# Variational retrieval of temperature and humidity profiles using rain rates versus microwave brightness temperatures

Emmanuel Moreau, Philippe Lopez, Peter Bauer, Adrian Tompkins, Marta Janisková, Frederic Chevallier

► To cite this version:

Emmanuel Moreau, Philippe Lopez, Peter Bauer, Adrian Tompkins, Marta Janisková, et al. Variational retrieval of temperature and humidity profiles using rain rates versus microwave brightness temperatures. Quarterly Journal of the Royal Meteorological Society, 2004, 130 (598), pp.827-852. 10.1256/QJ.03.118. hal-02948143

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# Variational retrieval of temperature and humidity profiles using rain rates versus microwave brightness temperatures

By EMMANUEL MOREAU, PHILIPPE LOPEZ*, PETER BAUER, ADRIAN M. TOMPKINS, MARTA JANISKOVÁ and FRÉDÉRIC CHEVALLIER

European Centre for Medium-Range Weather Forecasts, Reading, UK

(Received 7 July 2003; revised 20 October 2003)

## SUMMARY

This paper assesses the performance of two different approaches for the retrieval of temperature and humidity profiles from the satellite passive microwave measurements of the Tropical Rainfall Measuring Mission (TRMM) and of the Special Sensor Microwave Imager in rainy areas. Both methods are based on a one-dimensional variational retrieval (1D-Var) approach. The input to the first one is rainfall rates estimated by a standard retrieval algorithm from the raw brightness temperatures. In the second technique, 1D-Var directly uses the brightness temperatures. The retrieval experiments utilize new simplified physical parametrizations of convection and large-scale condensation specially designed for variational assimilation as well as a microwave radiative-transfer model.

Several tropical and midlatitude meteorological situations are studied. In all cases, it is found that both 1D-Var approaches are able to converge satisfactorily and produce consistent temperature and specific-humidity increments. However, the convergence of 1D-Var retrievals on surface rainfall rates, and to a lesser extent of 1D-Var retrievals on brightness temperatures, is reduced when precipitation in the background is produced through convection and not by large-scale processes. The results of the two 1D-Var methods are then compared in brightness temperature and rainfall rate space and are validated against independent observations from the precipitation radar on-board the TRMM. Finally, the sensitivity of the 1D-Var retrievals to the specification of observation-error statistics is studied.

**KEYWORDS:** Precipitation SSM/I TRMM 1D-Var

## 1. INTRODUCTION

The assimilation of observations related to clouds and precipitation has become an important issue for a number of operational weather-forecasting centres, including the European Centre for Medium-Range Weather Forecasts (ECMWF). In particular, the representation of the hydrological cycle in numerical weather-prediction models will be one of the major challenges in the next few years. Observations are already available from the Tropical Rainfall Measuring Mission (TRMM) since 1997, from the satellites of the USA's Defense Meteorological Satellite Program (DMSP) since 1987, from the Aqua mission since May 2002, and from the polar-orbiting satellites of the USA's National Oceanic and Atmospheric Administration. This study utilizes data from the TRMM Microwave Imager (TMI) and from the DMSP Special Sensor Microwave/Imager (SSM/I). In the coming years, an increasing amount of observations related to clouds and precipitation is likely to become available from lidars, radars, sounders and imagers installed on board satellite missions such as the Special Sensor Microwave Imager/Sounder (SSMIS; 2003), CloudSat (2004), the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations mission (CALIPSO; 2004), the National Polar-orbiting Operational Environmental Satellite System (NPOESS; 2006), the Global Precipitation Mission (GPM; 2008) and the Earth Clouds, Aerosols and Radiation Explorer (EarthCARE; 2008). Maximizing the benefits from this huge source of data will require the development of assimilation procedures capable of efficiently converting the information on clouds and precipitation into realistic increments applicable to the forecast model's variables (temperature, humidity, wind, and possibly clouds and precipitation themselves).

* Corresponding author: ECMWF, Shinfield Park, Reading, Berkshire RG2 9AX, UK.

e-mail: Philippe.Lopez@ecmwf.int

© Royal Meteorological Society, 2004.
Several studies have recently addressed the issue of precipitation assimilation using either a variational approach (e.g. Hou *et al.* 2000, 2001; Peng and Zou 2002) or other techniques like nudging (e.g. Macpherson 2001). At the ECMWF, Marécal and Mahfouf (2000) developed a unidimensional variational (1D-Var) method for correcting individual profiles of the ECMWF model’s control variables in order to decrease the discrepancies that often exist between the simulated surface rainfall rates (RRs) and corresponding retrievals obtained from TMI or SSM/I microwave brightness temperatures (TBs). Marécal and Mahfouf (2002) also demonstrated that an indirect ‘1D-Var + 4D-Var’ assimilation of TMI-derived RR s could improve the quality of humidity, temperature and wind forecasts in the tropics. In their approach, the background – observation departures on surface RR s are first converted into total column water vapour (TCWV) increments as an output from 1D-Var retrievals. The corresponding TCWV pseudo-observations are then assimilated in the 4D-Var system. Marécal and Mahfouf (2003) also showed that this indirect method is more robust than a direct 4D-Var assimilation of the TMI RR s, because of some inconsistencies between the inner and outer loops of the ECMWF 4D-Var assimilation system (Courtier *et al.* 1994). The ‘1D-Var + 4D-Var’ technique therefore seems more appropriate to assimilate such data until these inconsistencies are resolved.

Instead of performing the 1D-Var retrievals on surface RR s that are derived from multi-channel microwave TBs through various algorithms, the 1D-Var calculations could directly deal with the TBs. This would homogenize the approach and could even provide more accurate retrievals, since the sensitivities of TBs not only to rain, but also to cloud water and water vapour could be exploited (Moreau *et al.* 2003). This study explores this new method and compares its performance with that of the existing 1D-Var retrieval applied to derived RR s.

The plan of the paper is as follows. Section 2 introduces the 1D-Var technique used in this paper. This includes a description of two new parametrizations of moist processes that were specially designed for assimilation in order to make a compromise between physical realism and linear behaviour. The microwave radiative-transfer model (RTM) used in the 1D-Var computations is also briefly presented. Section 3 deals with the experimental set-up, which includes the description of the cases studied and of the observational and model datasets (including the specification of error statistics). The main results from the 1D-Var experiments are then detailed in section 4. A discussion of the comparison of 1D-Var on surface RR s with 1D-Var on TBs, as well as a validation of the 1D-Var outputs against independent satellite data, is proposed in section 5. This section also presents a study of the sensitivity of the 1D-Var retrievals to the specification of the observation-error statistics.

