A R E´ C A L A N D M A H F O U F

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Variational Retrieval of Temperature and Humidity Profiles from TRMM Precipitation Data VIRGINIE MAREĆAL

AND

JEAN-FRANÇOIS MAHFOUF

European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom (Manuscript received 15 November 1999, in final form 5 April 2000) ABSTRACT This paper examines the performance of a one-dimensional variational (1DVAR) assimilation of Tropical Rainfall Measuring Mission satellite-derived surface rainfall rates from the Microwave Imager TMI. Temperature and specific humidity profiles are retrieved that are consistent with both observed and model short-range forecast rain rates. Two atmospheric situations are examined from ECMWF short-range forecasts at T L319L31 resolution. They encompass tropical cyclones, frontal bands, and mesoscale convective systems. Results show that 1DVAR is generally able to find modified profiles within the range of forecast errors (specified from the operational ECMWF statistics) that provide a precipitation field close to observations. Increments of temperature with respect to the background state are small indicating that 1DVAR essentially adjusts specific humidity to modify precipitation amounts. Consistency checks have been defined in order to discard profiles producing too large departures from the observed rainfall rates or having too little sensitivity of model rain rates to changes in temperature and humidity. Moreover, the current 1DVAR approach is unable to modify profiles when the model rain rate is zero and the observed rain rate is nonzero. Sensitivity experiments are performed to the specification of observation errors. Observation errors are increased from 25% of the observed value to 50%. It is shown that 1DVAR is still able to assimilate some information from observations. To decrease the computational cost of 1DVAR, two simplifications are explored: a reduction of the control vector by not including temperature profiles and the use of one background field and one observation field instead of seven for each assimilation window. These two approaches will be further evaluated when 1DVAR products are assimilated in the ECMWF four-dimensional variational system.

1. Introduction The latent heat released by tropical rainfall plays a crucial role in driving the low-latitude atmospheric circulation and affects the weather around the world. However, conventional rainfall measurements are scarce in the Tropics. Satellite observation is thus the only effective way to provide continuous monitoring of precipitation events. The Tropical Rainfall Measuring Mission (TRMM), which was launched in November 1997, aims at measuring rainfall and energy (i.e., latent heat of condensation) exchanges of tropical and subtropical regions (between 308N and 308S). This is the first earth observation satellite to have on board a precipitation radar, in association with microwave (TRMM Microwave Imager, TMI) and visible–infrared imaging radiometers (Simpson et al. 1996; Kummerow et al. 1998). Accurate measurements of satellite-derived rainfall

Corresponding author address: Virginie Marećal, European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, Berkshire RG2 9AX, United Kingdom. E-mail: [email protected]

q 2000 American Meteorological Society

rates provided by TRMM could be used in numerical weather prediction (NWP) for improving the quality of operational analyses and forecasts. Indeed, most NWP models have difficulties in producing realistic weather elements, such as clouds and precipitation, at the beginning of the forecast period. This ‘‘spinup problem’’ corresponds to an imbalance of the hydrological cycle during the first hours of the forecast and is generally more pronounced in the Tropics than in the midlatitudes. Currently, no operational global NWP model assimilates tropical rainfall observations on a routine basis. Various feasibility studies have been performed with global NWP models at the European Centre for Medium-Range Weather Forecasts (ECMWF) (Heckley et al. 1990; Puri and Miller 1990), at the Florida State University (FSU) (Krishnamurti et al. 1984, 1993; Tsuyuki 1997), and at the National Centers for Environmental Prediction (NCEP) (Treadon 1996, 1997). The first approaches used in global models were based on diabatic normal mode initialization and on ‘‘physical initialization.’’ Even though there is evidence that diabatic normal mode balance based on Machenhauer’s scheme is not appropriate at low latitudes (Errico and Rash 1988), substantial impacts on the analyses (improved divergence field)

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were found by Heckley et al. (1990) and Puri and Miller (1990) using satellite rain rates derived from outwardgoing longwave radiation. Results from physical initialization (nudging of rain rate or rain-related quantities) studies also showed an improvement of the moisture analysis and a reduction of the spinup problem. More recent approaches, such as three-dimensional (3DVAR) or four-dimensional (4DVAR) variational assimilations, can explicitly take into account model and observation errors and provide an optimal initial state. Treadon (1997) has assimilated in the NCEP 3DVAR system satellite-derived rainfall rates from Geostationary Operational Environmental Satellite Precipitation Index and from Special Sensor Microwave/Imager (SSM/I) brightness temperatures. First results showed a reduction of the precipitation spindown and improved forecasts of the divergent wind at 200 hPa within the tropical belt. In parallel, Tsuyuki (1997) showed the feasibility of a simplified 4DVAR assimilation of SSM/I-derived precipitation rates using a low-resolution version of the FSU global spectral model. We propose in this paper a two-step approach to assimilate observed precipitation in a global NWP analysis system. First, a one-dimensional variational (1DVAR) assimilation of satellite-derived rain-rate observations is applied. It enables the retrieval of modified temperature and humidity profiles that provide precipitation rates consistent with both observations and a model short-term forecast. Second, 1DVAR analyzed products are introduced in a data assimilation system. The second step can be achieved more easily compared to a direct assimilation of precipitation rates since temperature and humidity variables are already operationally assimilated in 3DVAR or 4DVAR systems. The 1DVAR assimilation of precipitation has already been studied in a theoretical framework (i.e., using simulated observations) by Fillion and Errico (1997) and Fillion and Mahfouf (2000, hereafter FM2000). Fillion and Errico (1997) have shown that a 1DVAR method is able to make a significant adjustment of model precipitation rate within the range of realistic background temperature and specific humidity errors and rain-rate observation error. Their study was restricted to convective regimes, and large-scale precipitation processes were disregarded. FM2000 focused their study on the coupling of moist-convective and stratiform precipitation processes in 1DVAR. Three convective parameterizations were tested by FM2000. They showed that the vertical structure of increments produced by the 1DVAR strongly depends upon convective parameterization. They also demonstrated that the introduction of stratiform precipitation can control the minimization at the expense of convective precipitation for slightly supersaturated profiles. Treadon (1997) also used a 1DVAR framework to test a 3DVAR assimilation of satellite-derived rain rates. He showed that 1DVAR performed well on real data

