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  • View in gallery

    Flowchart describing the simulation of the SSM/I brightness temperatures from the 1DVAR control variables, which are temperature and humidity.

  • View in gallery

    Mean (solid line) and standard deviation (SD; dotted line) of (a) T and (b) lnQ background errors for GEM mesoglobal over the geographical area that extends from 40°S to 40°N for profiles for the TC Zoe case (0000 UTC 27 Dec 2002).

  • View in gallery

    Brightness temperature at 19 GHz V for TC Zoe 0000 UTC 27 Dec 2002. (a) F15 SSM/I observations and (b) GEM mesoglobal 12-h forecast.

  • View in gallery

    1DVAR first-guess fields of (a) IWV in kg m−2, (b) cloud LWP in kg m−2, (c) SRR in mm h−1, and (d) ratio in percent of convective to total SRR for TC Zoe 0000 UTC 27 Dec 2002.

  • View in gallery

    Analyzed SRR (mm h−1) for TC Zoe 0000 UTC 27 Dec 2002: (a) 1DVAR Tb SOE, (b) 1DVAR Tb LOE, (c) 1DVAR SRR, and (d) PATER retrievals from F15 SSM/I.

  • View in gallery

    Mean and SD of increments (analysis minus first guess) for 1DVAR experiments SOE, LOE, LOE2 (r = 0.0, 0.4, and 0.8), and SRR. The interchannel error correlation coefficient (r) is zero for experiments 1DVAR SOE, LOE, and SRR. The 1DVAR experiment LOE2 is similar to LOE but with a different a posteriori quality control (see text).

  • View in gallery

    The OP (solid line) and OA (dashed line) histograms for TC Zoe 0000 UTC 27 Dec 2002 for the 1DVAR Tb SOE. Observations (O) are in (a) PATER SRR retrievals (mm h−1), (b) Alishouse–Petty IWV retrievals in nonrainy areas (kg m−2), and (c) Weng and Grody (1994) LWP retrievals (kg m−2). All retrievals are for F15 SSM/I.

  • View in gallery

    (a) Mean and (b) SD of T (solid lines) and lnQ (dotted lines) increments (analysis minus first guess) normalized by SD of background errors (Fig. 2) for TC Zoe 0000 UTC 27 Dec 2002 and experiment 1DVAR Tb SOE.

  • View in gallery

    Background error in brightness temperature space for the SSM/I channels as a function of first guess SRR (mm h−1) for TC Zoe 0000 UTC 27 Dec 2002: (a) 19V GHz, (b) 19H GHz, (c) 22V GHz, (d) 37V GHz, and (e) 37H GHz.

  • View in gallery

    Background error in SRR space as a function of first-guess SRR (mm h−1) (small Zoe area). (a) 1DVAR SRR with all moist physical schemes (dotted line indicates 500% error and solid line 100%), (b) same as in (a) but with a linear scale instead of a logarithmic scale for the background error, and (c) same as in (b) but without deep convection.

  • View in gallery

    Estimate of IWV analysis error from inverse of Hessian of cost function for TC Zoe: (a) 1DVAR SOE, (b) 1DVAR LOE, and (c) 1DVAR SRR. The 10% error line is shown.

  • View in gallery

    Same as in Fig. 11, but for Typhoon Chaba.

  • View in gallery

    The (left) OP and (right) OA bias (dotted lines) and SD (solid lines) statistics for TC Zoe 0000 UTC 27 Dec 2002 (small area). Stars denote 1DVAR SRR and diamonds denotes 1DVAR Tb. Observations (O) for (a), (b) SRR, (c), (d) IWV in nonrainy areas, and (e)–(h) LWP are, respectively, PATER SRR retrievals (mm h−1), Alishouse–Petty IWV retrievals in nonrainy areas (kg m−2), and Weng and Grody (1994) LWP retrievals (kg m−2). All retrievals are for F15 SSM/I. For the moist physical scheme on the horizontal axis, 0 includes all three schemes, 1 does not include shallow-convection scheme, and 2 only has nonconvective scheme. The nonconvective scheme is (a)–(f) “Sundqvist” and (g), (h) CLOUDST.