## 2. THE 1D-VAR RETRIEVAL METHOD

### (a) General description

The objective of 1D-Var is to find an optimal model state that minimizes in a least-square sense the distance between a selected output quantity of the model and its observed equivalent, given a background constraint. In the present study, two control variables are considered in the model: temperature and specific humidity. They are given in the form of individual vertical profiles at a given time. The selected output quantity to be optimized is either the surface RR s (as in Marécal and Mahfouf 2000) or the multi-channel microwave TBs. The two corresponding 1D-Var retrieval methods will be hereafter referred to as 1D/RR and 1D/TB, respectively. Surface RR is an output
from the model's convection and large-scale condensation parametrizations and can be derived from observed microwave TBs via a retrieval algorithm. On the other hand, TBs can be simulated by applying the parametrizations of moist processes to the model's control variables followed by the application of a microwave RTM. The main inputs to the latter model are precipitation content, cloud water content, cloud cover, specific humidity, air and sea surface temperatures and surface wind speed. In the present study, the 1D-Var method searches for the model's state vector x that minimizes the following functional:

$$J(\mathbf{x}) = \frac{1}{2}(\mathbf{x} - \mathbf{x}_b)^{\mathrm{T}}\mathbf{B}^{-1}(\mathbf{x} - \mathbf{x}_b) + \frac{1}{2}(\mathrm{H}(\mathbf{x}) - \mathbf{y}_o)^{\mathrm{T}}\mathbf{R}^{-1}(\mathrm{H}(\mathbf{x}) - \mathbf{y}_o) \quad (1)$$

where $\mathbf{x}_b$ is the background model state, T is the transpose, and $\mathrm{H}(\mathbf{x})$ is the nonlinear observation operator that converts the model's variables into either surface RRs or microwave TBs. $\mathbf{y}_o$ denotes the corresponding observed quantity and $\mathbf{R}$ is the error covariance matrix of the observations, which includes the instrumental errors and the errors of H. Matrix $\mathbf{B}$ contains the background-error covariances for the model's control variables (temperature and specific humidity). The result of the 1D-Var retrieval is therefore a combination of the background and of the observation, weighted by the inverse of their respective error statistics.

The minimization of the functional is performed using the quasi-Newton descent algorithm (M1QN3) developed by Gilbert and Lemaréchal (1989). It requires the calculation of the gradient of the functional defined in Eq. (1):

$$\nabla J(\mathbf{x}) = \mathbf{B}^{-1}(\mathbf{x} - \mathbf{x}_b) + \mathbf{H}^{\mathrm{T}}\mathbf{R}^{-1}(\mathrm{H}(\mathbf{x}) - \mathbf{y}_o) \quad (2)$$

where $\mathbf{H}^{\mathrm{T}}$ is the transpose (or adjoint) of the Jacobian matrix of the nonlinear observation operator, H. In this study, H consists of new simplified parametrizations of convection and large-scale condensation that are described in the next subsections. In the case of 1D/TB, H also includes the RTM designed by Bauer (2002) and Moreau et al. (2002) that takes into account the scattering of microwave radiation in precipitating clouds.

### (b) Moist physics and the radiative-transfer model

(i) Convection. Previous studies advocated an improvement of the linearized version of the ECMWF's parametrizations of moist processes, especially convection, to be used in the full 4D-Var and in the simpler 1D-Var algorithm developed by Maréchal and Mahfouf (2000). A modified version of the operational convection scheme was therefore developed for the future variational assimilation of clouds and precipitation observations. The modified mass-flux convection scheme was designed such as to retain some basic similarities with the original mass-flux scheme from Tiedtke (1989) and at the same time to remove the high degree of complexity of the latter, which is a source of nonlinearities. Its detailed description and evaluation are presented by Lopez and Moreau (2004) but its main features will be summarized here.

All types of convection (shallow, mid-level, and deep) are treated in a similar way. In particular, the closure assumption that relates the model control variables to the subgrid-scale cloud-base mass flux is expressed through a single formulation based on the release of convective available potential energy in time. In contrast with the operational scheme, the equations that describe the vertical evolution of the updraught mass flux, $M_u$ (kg m$^{-2}$s$^{-1}$), and of the updraught thermodynamic variables, $\Phi_u$ (i.e. dry
static energy, specific humidity and cloud condensate), are uncoupled:

$$ \frac{\partial M_u}{\partial z} = (\epsilon - \delta) M_u \qquad (3) $$

$$ \frac{\partial \Phi_u}{\partial z} = -\epsilon(\Phi_u - \bar{\Phi}) \qquad (4) $$

where $\bar{\Phi}$ denotes field values in the environment, and $\epsilon$ and $\delta$ are respectively the fractional entrainment and detrainment rates (in m⁻¹). This uncoupling allows the removal of the iterative calculations involved in the operational code for updating the cloud-base mass flux, thereby leading to an easier development of the adjoint. The impact of this modification on the forecast is marginal.

Convection is assumed to be activated only if the bulk convective updraught vertical velocity remains positive at cloud base. The updraught is assumed to originate from the surface only if its initial vertical velocity, as calculated from the surface heat fluxes using Eq. (15b) of Holtslag and Moeng (1991), is positive. The departures of the updraught from the environment are also assumed to be dependent on the surface heat fluxes. If convection cannot be initiated from the surface, the convective ascent may originate from higher levels provided relative humidity exceeds 80%. In this case, the initial vertical velocity of the bulk updraught is set equal to 1 m s⁻¹. Regardless of whether the updraught originates from the surface or higher up, the vertical evolution of its kinetic energy is computed following Simpson and Wiggert (1969).

Based on Siebesma and Jakob (personal communication), $\epsilon$ is parametrized as $c_\epsilon/(z - z_{st}) + 10^{-5} \text{ m}^{-1}$ where $c_\epsilon$ is a coefficient equal to 0.5 and $z_{st}$ is the starting altitude of the updraught (in metres). For the detrainment, $\delta$ is set equal to $\epsilon$, except close to cloud top where a constant organized detrainment rate is added ($\delta = \epsilon + 2 \times 10^{-4} \text{ m}^{-1}$).

Simplified calculations of downdraughts and convective momentum transport based on the operational scheme (Tiedtke 1989) are also included in the new parametrization. More importantly, the precipitation formation rate is made proportional to the updraught cloud water content as in Tiedtke (1989).

Reasonably well-behaved physical parametrizations in terms of linearity are necessary to ensure the convergence of the 1D-Var retrievals. Therefore, the linear behaviour of the modified convection scheme compared with the operational one has also been assessed through the calculations of nonlinear residuals. For the nonlinear convection scheme, $M$, and its tangent-linear version, $M'$, the nonlinear residuals are defined as

$$ \mathrm{RES}_{\mathrm{NL}} = M(\mathbf{x} + \lambda\delta\mathbf{x}) - M(\mathbf{x}) - \mathbf{M}'[\mathbf{x}]\lambda\delta(\mathbf{x}) \qquad (5) $$

where $\mathbf{x}$ denotes input vertical profiles of temperature, $T$, and specific humidity, $q$. The residuals have been computed for the convective tendencies $\partial T/\partial t$ and $\partial q/\partial t$ that are produced by the convective parametrization. The calculations of Eq. (5) have been repeated for values of the perturbation scaling factor $\lambda$ ranging from -1 to -10⁻⁵ and from 10⁻⁵ to 1. The reference vector of perturbations $\delta\mathbf{x}$ was set equal to typical values of the standard deviation of the background model errors. An example of vertical profiles of such errors is shown in Fig. 1. Table 1 gives the correspondence between model levels and pressure levels. Figure 2 shows nonlinear residuals, averaged over 100 convective profiles, that are obtained when specific-humidity perturbations of varying size are applied on model level 60. Results are displayed for the lowest model level because input perturbations imposed there result in maximum changes in the profiles of convective tendencies. The x-axis gives the values of the unitless
<figure><img src="image_3.png" /><figcaption>Figure 1. Vertical profiles of typical values of the standard deviation of the model background errors on temperature (T) and specific humidity (q). Units are in K and g kg⁻¹ respectively.</figcaption></figure>