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although it is more successful for decreasing rather than increasing precipitation to get closer to observations. In the present work we use the version of 1DVAR developed by FM2000 based on the ECMWF mass-flux convective scheme. This study focuses on the behavior of 1DVAR in the context of real data assimilation. TMIderived surface rain rates have been selected for assimilation because of their good accuracy. Compared to SSM/I, the additional low-frequency emission channel at 10 GHz allows an almost linear relationship between rain rate and brightness temperature, the displacement of the water vapor absorption band at 22 GHz reduces saturation effects in the Tropics, and the lower flying orbit of the satellite reduces pixel heterogeneities, known as ‘‘beamfilling effects.’’ These three factors contribute to improve the retrieval of geophysical quantities from the set of TMI brightness temperatures with respect to SSM/I. Such improvement has been demonstrated quantitatively for retrievals of total column water vapor and surface wind speed in nonrainy areas by Mahfouf et al. (1998). This work on 1DVAR assimilation should be regarded as a first step. The second step, regarding the assimilation of 1DVAR products in the ECMWF 4DVAR system (Rabier et al. 2000; Mahfouf and Rabier 2000; Klinker et al. 2000), will be the subject of a forthcoming paper. Section 2 presents the 1DVAR method with special emphasis on modifications to FM2000’s original 1DVAR. The TRMM dataset to be assimilated in 1DVAR is described in section 3. Results of a control experiment are discussed in section 4. Section 5 is devoted to results of various sensitivity tests. Finally, the conclusions are given in section 6. 2. 1DVAR method a. Basis Let x be the vector (also denoted control variable) that represents the atmospheric state and R o the collocated surface rainfall rate estimated from TRMM data. The 1DVAR retrieval consists of finding an optimum state x that minimizes a distance between the model rainfall rate and R o , given a background constraint provided by the error covariances of a short-term forecast profile x b . If the background and observation errors are uncorrelated and each has a Gaussian distribution, the maximum likelihood estimator of the state vector x is the minimum of the following cost function:

1

2

1 1 R(x) 2 R o J(x) 5 (x 2 x b ) T B21 (x 2 x b ) 1 , 2 2 so 2

(1)

where B is the background error covariance matrix and s o the standard deviation of observation error (in theory, errors from the observation operator should also be included). In addition, R(x) is a calculation of surface rain rate from the atmospheric state x at the observation location. As in FM2000, the control variable vector con-


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tains 63 elements: profiles of temperature T and specific humidity q on 31 model levels and surface pressure PS . The minimization module uses a limited memory quasi-Newton optimization routine (M1QN3), which requires an estimation of the gradient of the cost function at each iteration (Gilbert and Lemarećhal 1989). A preconditioning procedure similar to Fillion and Errico (1997) is applied for improving the convergence. The gradient of J(x) is given by =J(x) 5 B21 (x 2 x b ) 1 R T

1

2

R(x) 2 R o . so2

(2)

The operator RT (x) is the transpose of the Jacobian vector of the partial derivatives of rainfall rates with respect to the input profile x (i.e., adjoint of the tangent linear of the moist physical processes). Given the low dimension of the control vector (63 variables here), the Jacobian elements are computed explicitly in finite differences by a perturbation method requiring 64 calls (63 perturbations and one reference) to the nonlinear observation operator R(x). More precisely, the component i of the Jacobian vector R is approximated by [R(x)] i 5

1]x2 5 ]R

i

R(x 1 «S xi ) 2 R(x) , «sxi

(3)

where s xi is the ith component of the standard deviation of background errors for the control vector x, S xi 5 [0, 0, 0, . . . , s xi , . . . , 0, 0, 0], and « 5 1025 . b. Observation operator The observation operator R(x) includes two components that are the moist-convective (noted CO hereafter) parameterization and the nonconvective precipitation or stratiform precipitation (noted ST hereafter) parameterization. A diagnostic parameterization of stratiform precipitation based on a moist-local adjustment process is used as in FM2000. In this scheme, the excess of moisture with respect to saturation at a given level is converted into nonconvective precipitation. The adjusted values of temperature and specific humidity correspond to a saturation state and are obtained by keeping the moist static energy constant. Evaporation of precipitation in subsaturated layers is accounted for. The convective parameterization is the ECMWF operational mass-flux scheme originally developed by Tiedtke (1989). The statistical cloud population is described through a single bulk entraining plume model. For deep convection, Tiedtke’s original moisture convergence closure is replaced by a closure based on the reduction of convective available potential energy toward zero over a given timescale, t , depending on model truncation (Gregory et al. 2000). Here t is 1400 s to be consistent with the ECMWF operational model. In the present study the convective scheme is not coupled to the operational prognostic cloud scheme (Tiedtke 1993).