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One-Dimensional Variational Data Assimilation of SSM/I Observations in Rainy Atmospheres at MSC

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  • 1 Data Assimilation and Satellite Meteorology Division, Meteorological Service of Canada, Dorval, Québec, Canada
  • | 2 Recherche en Prévision Numérique, Meteorological Service of Canada, Dorval, Québec, Canada
  • | 3 Data Assimilation and Satellite Meteorology Division, Meteorological Service of Canada, Dorval, Québec, Canada
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Abstract

Currently, satellite radiances in the Canadian Meteorological Centre operational data assimilation system are only assimilated in clear skies. A two-step method, developed at the European Centre for Medium-Range Weather Forecasts, is considered to assimilate Special Sensor Microwave Imager (SSM/I) observations in rainy atmospheres. The first step consists of a one-dimensional variational data assimilation (1DVAR) method. Model temperature and humidity profiles are adjusted by assimilating either SSM/I brightness temperatures or retrieved surface rain rates (derived from SSM/I brightness temperatures). In the second step, 1DVAR column-integrated water vapor analyses are assimilated in four-dimensional variational data assimilation (4DVAR). At the Meteorological Service of Canada, such a 1DVAR assimilation system has been developed. Model profiles are obtained from a research version of the Global Environmental Multi-Scale model. Several issues raised while developing the 1DVAR system are addressed. The impact of the size of the observation error is studied when brightness temperatures are assimilated. For two case studies, analyses are derived when either surface rain rate or brightness temperatures are assimilated. Differences in the analyzed fields between these configurations are discussed and shortcomings of each approach are identified. Results of sensitivity studies are also provided. First the impact of observation error correlation between channels is investigated. Second, the size of the background temperature error is varied to assess its impact on the analyzed column-integrated water vapor. Third, the importance of each moist physical scheme is investigated. Finally, the portability of moist physical schemes specifically developed for data assimilation is discussed.

Corresponding author address: Dr. Godelieve Deblonde, Meteorological Service of Canada, 2121 Trans-Canada Highway, Dorval PQ H9P 1J3, Canada. Email: godelieve.deblonde@ec.gc.ca

Abstract

Currently, satellite radiances in the Canadian Meteorological Centre operational data assimilation system are only assimilated in clear skies. A two-step method, developed at the European Centre for Medium-Range Weather Forecasts, is considered to assimilate Special Sensor Microwave Imager (SSM/I) observations in rainy atmospheres. The first step consists of a one-dimensional variational data assimilation (1DVAR) method. Model temperature and humidity profiles are adjusted by assimilating either SSM/I brightness temperatures or retrieved surface rain rates (derived from SSM/I brightness temperatures). In the second step, 1DVAR column-integrated water vapor analyses are assimilated in four-dimensional variational data assimilation (4DVAR). At the Meteorological Service of Canada, such a 1DVAR assimilation system has been developed. Model profiles are obtained from a research version of the Global Environmental Multi-Scale model. Several issues raised while developing the 1DVAR system are addressed. The impact of the size of the observation error is studied when brightness temperatures are assimilated. For two case studies, analyses are derived when either surface rain rate or brightness temperatures are assimilated. Differences in the analyzed fields between these configurations are discussed and shortcomings of each approach are identified. Results of sensitivity studies are also provided. First the impact of observation error correlation between channels is investigated. Second, the size of the background temperature error is varied to assess its impact on the analyzed column-integrated water vapor. Third, the importance of each moist physical scheme is investigated. Finally, the portability of moist physical schemes specifically developed for data assimilation is discussed.

Corresponding author address: Dr. Godelieve Deblonde, Meteorological Service of Canada, 2121 Trans-Canada Highway, Dorval PQ H9P 1J3, Canada. Email: godelieve.deblonde@ec.gc.ca

1. Introduction

In recent years, the assimilation of cloudy satellite radiances, which contain cloud and rain information, and retrieved surface rain rates (SRRs) has become an important area of research. Precipitating systems contribute substantially to the release of latent heat, which is an important part of the energy budget in the atmosphere and particularly so in the Tropics. Furthermore, the representation of the hydrological cycle in numerical weather prediction (NWP) models still needs more improvement and remains a major challenge.

The three- and four-dimensional variational data assimilation systems (3DVAR and 4DVAR, respectively) that run operationally at several NWP centers [e.g., the National Centers for Environmental Prediction (NCEP), the Met Office, Météo-France, European Centre for Medium-Range Weather Forecasts (ECMWF), the Japan Meteorological Agency (JMA), and the Meteorological Service of Canada (MSC)] are designed to maximize the extraction of the information content from the huge sources of satellite observations. These variational methods rely on so-called observation operators that calculate a model equivalent for each observation. As a result, the information on cloud and precipitation contained in the observations is converted to information on the state variables of NWP models.