<table><caption>TABLE 1. PRESSURE ON EVERY FIVE MODEL LEVELS BETWEEN LEVELS 20 AND 60. VALUES ARE GIVEN ASSUMING A SURFACE PRESSURE OF 1013.25 hPa.</caption><thead><tr><td>Model level number</td><td>Pressure (hPa)</td></tr></thead><tbody><tr><td>20</td><td>36</td></tr><tr><td>25</td><td>96</td></tr><tr><td>30</td><td>202</td></tr><tr><td>35</td><td>353</td></tr><tr><td>40</td><td>539</td></tr><tr><td>45</td><td>728</td></tr><tr><td>50</td><td>884</td></tr><tr><td>55</td><td>979</td></tr><tr><td>60</td><td>1012</td></tr></tbody></table>

scaling factor $\lambda$ that is applied to the input reference perturbation of specific humidity on model level 60. This normalization permits the comparison of the residuals obtained with the two parametrizations, and the smaller the values of the plotted field, the more valid the linear assumption. Figure 2 demonstrates that the nonlinear residuals are at least one order of magnitude smaller with the new scheme than with the operational one. Asymmetries about the y-axis result from the fact that large negative moisture perturbations at the surface turn off convection, thereby leading to a less linear behaviour of the scheme. Note that similar conclusions can be drawn when input perturbations are imposed on temperature instead of specific humidity (Lopez and Moreau 2004).

(ii) *Clouds*. Tompkins and Janisková (2003) have recently developed a new statistical diagnostic cloud scheme that is intended to be implemented in place of the simplified parametrization currently used in the ECMWF's 4D-Var tangent-linear and adjoint calculations (Janisková et al. 2002). A uniform probability density function (PDF) was
<figure><img src="image_4.png" /><figcaption>Figure 2. Nonlinear residuals of convective (a) and (b) temperature and (c) and (d) specific-humidity tendencies as functions of model levels (y-axis) and of the size of the input specific-humidity (q) perturbation imposed on model level 60 (x-axis). This size is labelled according to the values of the scaling factor λ applied to the input reference perturbation of specific humidity on model level 60. The displayed residuals have been averaged over 100 convective profiles. The field actually plotted is log₁₀ (RESNL/RESNLmax) where RESNLmax denotes the maximum value of the residual for a given pair of input/output parameters and for both studied convection schemes (i.e. for a given row of the figure): (a) and (c) operational scheme, (b) and (d) new scheme.</figcaption></figure>

chosen to describe the subgrid-scale fluctuations of temperature and total water in order
to make adjoint developments easier and to ensure a reasonable degree of consistency
with the scheme used in the full operational forecast model (Tiedtke 1993). The width of
the PDF is assumed to increase linearly when relative humidity, RH, varies from 100%
down to a critical threshold, RH₀, beyond which cloud formation occurs. Both the slope
of this linear dependence and RH₀ are functions of the normalized pressure vertical
coordinate P/Psurf, where P is the pressure on the current level, and Psurf is the surface
pressure. In particular, RH₀ ranges from 50% in the mid-troposphere to 85% at the
surface and at upper levels.

Large-scale cloud cover is expressed as a function of *RH* and *RH*₀, while con-
vective cloud cover depends on the diagnostic source term associated with convective
detrainment. The new simplified cloud scheme includes a parametrization of precipitation generation inspired from Sundqvist *et al.* (1989) and an original parametrization of precipitation evaporation based on the subgrid-scale variability of total water.

Tompkins and Janisková (2003) show that the linearity of the new simplified cloud scheme is suitable for its use in 1D-Var and 4D-Var.

(iii) *Microwave radiative transfer.* For simulating the radiative transfer in clouds and precipitation for large data volumes the fast modelling framework RTTOV\* (Eyre 1991; Saunders *et al.* 2002) has been chosen. The multiple scattering is simulated applying the 8-Eddington approximation that has proven to provide sufficient accuracy (Kummerow 1993) at microwave frequencies. All hydrometeor types (rain, snow, cloud liquid water and cloud ice) are assumed to be spherical and their optical properties are taken from pre-calculated look-up Mie tables to increase numerical efficiency (Bauer 2002). For the purpose of variational retrievals as well as to increase computational efficiency, the tangent-linear and the adjoint versions of the RTM are employed.

### 3. EXPERIMENTAL SET-UP

#### (a) *Meteorological events*

Three recent meteorologically interesting events have been identified for running 1D-Var: two tropical cyclones well sampled by both the TMI and the TRMM precipitation radar (PR) and one extratropical front with a good observational coverage from the SSM/I. These cases were correctly forecast by the ECMWF operational model in terms of location but not with regards to their structure and rain intensity.

The first selected situation features super-typhoon *Mitag* that developed over the western Pacific in early March 2002. 1D-Var has been tested at 1200 UTC 5 March 2002, when the super-typhoon reached its maximum intensity just east of the Philippines (estimated mean-sea-level pressure minimum of 930 hPa located at 14.2°N, 129.9°E).

The second event is tropical cyclone *Zoe* that developed over the western Pacific in the vicinity of the Fiji Islands, between 25 December 2002 and 31 December 2002. 1200 UTC 26 December 2002 was the date selected for running the 1D-Var. At that time, the storm was in its intensification phase and its central pressure minimum of 975 hPa was located at 10.7°S, 174.1°E.

The third case is an extratropical front that developed over the North Atlantic in January 2002 and which was dominated by large-scale precipitation. The 1D-Var experiment was run on the meteorological situation at 1200 UTC 9 January 2002.

#### (b) *Observations*

For super-typhoon *Mitag* and tropical cyclone *Zoe*, TMI data from the two TRMM satellite orbits at 1100 UTC 5 March 2002 and 1400 UTC 26 December 2002 have been used, respectively. In the case of the midlatitude front, the 1D-Var experiments have been based on SSM/I data at 1130 UTC 9 January 2002.

Various algorithms have been tested for retrieving surface rainfall amounts from the multi-channel microwave TBs observed by the TMI and SSM/I. For TMI data, both the TRMM standard rainfall product 2A12 (Kummerow *et al.* 2001) and the Precipitation radar Adjusted TMI Estimation of Rainfall (PATER; Bauer *et al.* 2001) retrievals have been used. PATER employs independent rainfall estimates from the TMI and PR over their common swath to derive a rain-dependent calibration curve. A similar principle

<footnote>* Radiative Transfer for TIROS (Television Infrared Observation Satellite) Operational Vertical sounder.</footnote>
is applied here in order to apply PATER to SSM/I data (PATER-SSMI, thereafter) by simulating the SSM/I observations with TMI data, that is by reducing the spatial resolution and selecting the common channels.

TMI surface RRs derived with 2A12 and PATER have been used in the cases of *Mitag* and *Zoe*. Three retrieval algorithms, namely PATER-SSM/I (described above), Bauer and Schluessel (1993), and Ferraro *et al.* (1996), have been utilized on SSM/I TBs in the midlatitude frontal case. All observed rainfall retrievals have then been averaged onto the Gaussian grid of the ECMWF model that corresponds to a T511 spectral truncation (i.e. to a grid-point resolution of about 40 km), so that observation and model points are co-located. The standard deviations of the observation errors, $\sigma_{obs}$, on the retrieved RRs have been arbitrarily set equal to 50% of the rainfall rate for Bauer-Schluessel and Ferraro. For the 2A12 algorithm, the rainfall-dependent errors derived by L'Ecuyer and Stephens (2002) have been applied. For PATER, the errors have been calculated according to Bauer *et al.* (2002). Finally, a minimum threshold of $0.05 \text{ mm h}^{-1}$ has been assigned to $\sigma_{obs}$ for all algorithms so as to keep a non-zero observation error at non-rainy points.