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Therefore, all detrained liquid water from convective clouds is assumed to evaporate instantaneously in the environment. The coupling of moist-convection and stratiform precipitation schemes follows the fractional-stepping (FS) approach (Beljaars 1991). This means that the convective precipitation operator is applied first. Then, the stratiform precipitation parameterization acts on an atmospheric state already affected by the convective precipitation operator. The alternative approach is called process splitting (PS) (Beljaars 1991) where convection and stratiform precipitation operators are applied to the same atmospheric state. The FS approach is chosen to be consistent with the ECMWF coupling of physical parameterizations. Moreover, FM2000 showed that using FS instead of PS strongly reduces the frequency of unrealistic supersaturations in the humidity profiles retrieved by the 1DVAR method. These supersaturations correspond to cases where the minimization mostly modifies stratiform precipitation at the expense of convective precipitation. Nevertheless, in a few cases the FS approach is also not able to control large supersaturations. This point will be discussed again in section 4. To perform the 1DVAR analysis, background profiles have to be collocated with the observations. This can be achieved by a horizontal interpolation of the background fields at the observation location. But applying a horizontal interpolation induces modifications of humidity and temperature profiles. FM2000 have studied the sensitivity of the CO parameterization to variations of specific humidity and temperature. Their Fig. 3 shows that surface rain rate is highly sensitive to humidity variations at the lowest model level. Preliminary tests have proved that, for some profiles, horizontal interpolation can lead to unrealistic changes in the amount of precipitation produced by the convection scheme when compared to the noninterpolated background precipitation. To avoid horizontal interpolation of model fields and reduce representativeness problems, rainfall observations have been averaged spatially at model resolution. c. Background profiles Since quantities retrieved using 1DVAR will be assimilated in the ECMWF operational 4DVAR analysis system, the 4DVAR configuration is used to define background fields. The current operational 4DVAR assimilation window of 6 h is divided into seven time slots in order to compare observations at the appropriate time. In this study, two atmospheric situations are examined. A set of seven background temperature and specific humidity profiles and surface pressure are produced from two short-range forecasts starting on 10 February 1998 at 1800 UTC and on 26 August 1998 at 0600 UTC from the ECMWF model with spectral truncation TL319 (corresponding to a 60-km resolution approximately) and 31 vertical levels.

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In FM2000, only the control variables (profiles of T and q and Ps ) were used in the observation operator to calculate surface rain rate. In the forecast model at each time step, tendencies from the dynamics and other physical processes modify the input profiles of the moist parameterizations. In the present study, we chose to include, as input to the 1DVAR, profiles of temperature and specific humidity tendencies after dynamical processes, radiation, turbulence, and gravity wave drag parameterizations are applied. More precisely, during a model time step Dt, the tendencies produced by adiabatic processes (]x/]t) adiab , dry physical processes (]x/]t)phys , and moist convection (]x/]t)conv are combined as follows to estimate the observation operator: R(x) 5 CO(x 1 D1x) 1 ST(x 1 D1x 1 D 2 x),

(4)

with D1x 5 [(]x/]t)adiab 1 (]x/]t)phys ]Dt and D 2x 5 (]x/]t)conv Dt. When time tendencies of specific humidity and temperature are not used (as in the sensitivity test presented in section 5), surface rainfall is computed from Eq. (4) by imposing D1x 5 0. Even though these tendencies are kept constant during the minimization they enable the precipitation field from the 1DVAR background state to be more consistent with the three-dimensional environment of the short-range model forecast. d. Background error statistics As in FM2000, the covariance matrix of background errors B is taken from the operational ECMWF 4DVAR system (Rabier et al. 1998; Derber and Bouttier 1999). The covariances are computed statistically using the National Meteorological Center (now known as NCEP) method (Parrish and Derber 1992) from differences between 48- and 24-h forecasts. As in the ECMWF 4DVAR analysis system, no cross correlations between background errors of specific humidity and temperature are considered. An example of vertical profiles for standard deviation of temperature and specific humidity is given in FM2000 (see their Fig. 2). Observations from TRMM being restricted to the the Tropics, background errors of ‘‘unbalanced’’ temperature background errors are chosen (Derber and Bouttier 1999). Thus, standard deviations over the vertical are less than 1 K. The vertical distribution of standard deviation for specific humidity exhibits a maximum at level 25 (around 850 hPa), an exponential decrease above, and lower values in the boundary layer. This profile has been empirically specified by Rabier et al. (1998). 3. Observations The observations from the TRMM mission selected for this study are the instantaneous surface rainfall rates estimated from the TMI radiometer that are provided operationally (product 2A12 level 4) by the National