Accurate measurements of satellite-derived SRR can be used in NWP models for improving the quality of 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. The first approaches that used precipitation observations to initialize NWP models were based on diabatic normal mode initialization and on “physical initialization” (nudging of rain rates or rain-related quantities). These initialization techniques showed an improvement of the moisture analysis and a reduction of the spinup problem (Krishnamurti et al. 1984; Puri and Miller 1990; Heckley et al. 1990). More recent approaches such as 3DVAR and 4DVAR assimilation can explicitly take into account model and observation errors to provide an optimal state. Encouraging results have already been obtained (Treadon 1997; Marécal and Mahfouf 2002; Mahfouf et al. 2005). Two important operational meteorological centres, NCEP and JMA, assimilate directly satellite- and radar-derived SRR, respectively, in their global 3DVAR (Treadon et al. 2003) and regional 4DVAR (Tsuyuki et al. 2003) systems. As well, very recently (since June 2005), ECMWF has operationally implemented the assimilation of brightness temperature in rainy atmospheres for the three lowest-frequency channels of the Special Sensor Microwave Imager (SSM/I) instrument (Bauer et al. 2006a, b). However, there remain a number of important issues to be solved regarding the usage of precipitation data in the variational context (Fillion 2002; Fillion and Bélair 2004; Errico et al. 2000; Marécal and Mahfouf 2003).

Space-based remote sensing of precipitation over the oceans has reached a high level of success with the advent of passive microwave (PMW) instruments such as the series of Defense Meteorological Satellite Project (DMSP) SSM/I instruments (since 1987) and in particular with the joint National Aeronautics and Space Administration–Japan Aerospace Exploration Agency (NASA–JAXA) Tropical Rainfall Measuring Mission (TRMM) instrument (launched in 1997; Kummerow et al. 2000). Other PMW instruments that are particularly suited for SRR retrievals are the Advanced Microwave Scanning Radiometer-EOS (AMSR-E) on board Aqua and the SSM/IS instrument on board the DMSP F16 (launched in 2003), which is the follow-on instrument to the SSM/I.

A two-step assimilation method was developed at ECMWF by Marécal and Mahfouf (2000, 2002) to assimilate satellite-derived SRR from the SSM/I and TRMM Microwave Imager (TMI) instruments. In step one, model temperature and humidity profiles are adjusted by assimilating retrieved SRR in a 1DVAR system that includes comprehensive moist physical processes. The retrieved humidity profiles are then vertically integrated to produce retrievals of integrated water vapor (IWV). In step two, 1DVAR IWV retrievals are assimilated in 4DVAR. Marécal and Mahfouf (2003) showed that the two-step method is more robust than a direct 4DVAR assimilation of SRR due to inconsistencies between the nonlinear and tangent linear/adjoint models of the ECMWF incremental 4DVAR assimilation system (Courtier et al. 1994) and to strong nonlinearities of the moist physical processes.

Moreau et al. (2004) extended the 1DVAR system to the assimilation of PMW brightness temperatures (Tb) instead of satellite-derived SRR. PMW Tb values have the advantage that they not only depend on precipitation but also on water vapor, cloud water, surface wind speed, and temperature (Moreau et al. 2003). In the case where SRR is assimilated, the observation operator is limited to moist physical schemes whereas when Tb is assimilated, a radiative transfer model (RTM) is also needed. At ECMWF, such a two-step assimilation method has been implemented operationally since June 2005 (Bauer et al. 2006a, b).

A one-dimensional variational data assimilation (1DVAR) system, developed to assimilate either SRR or Tb with the next version of the MSC Global Environmental Multi-Scale (GEM) weather forecast model (Côté et al. 1998), is presented in this paper. The control variables are profiles of temperature and natural logarithm of specific humidity. The 1DVAR system produces analyzed IWV, which will eventually be assimilated in the 4DVAR incremental assimilation system, operational at the Canadian Meteorological Centre (CMC) since 15 March 2005.

In this paper, several outstanding issues raised while developing the 1DVAR system are addressed for two case studies. Section 2 describes the different components of the 1DVAR system. Section 3 compares the analyzed fields obtained with three different configurations of the 1DVAR. First, results are presented when Tb is assimilated with small observation errors. Second, the same experiment is repeated but for larger observation errors that are comparable to the background error mapped to observation space. Finally, the 1DVAR is used to assimilate SRR. In section 4, results of sensitivity studies are provided. First the impact of observation error correlation between channels is investigated. Second, the size of the temperature background error is varied to evaluate its influence on the analyzed IWV increments. Finally, the impact of moist physical schemes is investigated in the 1DVAR system. Discussion and conclusions are provided in section 5.