When using TMI and SSM/I TBs directly in the 1D-Var retrieval, the observed value at each model grid point has been set equal to the value at the closest TMI pixel in each microwave channel, in order to avoid interpolation problems. For TBs, $\sigma_{obs}$ has been set to 3 K (6 K) for the vertically (horizontally) polarized channels, assuming no error correlations between channels. These $\sigma_{obs}$ values are supposed to represent the contributions from both the instrumental errors and the errors in the observation operator. The uncertainties in the RTM may arise both from the $\delta$-Eddington approximation (Moreau *et al.* 2002) as well as from the microphysical assumptions, in particular on drop size distributions and on the fall velocity, shape and density of hydrometeors. Larger values of $\sigma_{obs}$ are specified for the horizontal polarization to account for the higher natural variability of microwave TBs observed with this polarization.

Ideally, the observation errors in 1D-Var should also include the errors of the forward model, but these errors are currently neglected because they are difficult to estimate, especially for the moist physics parametrizations. The sensitivity of the 1D-Var retrievals to the specification of matrix **R** is assessed in section 5(c).

Finally, it should be mentioned that at the end of the minimization, each 1D-Var retrieval is quality controlled by rejecting it if the analysis – observation departure exceeds $\pm 3\sigma_{obs}$. In this case, the retrieval is reset to the background.

### (c) Background fields

The model's background fields that enter the 1D-Var have been obtained from three 12-hour T511 integrations of the ECMWF model that were started at 0000 UTC 5 March 2002 for *Mitag*, at 0000 UTC 26 December 2002 for *Zoe* and at 0000 UTC 9 January 2002 for the North Atlantic front. The input fields include the vertical profiles of temperature and water vapour, but also some temperature and specific-humidity tendencies, the surface heat fluxes and the surface momentum stress that are needed for running the modified convection scheme described in subsection 2(b)(i). For 1D/TB, the use of the RTM requires additional input fields, namely 10 m wind speed, surface temperature, and 2 m temperature and humidity. The covariance matrix of background errors **B** is taken from the operational ECMWF 4D-Var system (Rabier *et al.* 1997). Figure 1 shows typical values of the standard deviation of the model background errors on temperature and specific humidity which appear along the leading diagonal of matrix **B**. The assumption is made that the temperature and specific-humidity errors are uncorrelated.
<figure><img src="image_5.png" /><figcaption>Figure 3. 1D-Var on TMI-derived surface rainfall rates for the case of super-typhoon *Mitag* at 1200 UTC 5 March 2002: rain rates from (a) model background, (b) 2A12, (d) PATER, and (c) and (e) from the corresponding 1D/RR analysis. Units are in mm h⁻¹. The white hatched area indicates the region where surface convective rainfall represents more than 50% of the total surface rainfall. See text for further explanation.</figcaption></figure>

## 4. RESULTS

### (a) Super-typhoon Mitag

(i) 1D-Var on TMI surface rainfall retrievals. Figure 3 displays the results for the 1D-Var experiments on super-typhoon Mitag. Figure 3(a) shows the model background surface RRs, (b) the observed RRs as retrieved with the 2A12 algorithm from the TMI TBs, and (c) the 1D-Var retrievals of surface RRs. Figures 3(d) and (e) display the same quantities when the PATER algorithm is utilized. Although some heavy rain is correctly simulated by the model in the vicinity of the typhoon's core, panel (a) indicates that the background clearly differs from the observations in the western periphery of the storm.
Indeed, three regions with heavy precipitation that do not appear in the TMI observations are simulated by the model.

Figures 3(c) and (e) demonstrate that the 1D-Var procedure is able to correct the initial temperature and specific-humidity profiles in such a way that the 1D-Var retrieved rainfall gets rather close to the observations. This improvement on the picture of the simulated typhoon results from both the increase and the decrease of precipitation at places where the observed value is higher, respectively lower, than the model's background value. However, it should be emphasized that 1D/RR is efficient only at points where the background precipitation is non-zero. At locations where the initial simulated rainfall rate is zero, the Jacobian matrix **H** is zero, which implies that the minimization is ineffective. Such points are rejected a priori from the 1D-Var calculations. The rainfall patterns which appear to the south-east of the storm in Figs. 3(c) and (e) are created because very light precipitation (below 0.1 mm h⁻¹) is already present there in the background (Fig. 3(a)). In fact, 1D/RR performs rather well in the present case because the new simplified large-scale condensation scheme produces widespread very light rainfall amounts around the storm, which admits more observations to the retrieval procedure. The production of more widespread precipitation in the tropics, compared with the operational scheme, can be partly explained by the revised rainfall evaporation formulation used in the simplified parametrization (Tompkins and Janisková 2003). Earlier 1D-Var tests with a simpler parametrization that generated more confined precipitation led to much less satisfactory analysed surface RRs.

The convergence of the minimization is usually satisfactory, except at a few grid points for which the analysed RRs remain far from the observations due to nonlinearities in the observation operator H. One should also note that the observed TMI RRs from PATER are substantially lower than the ones obtained from 2A12. Such a difference was already demonstrated by Marécal *et al.* (2002). Generally, the agreement of the 1D-Var retrieved RRs with the corresponding observed values is better when using PATER retrievals than with 2A12 data, which can be explained by the fact that the background is closer to PATER than to 2A12.

Figure 4 displays the vertical profiles of temperature and specific-humidity increments averaged over the geographical domain of Fig. 3 when 1D/RR is applied to PATER rainfall retrievals. Increment profiles from 1D/RR on 2A12 RRs are not shown because they are very similar to those obtained from PATER data. The distinction is made between points for which the background RR is either larger or lower than the value retrieved from the TMI. These profiles clearly indicate that the reduction (increase) of the model surface rainfall by the 1D-Var retrieval is achieved through a drying and heating (moistening and cooling) of the troposphere below 300 hPa. This latter result can be readily explained by the fact that these increments are mainly obtained by activating the large-scale condensation scheme and not the convective parametrization. Such behaviour was previously found by Fillion and Mahfouf (2000). A simple conversion of the temperature increments into equivalent increments of saturation specific humidity shows that the 1D-Var corrections on temperature are approximately four times smaller than the corrections on specific humidity at all levels, which confirms the results of Marécal and Mahfouf (2000). However, this conclusion is likely to depend on the type of parametrizations used for describing moist processes.

Since the specific-humidity increments largely exceed those of temperature (when converted into saturation specific-humidity equivalents), the global impact of the 1D-Var procedure on the model state can be synthesized in Fig. 5, which shows the TCWV background field and the corresponding increments for the 2A12 and PATER experiments.
<figure><img src="image_6.png" /><figcaption>Figure 5. 1D-Var on TMI-derived surface rainfall rates for the case of super-typhoon Mitag at 1200 UTC 5 March 2002: (a) background field of TCWV, (b) TCWV increments from 1D/RR on 2A12 rain data, and (c) TCWV increments from 1D/RR on PATER rain data. Units are in kg m⁻². See text for further explanation.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6. 1D-Var on TMI microwave brightness temperatures for the case of super-typhoon *Mitag* at 1200 UTC 5 March 2002: 1D-Var retrieved surface rainfall rates in mm h⁻¹. See text for further explanation.</figcaption></figure>

corrections applied to the rain profiles but also to the profiles of cloud water. Preliminary tests with less channels showed that adding the 10 GHz and 37 GHz channels leads to a slight increase of the surface RRs inside the storm and to a further decrease in its vicinity. The minimization only fails to converge at a few points and in the north-west periphery of the super-typhoon's core. There, the departures between simulated and observed TBs reach such high values that the minimization problem becomes highly nonlinear so that no reasonable analysis increments can lead to the expected decrease of the simulated surface RRs.