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Aeronautics and Space Administration They are calculated from the profiling algorithm developed by Kummerow et al. (1996) using a cloud database coupled to a microwave radiative transfer model. To be consistent with the analysis dates/times chosen, the datasets selected correspond to TMI observations from 10 February 1998 at 2100 UTC to 11 February 1998 at 0300 UTC and from 26 August 1998 at 0900 UTC to 26 August 1998 at 1500 UTC. As in the operational 4DVAR assimilation system, the 6-h observations are divided into seven time slots. The first and the last slots are 0.5 h long and the other five slots are 1 h long. In the following, the February case is chosen to illustrate most of the 1DVAR results. For this case, a large variety of oceanic precipitation has been observed during the 6-h window: an intense tropical cyclone in the Indian Ocean, four frontal systems, and mesoscale convective systems over the east tropical Pacific and over the east Indian Ocean. For the August case, a hurricane (Bonnie), two fronts, and several tropical convective systems were sampled by TMI. Note that fronts are usually not observed in their full extent because of the limited TMI latitude coverage (between 2398 and 1398 latitude). TMI 2A12 instantaneous surface rain rates are provided for each high-resolution pixel corresponding to an area of about 7 3 5 km 2 . The swath width of 758.5 km is covered by 208 high-resolution pixels. Since TMI2A12 rain estimations over land are less reliable than over ocean (Kummerow et al. 1996), only observations over sea have been considered in this study. The TMI surface rain rates used in 1DVAR were obtained by averaging high-resolution rain rates at the model resolution (i.e., over the model grid boxes). This is done to be consistent with the surface precipitation calculated by the model, which represents the mean of precipitation and/or no-precipitation areas in the model grid box. Moreover, since the model is not able to represent smallscale precipitation, each mean surface rain rate corresponding to a precipitation coverage of less than 10% in a grid box is set to zero. In practice, this procedure filters out most mean rain rates smaller than 0.1 mm h21 . The mean resulting observation field is shown in Fig. 1 for the February case. This situation will be analyzed in detail in the following and the August case will be used to evaluate the generality of our results. For this figure and those following, three representative limited areas are chosen to illustrate the results obtained in this study. They correspond to the major types of meso- and large-scale precipitation systems encountered between 2408 and 1408 latitude: a tropical cyclone sampled on 11 February 1998 around 0140 UTC (Fig. 1a), mesoscale convective areas sampled on 11 February 1998 around 0240 UTC (Fig. 1b), and a front sampled on 10 February 1998 around 2305 UTC (Fig. 1c). To solve Eq. (1), it is necessary to set the observation error s o . Precipitation rate is a positive quantity and

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therefore its associated errors are not normally distributed. Errico et al. (2000) showed, in a theoretical context, that using nonnormal distributions has a significant impact on 1DVAR analyzed state. But because no estimation of TMI 2A12 surface rain-rate error and its associated distribution is yet available, a Gaussian distribution of errors was assumed for simplicity. A reasonable value of root-mean square of 25% of R o was chosen. However, this choice is inappropriate for low rain rates because of uncertainties on rain occurrence. Thus, a threshold value of 0.01 mm h21 was chosen for the smallest possible observation error. Since these choices are somewhat arbitrary, sensitivity of the results to the specification of observation errors was tested. This aspect is discussed in section 5. 4. Results Results discussed in this section will be considered in the following as the control experiment. a. 1DVAR background precipitation The surface rain-rate field obtained by applying the observation operator to the background profiles (called background rain rate and noted R b hereafter) is displayed in Fig. 2. This field is obtained from a one-time-step integration of the moist physical processes in standalone mode. It is important to verify that this 1DVAR background rain-rate field compares reasonably well with the instantaneous rain rate produced from the forecast model, and referred to hereafter as 4DVAR background rain-rate field. This 4DVAR background field is presented in Fig. 3. Comparison between the two background rain-rate fields shows that, for precipitation generated by convection, location and intensity are similar as illustrated by Figs. 2a,b and 3a,b. Frontal precipitation has similar intensity but is much less extended in the R b field (Figs. 2c and 3c). The main reason for this difference is the use of a diagnostic ST scheme. This scheme is less efficient at triggering stratiform precipitation than the operational prognostic ST scheme that uses additional dynamic information from the 3D forecast model (Tiedtke 1993; Gregory et al. 2000). Nevertheless overall, 1DVAR and 4DVAR background surface rain-rate fields are in good agreement. Comparison between R b (Fig. 2) and R o fields (Fig. 1) exhibits the following general differences. R Tropical cyclones are weaker and more diffuse in R b (see Figs. 1a and 2a). Location is generally similar although it is sometimes shifted by few grid points when the short-range model forecast (4DVAR background) is not accurate enough. R Mesoscale convection is often forecast and observed in the same area but not localized exactly at the same place in R b and R o as shown in Figs. 1b and 2b. R Frontal systems are less extended in R b than in R o

FIG. 1. Instantaneous mean observed surface rain rate in mm h21 (R o field) in the (a) south Indian Ocean, (b) central Pacific Ocean, and (c) North Pacific Ocean along TRMM track between 10 Feb 1998 at 2100 UTC and 11 Feb 1998 at 0300 UTC.

field (see Figs. 1c and 2c). The precipitation location in the R b field can also be slightly shifted with respect to the R o field, as is the case for tropical cyclones. This is the case for the frontal zone chosen here to illustrate results (see Figs. 1c and 2c). b. Analysis criteria Since the observation operator is highly nonlinear, the minimization procedure cannot always reach a realistic solution or even converge to any solution. These extreme and infrequent cases correspond to two situations: 1) The TMI-observed surface rain rate is much larger than the rain rate produced from the background state (by several orders of magnitude). In this case, the 1DVAR minimization is not able to find a solution.

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FIG. 2. Same as Fig. 1 but for 1DVAR background surface rain rate (R b field) for the control experiment.

FIG. 3. Same as Fig. 1 but for 4DVAR background surface rain rate.

2) The observation operator, when applied to the background state, has a very low sensitivity to temperature and specific humidity modifications leading the minimization to produce large unrealistic increments. This happens when background profiles produce very low surface rain rates and TMI observations indicate almost no rainfall.

sensitivity of the background state to modifications of temperature and specific humidity. Thus, the negative (positive) boundary value chosen in (5) allows us to avoid situations like type 1 (type 2).

To avoid analyzing these two types of profiles, all profiles producing background departures outside the following limits, 2100s Rb , R b 2 R o , 5s Rb ,

(5)

are discarded. Here s Rb is the background error expressed in the observation space (i.e., surface rain rate) using

s Rb 5 (R B RT )1/2 .