2. 1DVAR system

a. Method

The 1DVAR assimilation system can assimilate either Tb (1DVAR Tb) or SRR (1DVAR SRR) and its control variables are profiles of temperature (T) and natural logarithm of specific humidity (lnQ). The latter is also the humidity control variable of the CMC operational 4DVAR system. Using the 1DVAR, an optimal state is found that minimizes the expected analysis variance. The 1DVAR system computes the minimum of the cost function J(x) as:
i1520-0493-135-1-152-e1
where x is the control variable, xb is the background (i.e., a short-range forecast), yo are the observations, and 𝗕 is the background error covariance matrix. Here 𝗥 is the observation error matrix that has three components when Tb is assimilated (i.e., 𝗥 = 𝗢 + 𝗙RTM + 𝗙MP) and two components when SRR is assimilated (i.e., 𝗥 = 𝗢 + 𝗙MP). Here 𝗢 is the instrumental error, 𝗙RTM is the error of the RTM, and 𝗙MP is the error associated with the moist physical schemes. Here 𝗛(x), referred to as the observation operator, computes either Tb or SRR, depending on the type of observation that is assimilated. When Tb is assimilated, 𝗛(x) consists of two sequential operators: 1) hydrometeors profiles (cloud/ice water contents, rain/snow) are computed from x (temperature and humidity) by applying the moist physical schemes (section 2b) and 2) the RTM (section 2c) is applied to the output of the moist physical schemes to provide Tb (Fig. 1). For the 1DVAR SRR, the observation operator consists of moist physical schemes producing SRR.
The minimization of the cost function J(x) requires the computation of the gradient of the cost function
i1520-0493-135-1-152-e2
where H is the Jacobian matrix of derivatives of 𝗛 with respect to the control variables. The minimization of the cost function is performed with a quasi-Newton descent algorithm named M1QN3 developed by Gilbert and Lemaréchal (1989).

b. Background fields

The moist physical schemes in the 1DVAR system are consistent with those of a research version of the GEM model, referred to here as GEM mesoglobal, which will become the next operational global forecast model at CMC. GEM mesoglobal has 58 vertical levels, a horizontal resolution of 0.45° × 0.33°, a time step of 15 min, and a model top at 10 hPa. The moist physical processes are represented by three schemes, which consist of 1) a shallow-convection scheme (Bélair et al. 2005), which produces only nonprecipitating liquid clouds; 2) a deep-convection scheme (Kain and Fritsch 1990, 1993), which is based on the gradual release of the convective available potential energy; and 3) a nonconvective scheme (i.e., explicit clouds and precipitation; Sundqvist 1978, 1993; Pudykiewicz et al. 1992) for which cloud condensate is a prognostic variable. Jacobians for the moist physics are computed via a finite-difference technique (Marécal and Mahfouf 2000), which is computationally expensive but does not require adjoint coding. This approach is sound when performing scientific feasibility studies.

Another nonconvective scheme, named CLOUDST and developed by Tompkins and Janisková (2004) at ECMWF, is also available in the 1DVAR. This is a diagnostic scheme that was developed specifically for data assimilation. Efforts were made to render the scheme as simple and as linear as possible while keeping a realistic description of the physical processes.

As mentioned in section 2a, T and lnQ are the control variables of the 1DVAR system. However, several other fields are also needed for the simulation of hydrometeors. These are the time tendencies produced by dynamical and other physical processes (radiation, vertical diffusion) that are maintained constant during the minimization process. The moist physical schemes embedded in the 1DVAR system can accurately reproduce the hydrometeors fields obtained by the GEM mesoglobal forecasts.

Background fields for the 1DVAR are chosen to be 12-h forecasts because GEM mesoglobal has a globally averaged precipitation spinup of about 12 h. Following the operational 4DVAR configuration at CMC, the background error covariances are obtained with the “NMC” method (Parrish and Derber 1992) and background error correlations between T and lnQ are set to zero. The mean and standard deviation (SD) of the background error SD itself over the geographical area under study (40°S–40°N) are illustrated in Fig. 2 for the first of the two case studies presented in this paper: Tropical Cyclone (TC) Zoe on 0000 UTC 27 December 2002. This tropical cyclone was also chosen by Moreau et al. (2004). The mean of the T background error SD (Fig. 2) in the troposphere is less than 1 K except near the surface. The low variation of T background error SD over the study area follows from the low T variability over the tropical oceans. The mean of lnQ background error SD varies between about 0.25 (or about 25% if expressed as δQ/Q) at the surface to 0.45 (or about 45%) at 400 hPa. The background errors for the second case study, Typhoon Chaba on 1200 UTC 23 August 2004, are similar.