Figure 7 shows the vertical profiles of temperature and specific-humidity increments produced by 1D/TB. In terms of shape, they look rather similar to the profiles obtained with 1D/RR in Fig. 4. However, as far as their magnitude is concerned, temperature increments tend to be smaller with the TB approach and the maximum increment at level 50 (around 850 hPa) that was seen in Fig. 4(a) is not present in Fig. 7(a). On the other hand, the specific-humidity increments again dominate and are about twice as high when TBs are used in the minimization instead of surface RRs.

Figure 8 displays maps of the TCWV increments obtained from 1D/TB. Their spatial distribution looks very similar to the one shown in Fig. 5 for 1D/RR, with positive TCWV corrections in the core of the storm, and negative values around it. It is reassuring that two very different 1D-Var methods can lead to consistent increment patterns. However, their magnitude is much stronger with absolute values exceeding 20 kg m⁻² locally. As already mentioned, these larger TCWV increments can be explained by the fact that when working with TBs it is not only the amount of rain that is corrected, but also the amounts of water vapour and cloud water. The spotty increments to the south of the storm are due to the fact that observed TBs are interpolated onto the model grid using the closest-point approach.

### (b) *Tropical cyclone Zoe*

In the case of *Zoe*, Fig. 9 shows that 1D/RR increments improve the surface RRs with respect to the PATER retrieval. However, and contrary to the case of *Mitag* for
<figure><img src="image_8.png" /><figcaption>Figure 7. 1D-Var on TMI microwave brightness temperatures for the case of super-typhoon *Mitag* at 1200 UTC 5 March 2002: mean vertical profiles of the increments in (a) temperature (*T*) and (b) specific humidity (*q*). Same plotting conventions as in Fig. 4. See text for further explanation.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8. 1D-Var on TMI microwave brightness temperatures for the case of super-typhoon *Mitag* at 1200 UTC 5 March 2002: TCWV increments in kg m⁻². See text for further explanation.</figcaption></figure>

which an almost perfect match to the observations was achieved, the improvement is mainly visible in the core of *Zoe*, while the rainfall amounts in the spiralling rainbands in the western and southern periphery are not reduced enough. Even in the central part of the storm, the analysed rainfall amounts are still lower than the observed values.

Comparison of the hatched areas in Fig. 3(a) and Fig. 9(a) shows that for Mitag most of the simulated precipitation is fully resolved by the model (large-scale ascent), while in the case of Zoe, it mainly originates from the convection scheme (subgrid-scale processes). The fact that the convective parametrization is still more likely to
<figure><img src="image_10.png" /></figure>

Figure 9. 1D-Var on TMI-derived surface rainfall rates for the case of tropical cyclone *Zoe* at 1200 UTC 26 December 2002: rain rates from (a) model background, (b) PATER, and (c) from the corresponding 1D-Var analysis. Labels (WB) and (SB) indicate the western and southern rainbands mentioned in the text.

<footnote>14778764, 2004, 598, Downloaded from https://mete online library.wiley.com/doi/10.12566/ij.18 by Partial BibliOMARS IISU, Wiley Online Library on 12/21/2021. See the Terms and Conditions (https://online library.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use. OA articles are governed by the applicable Creative Commons License</footnote>
<figure><img src="image_11.png" /><figcaption>Figure 10. 1D-Var on TMI microwave brightness temperatures for the case of tropical cyclone *Zoe* at 1200 UTC 26 December 2002: 1D-Var retrieved surface rainfall rates in mm h⁻¹. See text for further explanation.</figcaption></figure>

suffer from nonlinearities than the large-scale condensation scheme explains the less successful performance of 1D/RR in the case of *Zoe*.

With 1D/TB the analysed RRs (Fig. 10) exhibit strong modifications compared with the background. Although observed surface RRs are not directly involved in 1D/TB minimization, the overall improvement on this field looks better than with 1D/RR. Indeed, when compared with PATER observations (Fig. 9(b)), one can see that 1D/RR is not able to remove the western and southern rainbands present in the background (respectively labelled WB and SB in Fig. 9(a)), and that it leads to a less satisfactory description of the north-eastern part of the storm. The intensity of the RRs reaches values larger than 10 mm h⁻¹ that correspond to positive TCWV increments of up to 10 kg m⁻² locally (not shown). TCWV increments are again much stronger and more widespread with 1D/TB than with 1D/RR (not shown). In contrast with the case of 1D/RR, the performance of 1D/TB is not much affected by the fact that convection dominates in the background. Nonlinearities in the convection scheme are indeed less detrimental in 1D/TB than in 1D/RR, because TBs are influenced by the full vertical profiles of cloud water and precipitation and are also directly related to specific humidity, beside any explicit link via the moist physics.

### (c) A mid-latitude front

Figure 11 shows the 1D/RR results for the midlatitude front. The model's background RRs are systematically higher than the three SSM/I observations. However, the geographical location and structure of the simulated frontal rainband matches the observations rather well. These characteristics are very similar to the general behaviour of the ECMWF forecasting system (Chevallier and Bauer 2003) that uses more elaborate moist physical parametrizations. Again, the 1D-Var method is able to correct the temperature and specific-humidity increments so that the 1D-Var retrieved surface RRs match almost perfectly the available SSM/I observations. Figures 11(b), (d) and (f) show, respectively, that significant differences are found between the rain fields retrieved with PATER-SSMI, Bauer-Schluessel and Ferraro, which is one of the major limitations of 1D/RR.
<figure><img src="image_12.png" /><figcaption>Figure 11. 1D-Var on SSM/I-derived surface rainfall rates for the case of the North Atlantic front at 1200 UTC 9 January 2002: rain rates from (a) model background, (b) PATER, (d) Bauer–Schluessel, (f) Ferraro, and (e), (g) from the corresponding 1D/RR analysis. Units are in mm h⁻¹. See text for further explanation.</figcaption></figure>

<footnote>14778764, 2004, 598. Downloaded from https://meteolibrary.wiley.com/doi/10.12566/ij.118 by Partial BibliOMARS IISU, Wiley Online Library on 12/10/2021. See the Terms and Conditions (https://meteolibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use. OA articles are governed by the applicable Creative Commons License</footnote>
<figure><img src="image_13.png" /><figcaption>Figure 12. 1D-Var on SSM/I-derived surface rainfall rates for the case of the North Atlantic front at 1200 UTC 9 January 2002: (a) background field of TCWV, and TCWV increments when using surface rainfall rates from (b) PATER, (c) Bauer-Schluessel and (d) Ferraro. Units are in kg m⁻². See text for further explanation.</figcaption></figure>

The vertical profiles of temperature and specific-humidity increments (not shown) exhibit a shape comparable with the one that was found for super-typhoon *Mitag*, but their amplitude is substantially lower due to the smaller surface rainfall departures between background and observations. Since the background field tends to overestimate the surface precipitation almost systematically over the domain, Fig. 12 shows that negative 1D-Var TCWV increments dominate, with extreme values reaching −5 kg m⁻² locally. The values and horizontal structure of the increments look very comparable with all three retrieval algorithms.