(6)

A low (high) value of s Rb corresponds to a low (high)

c. Analyzed precipitation Once the 1DVAR method has been applied to background fields, analyzed profiles of temperature and specific humidity are available. To verify the behavior of the 1DVAR assimilation the analyzed precipitation field (noted R a hereafter) is compared to observations. The R a field is obtained by applying the observation operator to the 1DVAR analyzed temperature and specific humidity profiles. If 1DVAR assimilation is efficient, the R a field should be close to the TRMM surface rain-rate field at the level of prescribed accuracy for the observations. The R a field for the February case is displayed in Fig. 4.


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cipitation structure in the R a field is similar to observations. Surface rain rate is increased by about 7 mm h21 in the central part of the cyclone but still remains weaker than R o . Part of the difference can be explained by the large observation error associated with intense observed rain rates, which leads to reduced weight for the observation term in the cost function with respect to the background term. These results agree with Treadon (1997) who found, using a constant observation error of 0.5 mm h21 , that his 1DVAR was better at handling a decrease of precipitation with respect to background than an increase. The second type of profile to be discussed corresponds to R b 5 0. In this case, s Rb is zero because there is no sensitivity of the background state to small increments of temperature and specific humidity. In other terms, 1DVAR assimilation is not able to trigger precipitation by modifying temperature and humidity profiles. Consequently, profiles with R b 5 0 cannot be analyzed by 1DVAR when precipitation exists in the observation field (see for instance Figs. 1b, 2b, and 4b). In the frontal zone displayed in Fig. 4c, this leads to a global decrease of precipitation as a consequence of a phase shift in the precipitation patterns between R b (Fig. 2c) and R o (Fig. 1c) fields. Treadon (1997) found the same behavior of his 1DVAR for nonrainy background profiles. d. Statistics

FIG. 4. Same as Fig. 1 but for 1DVAR analyzed surface rain rate (R a field) for the control experiment. Background surface rain rate is used for profiles not satisfying the criteria of Eq. (5).

For the interpretation of the results, two types of profiles are discussed separately. First, when 1DVAR background precipitation is greater than zero (R b . 0), the 1DVAR analysis is generally able to increase, decrease, and even switch off precipitation as required, except for cases of heavy precipitation where analyzed rain rates never reach observed values. This result is well illustrated for the tropical cyclone (Fig. 4a) where the pre-

A more general view of the 1DVAR behavior for the whole 6-h dataset is given in Table 1. For the February case, the number of profiles ‘‘missed’’ by 1DVAR (R o ± 0 and R b 5 0) is comparable to the total number of profiles analyzed by 1DVAR and mostly corresponds to frontal and mesoscale convective precipitation. This number is usually smaller for other cases, as illustrated by the August case, the number of fronts generally sampled in 6 h being smaller (one or two). For both case studies, the number of points that do not satisfy the 1DVAR analysis criteria [Eq. (5)] is small compared to the total number of 1DVAR analyzed points. It was verified that all rejected profiles correspond to cases where 1DVAR analysis would provide unrealistically large increments in absolute value or no increments at all, leading to no modification of R b while expected to get closer to R o . Figure 5 displays the histograms of (R b 2 R o ) and

TABLE 1. Results of 1DVAR analysis for 6-h periods for the control and test experiments. Given in parenthesis are the number of profiles analyzed where ST parameterization is active. Expt Control Control Test 4 Test 4

Analysis date and time (UTC) 11 26 11 26

Feb 1998 0000 Aug 1998 1200 Feb 1998 0000 Aug 1998 1200

No. of analyzed profiles (ST) 2685 2667 2238 2260

(383) (151) (227) (105)

No. of R b 5 0 and R o . 0 profiles

No. of profiles rejected by Eq. (5)

2680 2196 2330 1959

80 10 63 8

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FIG. 5. Histogram of (a) (R b 2 R o ) in mm h21 and (b) (R a 2 R o ) in mm h21 for the control experiment (11 Feb 1998 at 0000 UTC).

(R a 2 R o ). The corresponding bias and standard deviation are given in Table 2. This figure clearly shows that 1DVAR assimilation has been able to provide an analyzed state closer to observations with a reduction

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of the standard deviation from 2.01 to 0.80 mm h21 . The histogram for (R b 2 R o ) extends further in negative values (minimum 5 217.3 mm h21 ) than in positive values (maximum 5 7.1 mm h21 ). This is related to the model behavior that always produces precipitation areas that are extensive and less intense than observed for very active convective systems, as shown in Fig. 2 and Fig. 1. The 1DVAR analysis produces an important reduction of the histogram range with minimum and maximum values for (R a 2 R o ) of 212.9 and 2.5 mm h21 , respectively. Very similar results are found for the August case as illustrated by the bias and standard deviation of (R b 2 R o ) and (R a 2 R o ), given in Table 2, which are close to those found for the February case. Nevertheless, some profiles remain nearly unchanged by 1DVAR analysis leading to R a values close to R b and therefore far from the observation. This behavior is illustrated in the scatterplot of Fig. 6a where two families of points clearly appear. The first family corresponds to profiles close to the unit slope line where the 1DVAR analysis has modified the background state so that it provides a surface rain rate close to observations. Note that the fit to the unit slope line deteriorates when (R o 2 R b ) increases because 1DVAR is less efficient in increasing precipitation and to the fact that we specify larger observation errors for high rain rates. The second family of points in Fig. 6a corresponds to (R a 2 R b ) values close to zero where no rain-rate information from the observations is assimilated by 1DVAR. For these profiles, the nonlinear behavior of the physical parameterizations is such that the 1DVAR has been unable to find a minimum for the cost function. In the context of 4DVAR assimilation of 1DVAR products, this second type of profile should not be retained because they would provide an information about the model background instead of an information about the observations. Thus, a quality control of the 1DVAR products based on a criterion excluding this second type of profiles will be applied before 4DVAR assimilation. Figure 6b presents 1DVAR products after a quality control has been applied. By retaining only analyzed points where R o 2 s o , R a , R o 1 s o , all the profiles were R a ø R b are clearly rejected, but a significant number of profiles are kept (78%).