c. Radiative transfer model

The RTM is a version of radiative transfer for Television and Infrared Observation Satellite (TIROS) Operational Vertical Sounder (TOVS) (RTTOV) (Eyre 1991; Saunders et al. 2002) that includes scattering processes in precipitating atmospheres with the delta-Eddington approximation (Kummerow 1993; Bauer 2002; Moreau et al. 2002). RTTOV also includes a fast emissivity model of the ocean surface (Deblonde and English 2000; Ellison et al. 2003). Jacobians of Tb with respect to the hydrometeor profiles are computed with the adjoint code included in RTTOV. Because of the high variability of the surface emissivity over land and the inability to model the emissivity accurately enough, the assimilation of Tb is limited to ocean surfaces.

d. Observations and associated errors

The SSM/I instrument has four channels with frequencies of 19 GHz (V, H), 22 GHz (V), 37 GHz (V, H), and 85 GHz (V, H) where V and H stand for vertical and horizontal polarizations, respectively. For frequencies less or equal to 37 GHz, the emission produced by liquid rain dominates and increases the brightness temperature. For cold clouds (i.e., containing frozen precipitation-sized hydrometeors) at 85 GHz, the upwelling radiation is dominated by scattering from ice hydrometeors leading to a decrease in brightness temperature. The SRR can be related to the amount of ice scattering but the relationship is by no means direct. Observations at 85 GHz are not considered here for the same reasons as described in Moreau et al. (2004).

In the 1DVAR Tb, the observation error 𝗥 (section 2a), is given two sets of values. In the first case, 𝗥 only includes 𝗢 and 𝗙RTM and is 3, 6, 3, 3, and 6 K, respectively, for the SSM/I channels at 19V, 19H, 22V, 37V, and 37H GHz (as in Moreau et al. 2004). The higher 𝗥 values for horizontally polarized channels follow from the higher Tb dynamic range due to the lower surface emissivity of the horizontally polarized channels compared to that of the vertically polarized channels. Hereafter, these errors are referred to as small observation errors (SOEs). In precipitating atmospheres, setting such low values for observation error leads to a very strong assimilation of the observations (i.e., large weight given to the observations) and hence the background fields become almost irrelevant (see the appendix). In the second case, the observation errors are set to the background errors computed in observation space (i.e., H𝗕HT), which will be shown to be substantially larger than SOE. This choice leads to a more balanced assimilation: the weights given to the background field and to the observations are more similar (see the appendix). This set of errors is referred to as large observation errors (LOEs).

SRR retrievals are computed with the Precipitation Radar Adjusted TRMM Microwave Radiometer Estimation of Rainfall (PATER) scheme (Bauer et al. 2002). This algorithm initially developed for TMI has been extended to SSM/I (Mahfouf et al. 2005). Observation errors for SRR are as defined in Bauer et al. (2002) and used in Mahfouf et al. (2005).

There are advantages and disadvantages to assimilate Tb over SRR. The 1DVAR SRR is computationally cheaper since it does not include a RTM. However, a 1DVAR solution cannot be obtained if the model background SRR is zero (i.e., precipitation cannot be triggered, see section 3c). Furthermore, the quality of the satellite-derived SRR depends on the particular retrieval algorithm and contains regional biases (Bauer et al. 2002). The advantages of the 1DVAR Tb result from the fact that the 1DVAR can be solved in a consistent manner inside and outside of the precipitating regions since Tb is sensitive to water vapor, cloud water, and precipitation. A disadvantage is that one also needs a more sophisticated observation operator to model both cloud and precipitation profiles and their respective fractional coverage.

e. Selection of profiles for assimilation

1) Assimilation of Tb

Profiles are only assimilated where satellite data are available. A nearest-neighbor algorithm assigns a Tb observation to the closest model grid point. This avoids having to interpolate model profiles to the observation point. Profiles are selected only if either the background SRR or the satellite-derived SRR (PATER algorithm) is greater than 0.1 mm h−1. The reduced profile set is then split into two sets (E. Moreau 2003, personal communication) based on the value of P, which is a measure of visibility of the sea surface relative to the expected value in the absence of clouds (Petty 1994):
i1520-0493-135-1-152-e3
where TCloudyb is the cloudy-sky brightness temperature and TClearb is the model clear-sky brightness temperature of the same profile. Here P equal to 0 implies a completely opaque rain cloud while P equal to 1 implies a cloud-free ocean. If either the background or observation value of P is greater than 0.15 then the 37-GHz channels are used in the 1DVAR (scattering effects are larger at these higher frequencies) in addition to the three lowest-frequency channels (i.e., 19V, H GHz and 22V GHz), otherwise, for the more opaque atmospheres, only the three lowest-frequency channels are used. This optimizes the number of profiles for which a converging solution is found.