Figure 13 displays the analysed surface RRs for the midlatitude front with 1D/TB using SSM/I observations. The main change with respect to the background field shown in Fig. 11(a) is the reduction of the intensity in the southern part of the frontal rainband (below 40°N) and the slight reshaping of the front. These modifications in the analysis surface RRs are also present in Fig. 14 that shows positive 1D-Var TCWV increments in the western part of the rainband that locally exceed 4 kg m⁻², with negative increments over the rest of the domain. The analysed RRs agree fairly well with all three RR retrievals and in particular with the Ferraro algorithm (Fig. 11(e)). It should be mentioned that the unchanged light-rainfall values in the wake of the front correspond to grid points that were not treated in 1D/TB to save computational time (minimum threshold of 0.3 mm h⁻¹ in the background).
<figure><img src="image_14.png" /><figcaption>Figure 13. 1D-Var on SSM/I microwave brightness temperatures for the case of the North Atlantic front at 1200 UTC 9 January 2002: 1D-Var retrieved surface rainfall rates in mm h⁻¹. See text for further explanation.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 14. As Fig. 13 but for TCWV increments in kg m⁻². See text for further explanation.</figcaption></figure>

## 5. EVALUATION

It was shown that 1D/RR and 1D/TB lead to consistent results both in terms of analysed RRs and TCWV increments. This section deals with the evaluation of the two methods based on SSM/I and TRMM data that are the only reliable observations available in precipitation areas over ocean. First, the two methods are statistically compared in TB and RR space. Then, the consistency of the vertical profiles of analysed RRs is assessed using measurements from the TRMM/PR for tropical cyclones Mitag
and *Zoe*. Finally, an assessment of the impact of the observation-error statistics on the 1D/TB retrievals is presented.

### (a) Evaluation in TB and RR space

For each of the three meteorological cases, TBs have been computed from the background fields and from the analyses given by the two methods, using the RTM described in subsection 2(b)(iii). The biases and the standard deviations of background – observation and analysis – observation departures are shown in Table 2 for *Mitag*, *Zoe* and the midlatitude front and for each available microwave frequency. Statistics refer to the common points where both methods successfully converged. In all histograms the standard deviation of the analysis departures both for 1D/TB and 1D/RR is reduced with respect to the background departures. The reduction is always three to four times larger with 1D/TB than with 1D/RR for all channels. This is due to the fact that TBs are directly involved in the cost function of 1D/TB.

The biases of 1D/RR and 1D/TB analyses are generally reduced when compared with the background. The amount depends on the meteorological case and microwave channel. In the *Mitag* case, the bias of the background departure is small compared with the other cases, with values lower than 5 K except at 37 GHz. The two analyses and the background exhibit similar biases, except at 37 GHz where a noticeable decrease is found. In fact, the larger negative biases of background departures for the 37 GHz channels are an artefact of the saturation of the microwave signal in the rainy core of the storm that implies the absence of offsetting large positive biases, as is the case in the other channels.

In the midlatitude case, the bias of the background departures shows large negative values, which corresponds to the presence of an excessive amount of precipitation and cloud liquid water in the background with respect to the observations. *Zoe* exhibits opposite features with large positive background biases of up to 15 K at several frequencies. For both cases, 1D/TB analysis biases are reduced and do not exceed a few K. A significant reduction of the bias is found with 1D/RR in the frontal case, but only a minor one in the case of *Zoe*. In the latter, the bias and the standard deviation of the surface rainfall analysis departures remain large, which indicates a poor convergence during the minimization. For *Mitag* and the midlatitude case, largely dominated by the large-scale condensation scheme, an almost perfect agreement is found between observed and analysed surface RRs.

The smaller errors of the TB analysis departures with 1D/TB compared with 1D/RR do not necessarily correspond to smaller differences in terms of surface RRs, especially for *Mitag*. This is due to the ambiguous link that exists between surface RRs and TBs that integrate information on the whole atmospheric column. The uncertainty on the surface RRs obtained with the 2A12, PATER, Bauer-Schluessel or Ferraro algorithms can also provide an explanation.

### (b) Comparison with TRMM/PR rainfall rates

Vertical profiles of RRs (2A25 product; Iguchi *et al.* 2000) measured by the PR on board the TRMM platform have been used to validate the analysed RR profiles from 1D/RR and 1D/TB. The initial 2A25 product with a horizontal (vertical) resolution of 1.4 km (250 m) has been averaged to the model grid resolution (about 40 km (60 levels)). Figure 15 shows the mean vertical profiles of RRs for *Mitag* and *Zoe*. In both cases, the shape of the rain profile as observed from the PR is characterized by nearly constant values on the vertical below 600 hPa. In contrast, the mean analysed rain flux for
<table><caption>TABLE 2. 1D-VAR STATISTICS FROM BACKGROUND (GUESS) AND ANALYSIS FOR THE 1D/TB AND THE 1D/RR EXPERIMENTS (SEE TEXT) FOR THE THREE STUDIED CASES. BIASES AND STANDARD DEVIATIONS (BOLD FONT) OF OBSERVATION – BACKGROUND, AND OBSERVATION – ANALYSIS DEPARTURES IN TERMS OF BRIGHTNESS TEMPERATURES AND IN TERMS OF SURFACE RAINFALL RATES.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Typhoon Mitag</th><th colspan="3">Cyclone Zoe</th><th colspan="3">Front</th></tr><tr><th>Guess</th><th>1D/TB</th><th>1D/RR</th><th>Guess</th><th>1D/TB</th><th>1D/RR</th><th>Guess</th><th>1D/TB</th><th>1D/RR</th></tr></thead><tbody><tr><td>Channels* (GHz)</td><td></td><td></td><td></td><td>Observation – model TB bias and standard deviation (K)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10 V</td><td>-1.6</td><td>11.4</td><td>-2.0</td><td>-0.8</td><td>7.0</td><td>6.0</td><td>-1.0</td><td>3.4</td><td>3.0</td><td>11.2</td></tr><tr><td>19 V</td><td>0.9</td><td>14.4</td><td>2.0</td><td>2.2</td><td>8.6</td><td>10.6</td><td>13.6</td><td>1.3</td><td>3.0</td><td>7.8</td><td>12.6</td></tr><tr><td>22 V</td><td>-0.1</td><td>8.2</td><td>1.1</td><td>1.7</td><td>4.8</td><td>5.7</td><td>6.0</td><td>1.1</td><td>2.7</td><td>5.0</td><td>6.1</td></tr><tr><td>37 V</td><td>-5.3</td><td>15.9</td><td>-1.4</td><td>4.3</td><td>9.6</td><td>10.2</td><td>12.8</td><td>3.4</td><td>9.4</td><td>9.9</td><td>13.2</td></tr><tr><td>10 H</td><td>-4.7</td><td>20.4</td><td>-5.4</td><td>5.0</td><td>-3.2</td><td>12.4</td><td>10.1</td><td>25.9</td><td>-2.7</td><td>6.5</td><td>4.5</td><td>20.7</td></tr><tr><td>19 H</td><td>-2.1</td><td>26.7</td><td>-0.2</td><td>4.4</td><td>3.5</td><td>16.1</td><td>15.5</td><td>26.1</td><td>-1.6</td><td>5.2</td><td>10.0</td><td>23.9</td><td>-10.2</td></tr><tr><td>37 H</td><td>-10.2</td><td>32.1</td><td>-2.7</td><td>5.8</td><td>1.7</td><td>19.4</td><td>16.7</td><td>25.7</td><td>1.0</td><td>11.9</td><td>14.3</td><td>25.6</td><td>-17.5</td><td>30.1</td></tr><tr><td>Surface rainfall</td><td>-0.04</td><td>1.58</td><td>-0.45</td><td>1.73</td><td>-0.02</td><td>0.11</td><td>2.6</td><td>4.6</td><td>1.2</td><td>3.0</td><td>2.0</td><td>3.5</td><td>-0.86</td><td>1.09</td><td>-0.42</td><td>0.91</td><td>-0.02</td><td>0.22</td></tr></tbody></table>