TABLE 2. Statistics on 1DVAR results for the control and test experiments. In test 1, the observation error for moderate and high rain rates is doubled. In test 2, the observation error threshold for light rain rates is decreased. In test 3, the control vector only includes specific humidity profiles. In test 4, observations are taken at the middle of the 6-h assimilation window. R b 2 R o in mm h21

R a 2 R o in mm h21

Expt

Analysis date and time (UTC)

Bias

Std dev

Bias

Std dev

Control Control Test 1 Test 2 Test 3 Test 4 Test 4

11 Feb 1998 0000 26 Aug 1998 1200 11 Feb 1998 0000 11 Feb 1998 0000 11 Feb 1998 0000 11 Feb 1998 0000 26 Aug 1998 1200

0.12 0.22 0.12 0.12 0.13 0.21 0.43

2.01 2.18 2.01 2.01 2.02 1.96 1.78

20.11 20.09 20.22 20.11 20.12 20.12 20.04

0.80 1.09 1.25 0.80 0.82 1.04 0.99

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In their study, FM2000 have shown that for some profiles the minimization is controlled by ST parameterization creating unrealistic stratiform precipitation where convective precipitation is expected. Table 1 shows that ST parameterization is involved in the minimization of a few hundred profiles depending on the number of fronts overpassed by TRMM in 6 h. Even though supersaturations can be created by the 1DVAR for increasing stratiform precipitation, the pathology shown by FM2000 has not been evidenced in the present results. This may suggest that such problem is marginal, at least with the ECMWF mass-flux convection scheme. e. Temperature and humidity increments The 1DVAR provides analyzed profiles of temperature and specific humidity from which increments between the analyzed and the background states (x a 2 x b ) can be derived. They are represented in Fig. 7 where the R o $ R b profiles (denoted MORE profiles) and the R o , R b profiles (denoted LESS profiles) have been treated separately. For both types of profiles, mean temperature increments in absolute value are small with values below 0.10 K for all vertical levels (Fig. 7a). Mean specific humidity increments (Fig. 7b) reach 0.39 g kg21 (20.13 g kg21) with a maximum (minimum) around model level 24 for MORE (LESS) profiles. The vertical structure of relative humidity increments (Fig. 7c) shows that, to increase precipitation (MORE profiles), 1DVAR analysis moistens the troposphere up to level 7 (around 130 hPa) and slightly dries the stratosphere. This behavior is a consequence of the negative temperature correlations of background errors between the troposphere and the stratosphere. The contrary is found for LESS profiles. A similar result was found by Treadon (1997) but with more drying and less moistening. Differences may be related to the use of a different convection scheme, different background errors, and different rain-rate observation errors. The vertical distribution of specific humidity (Fig. 7b) and of relative humidity (Fig. 7c) increments below model level 7 are well correlated with the vertical distribution of background errors for q (see Fig. 2 in FM2000). This is because 1DVAR mostly modifies specific humidity profiles where corresponding background errors are large. The fact that temperature increments are low and that relative humidity increments are also correlated with specific humidity increments means that 1DVAR minimization is more sensitive to modifications of specific humidity than to modifications of temperature to adjust precipitation. Thus 1DVAR products related to specific humidity rather than to temperature should be preferred for 4DVAR assimilation. 5. Sensitivity tests a. Time tendencies of specific humidity and temperature The main difference between the present 1DVAR assimilation and FM2000 version is that time tendencies

FIG. 6. Scatterplot of (Ra 2 Rb) in mm h21 as a function of (Ro 2 Rb) in mm h21 for the control experiment (11 Feb 1998 at 0000 UTC).

of temperature and specific humidity related to processes other than CO and ST (i.e., dynamics, radiation, gravity wave drag, vertical diffusion) have been used to update vertical profiles in the observation operator. To show the impact of this choice, R b is recalculated by setting to zero these tendencies as in FM2000 [D1x 5 0 in Eq. (4)]. Results are displayed in Fig. 8. Compared to the R b control field (Fig. 2), convective precipitation is enhanced for heavy rates and slightly reduced for light rates. Stratiform precipitation is largely reduced so that the frontal precipitation zone does not exist anymore (Fig. 8c). Comparison with the 4DVAR background rain-rate field produced by the 3D model (Fig. 3) shows even more differences. Thus, the R b field calculated

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FIG. 7. Profiles of (a) mean temperature increments, (b) mean specific humidity increments, and (c) mean relative humidity increments. The control and tests experiments are all from the Feb case. LESS and MORE are explained in the text.

without thermodynamic time tendencies is clearly less consistent with the 4DVAR background rain-rate field. This is because these time tendencies provide a major piece of information to CO and ST schemes concerning the environment in which precipitation is formed. They are essential to ensure that CO and ST parameterizations behave similarly in the one-dimensional and three-dimensional contexts. b. Observation error As discussed in section 3, the observation error used in Eq. (1) is based on rough estimation. A first sensitivity test (experiment noted test 1) has been run with an observation error set to 0.50R o instead of 0.25R o and no modification of the threshold error for low observed surface rain rates. This increase of the observation error can also be interpreted as an indirect way to take into account errors coming from the observation operator. Test 1 produces an increase of the negative bias and a larger standard deviation of the analysis departure (R a

FIG. 8. Same as Fig. 1 but for 1DVAR background surface rain rate calculated without using time tendency profiles of temperature and specific humidity.