2) Assimilation of SRR

SRR observations are mapped from observation space into model grid space by averaging all observations within a grid box. When SRR is assimilated, profiles with zero background field SRR are eliminated.

f. Quality control of 1DVAR solutions

Profiles are rejected during the minimization when the increments become more than 20 times larger than the background error SD (1.3% and 0.8%, respectively, of profiles for the Zoe and Chaba cases presented in section 3a). Quality control (QC) on the 1DVAR solution ensures that profiles are rejected when the deviation between observation and analysis is large. The QC after 1DVAR SRR is the same as in Marécal and Mahfouf (2000). For the 1DVAR Tb, profiles are rejected when
i1520-0493-135-1-152-e4
where Tbobs and Tanalyzedb are vectors and α = 3.

3. Results

Brightness temperatures for the SSM/I 19V-GHz channel computed from the GEM mesoglobal fields are illustrated in Fig. 3b for TC Zoe on 0000 UTC 27 December 2002 and for the geographical area that extends from 20°S–19°N to 139°E–179°W. SSM/I F15 Tb observations (Fig. 3a) show a much tighter spiral than in the model simulations. Still several features such as cloud systems, rainbands, and areas with drier air are in good agreement and particularly so for the left orbit.

For each of the two study cases (Zoe and Chaba), the 1DVAR system is solved for in three cases. For the first two cases, Tb is assimilated for two sets of observation errors as defined in section 2d (i.e., SOE and LOE). LOE, estimated from the background error in brightness temperature space, is discussed below in section 3b. For the third case, SRR is assimilated instead of Tb. The three cases throughout the paper are referred to respectively as 1) 1DVAR Tb SOE, 2) 1DVAR Tb LOE, and 3) 1DVAR SRR.

Background fields computed with the 1DVAR and for the zone covering TC Zoe are illustrated in Fig. 4. Background (also referred to as first guess) IWV, cloud liquid water path (LWP), and SRR are illustrated in Figs. 4a–c, respectively. Figure 4d shows the ratio of convective to total SRR. Regions where this ratio exceeds 80% clearly indicate areas where deep-convection dominates. In particular, there is a rainband to the south of the system where SRR is between 6 and 12 mm h−1. According to the brightness temperatures displayed in Fig. 3a this rainband is not observed.

a. 1DVAR Tb SOE

Innovation [observation minus first guess (OP)] bias and SD for the entire study areas (Zoe and Chaba) are given in Table 1 for both the more (less) opaque atmosphere dataset where three (five) channels are used to obtain a solution. The large values of bias and SD are consistent with the large differences between model and observations noticed in Fig. 3 for TC Zoe. These values are significantly larger than in Moreau et al. (2004) indicating a less accurate forecast with the GEM model. It should be noted, however, that the time of TC Zoe in Moreau et al. (2004) is 12 h earlier (than for the case study presented here) and that the observations are assimilated from the TMI instrument. Moreover, Moreau et al. (2004) have not split the results in two categories. Table 1 shows that the biases are much larger for the three channel datasets than for the five channel datasets.

Values of observation minus analysis (OA) are also listed in Table 1. The large ratio of OP to OA SD values denotes a very strong assimilation of the observations (see the appendix). As expected, the analyses draw strongly toward the observations. The OA SD when a posteriori QC is not applied (section 2f) increases by a factor of somewhat less than 2 for the three-channel dataset (including the most opaque profiles) while the SD for the five-channel dataset increases slightly.

Figure 5a illustrates the analyzed fields of SRR for TC Zoe. In Fig. 5d, PATER SRR retrievals are also illustrated for comparison. The analyzed field of SRR is much closer to the observations than the first guess (Fig. 5c). Note that only SRR larger than 0.1 mm h−1 are illustrated. LWP retrievals using the Weng and Grody (1994) retrieval algorithm are in quite good agreement with the analyzed field (not shown). Finally, large IWV increments ranging between −20 and 15 kg m−2 result from the assimilation of Tb. The humidity is increased (decreased) inside (outside) the region where the TC is located in the observations. Mean and SD of the IWV increments for the TC Zoe case are provided in Fig. 6. The values are similar for the Typhoon Chaba case. Negative increments (i.e., drying of the background state) that reduce or switch off precipitation dominate for the 1DVAR SRR whereas the size of the increments for the 1DVAR Tb SOE is smaller by a factor somewhat larger than 1/2.