*V: vertical, H: horizontal.
<figure><img src="image_16.png" /><figcaption>Figure 15. Horizontally averaged vertical profiles of rainfall rates from TRMM/PR observations, from the background (Guess), from 1D/RR and from 1D/TB. Results are shown for (a) *Mitag* and (b) *Zoe*. See text for further explanation.</figcaption></figure>

*Mitag* increases from 1.5 mm h⁻¹ at 600 hPa to 2.5 mm h⁻¹ (respectively 4.0) at 900 hPa with 1D/RR (respectively 1D/TB). This particular shape is due to the fact that the large-scale condensation scheme is dominant in this case (see Fig. 3(a)) and that it produces precipitation at all vertical levels. On the other hand, for cyclone *Zoe* (see Fig. 9(a)), the convective scheme is more active and precipitation generation mainly occurs above 600 hPa, with a nearly constant rainfall profile below this level, as in the PR observations. The shape of the analysed rain profile seems to depend on which moist physical parametrization is the most active in the background. It should be noted that the partitioning between convective and large-scale precipitation remains comparable in the 1D-Var retrieval and in the background.

The biases and the standard deviations of the observation – analysis departures shown in Table 2 indicate that 1D/RR performs better for *Mitag* than for *Zoe*. This better behaviour in the *Mitag* case is also seen in Fig. 15(a). For the 1D/RR retrieval, which optimizes the surface RR, the analysed precipitation flux closely matches the radar observations below 800 hPa. In the 1D/TB method, the analysed precipitation profile is computed so as to optimize the TBs, which are mainly related to the integrated rain content and not to the surface rainfall. Hence a strong tilt of the analysed rain profile can be seen in Fig. 15(a). This is required to ensure a good agreement between the simulated and observed integrated rain contents. In the case of *Zoe*, the analysed
<table><caption>TABLE 3. IMPACT OF THE INCLUSION OF TMI CHANNEL ERROR CORRELATIONS ON THE 1D-VAR RETRIEVALS (SEE TEXT) IN THE CASE OF SUPER-TYPOON Mitag: BIASES AND STANDARD DEVIATIONS (σ, BOLD FONT) OF OBSERVATION - BACKGROUND, AND OBSERVATION - ANALYSIS DEPARTURES FOR ALL CHANNELS USED, AND MEANS AND STANDARD DEVIATIONS OF INTEGRATED CLOUD AND RAIN WATER CONTENTS</caption><thead><tr><th rowspan="2">Channel* (GHz)</th><th colspan="6">Correlation</th></tr><tr><th>Background</th><th>None</th><th>0.5</th><th>0.8</th><th colspan="2">2 × σ<sub>obs</sub></th></tr></thead><tbody><tr><th colspan="7">Observation - model TB bias and standard deviation (K)</th></tr><tr><td>10 V</td><td>-1.8</td><td><strong>11.6</strong></td><td>-2.0</td><td><strong>3.1</strong></td><td>-2.4</td><td><strong>3.4</strong></td><td>-2.8</td><td><strong>3.9</strong></td><td>-1.8</td><td><strong>3.7</strong></td></tr><tr><td>19 V</td><td>0.6</td><td><strong>14.6</strong></td><td>1.9</td><td><strong>2.4</strong></td><td>0.8</td><td><strong>3.2</strong></td><td>0.1</td><td><strong>4.1</strong></td><td>1.9</td><td><strong>2.5</strong></td></tr><tr><td>21 V</td><td>-0.2</td><td><strong>8.3</strong></td><td>1.1</td><td><strong>1.6</strong></td><td>0.1</td><td><strong>2.3</strong></td><td>-0.6</td><td><strong>3.0</strong></td><td>1.0</td><td><strong>1.8</strong></td></tr><tr><td>37 V</td><td>-5.4</td><td><strong>16.2</strong></td><td>-1.5</td><td><strong>3.8</strong></td><td>-2.8</td><td><strong>4.0</strong></td><td>-3.4</td><td><strong>4.6</strong></td><td>-1.9</td><td><strong>3.9</strong></td></tr><tr><td>10 H</td><td>-5.1</td><td><strong>20.7</strong></td><td>-5.4</td><td><strong>5.5</strong></td><td>-6.1</td><td><strong>6.3</strong></td><td>-6.7</td><td><strong>7.1</strong></td><td>-4.9</td><td><strong>6.6</strong></td></tr><tr><td>19 H</td><td>-2.9</td><td><strong>27.0</strong></td><td>-0.4</td><td><strong>4.7</strong></td><td>-2.4</td><td><strong>6.2</strong></td><td>-3.8</td><td><strong>7.5</strong></td><td>-0.4</td><td><strong>5.1</strong></td></tr><tr><td>37 H</td><td>-10.7</td><td><strong>32.6</strong></td><td>-3.0</td><td><strong>5.4</strong></td><td>-5.7</td><td><strong>6.1</strong></td><td>-7.1</td><td><strong>7.5</strong></td><td>-3.9</td><td><strong>6.1</strong></td></tr><tr><th colspan="7">Mean and standard deviation (kg m<sup>-2</sup>)</th></tr><tr><td>Integrated contents</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Rain water</td><td>0.22</td><td><strong>0.32</strong></td><td>0.23</td><td><strong>0.44</strong></td><td>0.24</td><td><strong>0.45</strong></td><td>0.25</td><td><strong>0.45</strong></td><td>0.22</td><td><strong>0.39</strong></td></tr><tr><td>Cloud water</td><td>0.64</td><td><strong>0.63</strong></td><td>0.62</td><td><strong>0.84</strong></td><td>0.64</td><td><strong>0.84</strong></td><td>0.64</td><td><strong>0.81</strong></td><td>0.62</td><td><strong>0.82</strong></td></tr></tbody></table>