2 R o ) compared to the control (see Table 2). For moderate and heavy observed rain rates, the observation term in the cost function is smaller for test 1 leading to a weaker constraint from observations in the minimization. This effect also appears on the MORE profiles of specific humidity, temperature, and relative humidity increments, which have a similar vertical structure but with lower values by about 30% compared to the control experiment (see Fig. 7). Although there is an impact on the results when a higher observation error for moderate and heavy rain rate is assumed, 1DVAR is still able to assimilate information from observations as shown by the reduction of the standard deviation and by the nonnegligible humidity increments obtained. However, it is likely that this specified observation error is still too small (or that the background errors are too large) since for most of the profiles s Rb is larger than s o . A second test on the observation error has been done.

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It concerns the threshold value imposed for light rain rates. Since the 2A12 algorithm, from which mean surface rain rates are obtained, generally performs well in identifying nonrainy pixels, a smaller threshold error for low mean observed rain rates was set: 0.001 mm h21 instead of 0.01 mm h21 (experiment noted test 2). Results for (R a 2 R o ) bias and standard deviation are nearly identical for test 2 and control (see Table 2). Profiles of temperature and humidity increments are also very close except for some profiles where test 2 provides very large increments in absolute values for temperature (up to 7 K) and specific humidity (up to 3.5 g kg21 ). For these ‘‘pathological’’ profiles, the weight given to the observation term in the cost function is obviously too large and therefore almost relaxes the background constraint. Thus, 1DVAR analysis cannot accept a too small observation error for light rain rates. c. Reduced control vector It was shown in section 4 that 1DVAR analysis essentially modifies specific humidity profiles to improve model rain rate while providing very small increments of temperature. Since in the Tropics background errors for temperature are small, it has been checked that the standard deviation of increments normalized by the standard deviation of forecast errors [std(Dx)/s x ] are much smaller for temperature than for specific humidity. For the February case, the value of std(Dq)/s q is 6 times larger than std(DT)/s T for the MORE profiles and 3 times larger for the LESS profiles. Thus, a possibility to decrease the computational cost of 1DVAR is to reduce the control vector by not modifying the temperature profiles. A test (experiment noted test 3) was done in which temperature profiles are only used as background information in the observation operator. Results given in Table 2 show that the bias and the standard deviation of (R a 2 R o ) are both very close to those of the control experiment. The difference between the control and test 3 experiments for relative humidity increments (Fig. 7c) is rather small showing that the minimization finds a similar solution in terms of atmospheric state. However, mean increments of specific humidity for test 3, displayed in Fig. 7b, are larger by about 50% in absolute values than the control experiment for both the LESS and MORE profiles. Since in this case the minimization can only act on specific humidity, modifications have to be larger than when temperature is also used in the control vector for producing equivalent rainfall increments. Using only specific humidity profiles and surface pressure as control variables can reduce 1DVAR computational cost but modifies noticeably the analyzed specific humidity state. d. Number of time slots The impact of assimilating the 6-h dataset at the middle time of the assimilation window instead of at the

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FIG. 9. Same as Fig. 1c but for the R o field used in test 4 (see text for explanations).

appropriate observation time (seven time slots) is examined (experiment noted test 4). The mean surface rain rate used for test 4 has been calculated as described in section 3 but using the whole 6-h dataset. In areas that are sampled several times by TMI in 6 h (subtropical regions) the resulting field is the mean of the different overpasses. For fast moving systems, this can lead to a slightly different rainfall field, as illustrated by the front case (Fig. 9) which was sampled at three different times during the 6-h window: 10 February 1998 around 2305 UTC, 11 February 1998 around 0045 UTC, and 11 February 1998 around 0220 UTC. The resulting field is more extensive and less intense than that obtained only using the overpass on 10 February 1998 around 2305 UTC (Fig. 1c). Test 4 evaluates the hypothesis that precipitation does not vary much in intensity and location within 6 h. Although this hypothesis is not always fulfilled as illustrated by Fig. 9, the impact on the 1DVAR analysis tends be limited. The rainfall field analyzed for test 4 is displayed in Fig. 10. Comparison between R a fields for the control and test 4 experiments exhibits small differences that are related to the use of a slightly different background field and of a slightly different observation field for the front case. General results given in Table 1 show that the number of points analyzed in test 4 is reduced by 17% for the February case and by 15% for the August case because of a reduced number of observations. The bias (R b 2 R o ) is increased mainly because of the use of the background field at a single time and also because of the slight decrease of R o values in subtropical regions. The bias and standard deviation of (R a 2 R o ) for test 4 are close to those found in the control experiment. Increments for temperature, specific humidity, and relative humidity for LESS profiles are the same between test 4 and the control experiment. The test 4 increments for the MORE profiles are slightly smaller in absolute value because in this case the R o field is slightly weaker in regions that are sampled more that once. All these results indicate that the reduction of the number of time slots from seven to one does not

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FIG. 10. Same as Fig. 1 but for 1DVAR analyzed surface rain rate (R a field) for test 4. Background surface rain rate is used for grid points not satisfying in the criteria of Eq. (5).