The performance of the 1DVAR Tb SOE for Zoe (Fig. 7) and Chaba (not shown) is evaluated by comparing the analyses with three SSM/I-retrieved fields. First, in Fig. 7a, histograms of OP and OA SRR are plotted where the observations are PATER SRR retrievals. The background SRR field can be decreased or increased. Evaluation statistics (bias and SD) for OP and OA are listed in Table 2 for both Zoe and Chaba. The SRR OA SD is reduced by a factor of around 2 compared to that of OP.

Second, in Fig. 7b, OP and OA distributions are presented where observations are IWV retrievals obtained with the regression algorithm of Alishouse et al. (1990) and later improved by Petty (see Colton and Poe 1994). IWV retrievals can only be retrieved when there is no precipitation in the observations. Therefore, the number of observations is about 1300 compared with about 4500 for the entire study area. Statistics (Table 2) show that this 1DVAR system properly retrieves IWV even when the background field initially contains clouds and/or precipitation. When there is no precipitation in the observations, the problem becomes equivalent to retrieving IWV in a nonrainy atmosphere, which the SSM/I instrument can do quite accurately (∼2 kg m−2; see Deblonde and Wagneur 1997).

Third, in Fig. 7c, the observations used are LWP retrievals from the Weng and Grody (1994) algorithm. This algorithm retrieves LWP also in precipitating atmospheres. The OA SD is reduced by a factor of 2 compared to that of the OP (Table 2) and a much lower bias is obtained in the analyzed fields (factor of 2). The 1DVAR system efficiently extracts the LWP information contained in the observations.

The mean and SD of the temperature and humidity increments normalized by the background error SD (Fig. 2) are illustrated in Fig. 8 for the Zoe case. The mean of the normalized humidity increments is similar to that of the normalized temperature increments. A high ratio of normalized increments of humidity to temperature is needed to justify only assimilating 1DVAR IWV retrievals in the 4DVAR system and thus neglecting temperature increments (Marécal and Mahfouf 2000). This issue is discussed further in section 4b.

b. 1DVAR Tb LOE

As discussed in section 2b, components of the 1DVAR can be used to compute the background error covariance in observation space or H𝗕HT also referred to as equivalent error. The square root of the diagonal elements of H𝗕HT as a function of background SRR is illustrated in Fig. 9 for the Zoe case (similar results for the Chaba case are not shown). For the three-channel dataset, the maximum background SRR is 16.4 mm h−1, while for the five-channel dataset it is 6.44 mm h−1. For the 19-GHz channels, the equivalent error increases (in an average sense) with background SRR for SRR up to about 4 mm h−1 and then decreases beyond that value. This decrease is likely due to the fact that the SSM/I Tb sensitivity to SRR saturates for high rain rates. At low rain rates, the equivalent error is higher for the horizontally polarized channel because it has a higher dynamical range due to the lower surface emissivity of the ocean. At high rain rates, when the surface is hidden, the equivalent error is similar for both polarizations. For the 22-GHz channel, the equivalent error is more constant up to around 4 mm h−1. The spread in equivalent error (noticeable around 4 mm h−1) for frequencies of 19 and 22 GHz is considerably reduced when the deep-convection scheme is removed and in particular for the 22-GHz channel. Indeed, the Kain–Fritsch deep-convection scheme is known to be a very nonlinear moist physical scheme (Fillion and Bélair 2004).

Given the large variations of the equivalent error with rain intensity, LOE is estimated by computing the average of the equivalent error over the study areas and the corresponding rough estimate values are listed in Table 3. The same values for LOE are used for both the Zoe and Chaba cases. It should be noted that the three-channel dataset has mostly rainy profiles (96.7% for the Zoe case and 96.32% for the Chaba case). As a result of giving similar errors to the observation and background fields, the assimilation is weaker and produces smaller IWV increments that now range between −12.0 and 8.0 kg m−2 for the Zoe case. This weaker data assimilation is also reflected on the analyzed field of SRR (Fig. 5b). The evaluation statistics listed in Table 2 confirm that the analyzed LWP field is quite similar to that of the SOE experiment. For the 1DVAR LOE Zoe case, the ratios of the mean and SD of the normalized humidity increments to those of temperature (ratio of around 3.5 for the mean and 5 for the SD) are larger than those of the 1DVAR SOE Tb experiment (ratio of around 1.0 for the mean and 2.5 for the SD). A similar conclusion can be drawn for the Chaba case. This would justify a two-step assimilation method as described in section 2a, at least in the tropical belt where the temperature errors are dominated by the unbalanced component. However, for the IWV analysis in clear sky (Table 2), a larger bias remains in both study cases (Zoe case: −3.7 kg m−2 for LOE compared to −1.37 kg m−2 for SOE) and the SD is higher (Zoe case: 3.81 kg m−2 compared to 2.83 kg m−2).