<table><caption>TABLE 4. SAME AS IN TABLE 3, BUT FOR TROPICAL CYCLONE Zoe</caption><thead><tr><th rowspan="2">Channel (GHz)</th><th colspan="6">Correlation</th></tr><tr><th>Background</th><th>None</th><th>0.5</th><th>0.8</th><th colspan="2">2 × σ<sub>obs</sub></th></tr></thead><tbody><tr><th colspan="7">Observation - model TB bias and standard deviation (K)</th></tr><tr><td>10 V</td><td>3.4</td><td><strong>14.6</strong></td><td>-1.0</td><td><strong>3.9</strong></td><td>-1.0</td><td><strong>4.7</strong></td><td>-1.1</td><td><strong>5.0</strong></td><td>-0.5</td><td><strong>4.4</strong></td></tr><tr><td>19 V</td><td>8.1</td><td><strong>14.3</strong></td><td>1.8</td><td><strong>3.1</strong></td><td>1.5</td><td><strong>3.8</strong></td><td>1.4</td><td><strong>4.5</strong></td><td>2.0</td><td><strong>3.0</strong></td></tr><tr><td>21 V</td><td>4.9</td><td><strong>6.4</strong></td><td>1.6</td><td><strong>2.7</strong></td><td>1.3</td><td><strong>3.0</strong></td><td>1.1</td><td><strong>3.3</strong></td><td>1.7</td><td><strong>2.4</strong></td></tr><tr><td>37 V</td><td>8.0</td><td><strong>13.4</strong></td><td>2.5</td><td><strong>8.4</strong></td><td>2.3</td><td><strong>9.0</strong></td><td>1.8</td><td><strong>8.8</strong></td><td>2.6</td><td><strong>8.3</strong></td></tr><tr><td>10 H</td><td>5.2</td><td><strong>27.2</strong></td><td>-2.5</td><td><strong>7.4</strong></td><td>-2.6</td><td><strong>8.8</strong></td><td>-2.7</td><td><strong>9.1</strong></td><td>-1.7</td><td><strong>8.3</strong></td></tr><tr><td>19 H</td><td>10.4</td><td><strong>27.6</strong></td><td>-0.4</td><td><strong>5.5</strong></td><td>-1.0</td><td><strong>6.4</strong></td><td>-1.3</td><td><strong>7.6</strong></td><td>-0.2</td><td><strong>5.7</strong></td></tr><tr><td>37 H</td><td>11.0</td><td><strong>27.7</strong></td><td>0.1</td><td><strong>9.2</strong></td><td>-0.6</td><td><strong>10.3</strong></td><td>-1.3</td><td><strong>10.9</strong></td><td>0.2</td><td><strong>9.2</strong></td></tr><tr><th colspan="7">Mean and standard deviation (kg m<sup>-2</sup>)</th></tr><tr><td>Integrated contents</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Rain water</td><td>0.44</td><td><strong>0.62</strong></td><td>0.56</td><td><strong>0.64</strong></td><td>0.58</td><td><strong>0.64</strong></td><td>0.57</td><td><strong>0.64</strong></td><td>0.54</td><td><strong>0.60</strong></td></tr><tr><td>Cloud water</td><td>0.62</td><td><strong>0.51</strong></td><td>0.92</td><td><strong>0.92</strong></td><td>0.94</td><td><strong>0.91</strong></td><td>0.92</td><td><strong>0.89</strong></td><td>0.90</td><td><strong>0.90</strong></td></tr></tbody></table>

rain profiles from 1D/RR and 1D/TB are closer to the radar observations than the background. However, the 1D/TB retrieval better matches the radar observations than 1D/RR, which is consistent with the results of the comparison with PATER surface RR shown in Figs. 9 and 10 and with observed TBs (Table 2).

(c) *Sensitivity to observation-error statistics*

In order to assess the impact of the uncertainties on the definition of the **R** matrix, some tests have been run with 1D/TB including error correlations between the TMI channels or larger standard deviations of observation errors.

Since inter-channel error correlations cannot be accurately known, two values of 0.5 and 0.8 that are representative of a low and a high level of correlation, respectively, have been tested. The corresponding biases and standard deviations of the departures between background or 1D-Var retrieval and observations as well as statistics of integrated cloud and rain water are summarized in Table 3 and Table 4 for Mitag and Zoe, respectively. Statistics have been computed over the points where the convergence was successful. As expected, the inclusion of inter-channel error correlations brings the 1D-Var retrievals closer to the background, but this impact appears marginal.

<footnote>*V: vertical, H: horizontal.</footnote>
Table 3 and Table 4 also show statistics obtained when the standard deviations of the observation errors are doubled. Again the impact on the quality of the 1D-Var results remains weak.

Therefore, it appears that the 1D-Var retrievals are not very sensitive to the specification of matrix **R**. This can be explained by the fact that because of the rather large errors specified in the current **B** matrix, the contribution of the background term, *J*<sub>b</sub>, in the total cost function (Eq. (1)) is usually very small. This was confirmed by some experiments in which *J*<sub>b</sub> was set to zero. Although *J*<sub>b</sub> does not currently play a significant role in the 1D-Var retrievals, future improvements of the forecasting system will increase its contribution in **J** to some extent. Similar results were obtained with 1D/RR.

## 6. CONCLUSIONS

In this study, 1D-Var retrieval experiments based on retrieved surface RRs as observed from TMI or SSM/I have been compared with 1D-Var retrieval experiments directly performed on the microwave TBs, for two tropical cyclones and an extratropical front. 1D/RR and 1D/TB include new simplified parametrizations of moist processes that perform similarly to the parametrizations used in the operational forecast model for the simulation of cloud and precipitating systems, as can be seen from comparing the results presented here with the ones documented by Chevallier and Bauer (2003).

Both 1D-Var methods produce consistent temperature and specific-humidity increments that correct either the model's surface RRs or the simulated TBs towards equivalent TMI or SSM/I observations. However, 1D/RR performs better when large-scale precipitation dominates in the background, due to the stronger nonlinearities that sometimes exist in convective conditions. 1D/TB seems less affected by the nature of precipitation in the background. In other respects, the 1D-Var retrievals are not very sensitive to the definition of the observation-error statistics, both in terms of standard deviations and error correlations. This can be explained by the current weak contribution of the background term in the cost function.

To summarize the respective advantages and drawbacks of the two 1D-Var methods, 1D/RR (including the surface RR retrieval) has the main advantage of being computationally cheaper than 1D/TB (that requires substantial additional time for the radiative-transfer calculations). The two major drawbacks of 1D/RR lie in its inefficiency whenever the background RR is zero and in the dependence on the performance of the selected rainfall algorithm. On the other hand, the fact that microwave TBs are sensitive not only to precipitation but also to water vapour and cloud water makes it possible to correct the model's control variables outside and inside rainy areas of the background in a consistent manner. 1D/TB also offers the possibility of selecting which channels should be assimilated according to the meteorological conditions (e.g. clear or cloudy sky, light or heavy precipitation). This flexibility also applies if one considers the changing availability of individual channels during the lifetime of a given instrument. One limitation of both 1D/RR and 1D/TB is their current inapplicability over land due to the difficulty of dealing with heterogeneities in surface emissivity. Furthermore, due to the lack of high-accuracy precipitation measurements over the ocean (except maybe for the TRMM/PR), 1D-Var retrieval errors cannot be properly estimated.

An important issue that will deserve some attention in the future relates to the biases of the forward model (moist physics and/or radiative transfer), which may depend on the meteorological situation and which are not dealt with at the moment in the 1D-Var retrievals.
Since neither of the two 1D-Var methods clearly outperforms the other, their respective potential for the future operational assimilation of precipitation at the ECMWF is currently under investigation in '1D-Var+4D-Var' experiments (Marécal and Mahfouf 2002) in which pseudo-observations of TCWV produced by 1D/RR or 1D/TB on TMI and SSM/I data are assimilated in the ECMWF forecasting system. Taking advantage of the dynamical coherence of 4D-Var, these experiments will also allow the consistency of the 1D-Var retrievals with all other observations to be checked.

## ACKNOWLEDGEMENTS

We would like to thank Virginie Marécal and Jean-François Mahfouf for their pioneering work on precipitation assimilation at the ECMWF, and for their kind support. The two anonymous referees should also be acknowledged for their helpful comments about this paper. We are also grateful to the INRIA (Institut National de Recherche en Informatique et en Automatique) for providing the M1QN3 minimization code. This study was supported by the European Space Agency projects EuroTRMM-2 (#12651/97/NL/NB) and EGPM (#3-10600/02/NL/GS).

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