significantly change the 1DVAR results while it allows an easier implementation in 4DVAR. The impact of this configuration should be tested in the 4DVAR assimilation framework. 6. Conclusions This paper presents results from a 1DVAR assimilation of TMI-derived surface rain-rate fields. The 1DVAR method allows retrieval of temperature and humidity profiles that provide precipitation close to observations while remaining consistent with a model short-range forecast. A 1DVAR scheme has been adapted from Fillion and Mahfouf (2000) using the ECMWF mass-flux convection scheme. A diagnostic large-scale condensation scheme (ST) is coupled to the convection scheme in a fractional-stepping mode (Beljaars 1991). A major dif-

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ference from the FM2000 version is the use of rain-rate observations from the TMI radiometer instead of simulated observations. The introduction in the observation operator of the background profiles of temperature and specific humidity time tendencies due to other processes (dynamics, radiation, turbulence, and gravity wave drag) has provided information on the environment of the considered profiles. They proved to be essential in order to get consistency between 1DVAR and 4DVAR background precipitation fields (i.e., model short-range forecast). Nevertheless, frontal precipitation is always weaker and less extended in the 1DVAR background field due to the use of a diagnostic ST scheme in 1DVAR, which is less efficient than the operational prognostic ST scheme that uses more explicitly the model dynamics. When the background rain-rate field is non-zero, results have shown that 1DVAR assimilation is generally able to increase, decrease, and switch off precipitation to get closer to observations. For heavy precipitation, analyzed rain rates are always lower than observed ones due to the model inability to produce very high rain rates and to the specification of large observation errors. Results also clearly showed that some profiles remain almost unchanged by 1DVAR while a modification of the surface rain rate is expected. These profiles should not be assimilated in 4DVAR and a quality control has been proposed before assimilation of 1DVAR products in 4DVAR. When the background precipitation is zero, 1DVAR is not able to trigger precipitation where it exists in the observations. This is a major weakness of the 1DVAR approach that will need to be addressed in future developments. The 1DVAR analysis provides mean temperature increments less than 0.1 K over the whole atmosphere while mean specific humidity increments are greater than 0.35 g kg21 around model level 25 (about 800 hPa). As a consequence, 1DVAR products related to specific humidity such as total column water vapor could be assimilated in 4DVAR without significant loss of information. Because no precise information on observation error is yet available to quantify the uncertainty of the precipitation retrievals, sensitivity tests on the mean surface rain-rate error values have been done. An increase of the error from 0.25R o to 0.50R o for moderate and heavy rain rates leads to a 30% decrease of the specific humidity increments and to an increase of the standard deviation for (R a 2 R o ). Nevertheless, there is a noticeable improvement of the model precipitation with respect to observations even when a large value is used. A reduction of the threshold observation error value used for little and zero rain rates does not change the results except for A few profiles where unrealistic increments of temperature and specific humidity are found. To avoid this problem, a threshold value of at least 0.01 mm h21 should be used. This 1DVAR retrieval is the first step toward the use of TMI data in the ECMWF 4DVAR system. The second

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step is the assimilation of 1DVAR products in the ECMWF 4DVAR assimilation system. In order to reduce the computational cost of 1DVAR, two approaches have been tested. The first one uses a reduced control vector by not modifying temperature profiles. Specific humidity increments are significantly larger than when temperature is also adjusted during the minimization. In the second test, one single background field and one single observation field are used instead of seven for each assimilation window. Although it assumes the steadiness of precipitation over a period of 6 h, which is not always true, results are very similar to the control experiment. Further work needs to be done to evaluate the impact of such possible simplifications on the 4DVAR assimilation of the 1DVAR products. In the absence of information on observation errors and for simplicity, a Gaussian distribution of the observation error has been assumed in 1DVAR. Errico et al. (2000) showed that 1DVAR results depend on the form of the distribution chosen, which is likely not Gaussian. New tests of 1DVAR retrieval will be done when a reliable estimation of the distribution form and parameters of the errors is available from TMI data. In this study, the observation operator was assumed to be error free because no estimation of this error has yet been done. Statistics of background errors were specified from the current ECMWF 4DVAR system. They may need to be revised because it is likely that shortrange forecast errors are significantly different in rainy situations than in clear-sky conditions. These two aspects will be improved in the future. Indeed, precipitation observations from TRMM represent an ideal dataset to assess the accuracy of moist parameterization schemes. Acknowledgments. TRMM is a joint NASA–NASDA mission (spacecraft launched in November 1997). We acknowledge NASA and NASDA for opening the TRMM data to EuroTRMM, a consortium of scientists from the Centre d’E´tude des Environnements Terrestre et Plane´taires (France), German Aerospace Research Establishment (Germany), Istituto di Fisica dell’Atmosfera (Italy), Max-Planck-Institut fu¨r Meteorologie (Germany), Rutherford and Appleton Laboratory (United Kingdom), University of Essex (United Kingdom), Universite´ Catholique de Louvain (Belgium), University of Munich (Germany), and European Centre for MediumRange Weather Forecasts (United Kingdom). EuroTRMM is funded by the European Commision and the European Space Agency. We thank F. Bouttier for providing us the statistics of forecast errors from the ECMWF operational data assimilation system. Thanks are also due to the Institut National de Recherche en Informatique et Automatique for providing the M1QN3 minimization code used in this study. Tony Hollingsworth, Martin Miller, Luc Fillion, and Ron Errico provided valuable suggestions for improving the earlier versions of this paper.

3865 REFERENCES

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