To ensure that both the 1DVAR LOE and SOE have the same a posteriori QC, 𝗥 in Eq. (4) was set equal to the SOE value in both cases. A posteriori QC can also be made less strict for the LOE experiment by setting 𝗥 in Eq. (4) equal to the LOE values. This experiment is referred to as 1DVAR Tb LOE2. The histogram of IWV increments (not shown) and statistics shown in Fig. 6 both confirm that the IWV increments are similar for both the LOE and LOE2 experiments.

c. 1DVAR SRR

In the 1DVAR SRR experiment, PATER SRR is assimilated with Bayesian retrieval error estimates as observation error (Bauer et al. 2002). Increments of IWV can be compared to those of the 1DVAR Tb SOE and LOE experiments (Fig. 6). IWV increments of the 1DVAR SRR span the same range as the 1DVAR Tb SOE experiment but negative increments dominate more. In the 1DVAR SRR experiment, there are fewer points since profiles with background SRR less than 0.0001 mm h−1 are not considered. Figure 5c illustrates analyzed SRR from the 1DVAR SRR. The area where SRR is greater than 6 mm h−1 is much larger when SRR is assimilated instead of Tb. Since SRR is the observable that is assimilated, it is also expected that the analyzed fields agree more with these observations (Fig. 5d).

Equivalent error (background error projected in the observation space) can also be computed by the 1DVAR SRR and is illustrated in Figs. 10a (logarithmic scale for equivalent error) and 10b (linear scale for equivalent error) for the smaller Zoe area (i.e., from 20°S–19°N to 139°E–179°W). The equivalent error increases with SRR, and as for Tb equivalent error (Fig. 9), has a large spread around 4 mm h−1, which decreases for larger SRR. In fact, when the deep-convective scheme is excluded in 𝗛(x) (Fig. 10c), the spread diminishes considerably as was the case for Tb equivalent error (not shown). By comparing the SRR equivalent error (Fig. 10a) with the PATER SRR error (Fig. 5a in Mahfouf et al. 2005), one can conclude that the SRR assimilation will be weighted strongly toward the observations for SRR less than ∼8 mm h−1, while for SRR greater than ∼8 mm h−1 the weights given to the background and observations (at least in a linear or weakly nonlinear sense) become similar.

The SD of IWV for the 1DVAR Tb SOE is considerably larger than that of the 1DVAR SRR (Fig. 6). Evaluation statistics listed in Table 2 show that the 1DVAR SRR–analyzed LWP has improved considerably less with respect to the background than the other 1DVAR types. For the 1DVAR SRR, no direct information on clouds is contained in the observations as is the case for Tb. Similarly, analyzed IWV with the 1DVAR SRR in nonrainy areas has a high bias (Zoe case: −3.18 kg m−2) and SD (Zoe case: 4.84 kg m−2). Unlike Tb, SRR does not include direct information on IWV, which is available only through the moist physical schemes. As mentioned above, since SRR is the observable that is assimilated, it is also expected that the analyzed SRR field agrees more with the SRR observations. This is indeed the case: the 1DVAR SRR has the lowest OA bias and SD for SRR (Table 2). The mean and SD ratios of the normalized humidity increments to those of temperature are around 2.8 and 3, respectively, for the Zoe case. Such ratio values would also allow for neglecting the inclusion of 1DVAR temperature increment information into the 4DVAR.

When the SRR of the background is zero, the 1DVAR can not be applied because the derivate of SRR with respect to specific humidity is identically zero. This is not the case in brightness temperature space where the derivative of Tb with respect to humidity is generally nonzero. For the Zoe case, for example, SRR was created for 654 profiles with a maximum value of 2.93 mm h−1 in the 1DVAR Tb LOE. An example of creation of precipitation can be noticed in Fig. 5a in the geographical area comprised between 10°–5°S and 170°–175°E.

d. Estimate of IWV analysis error

An estimate of the analysis error can be computed by taking the diagonal elements of the inverse of the Hessian (i.e., second derivative of the cost function) (Rodgers 2000) and integrating over the depth of the atmosphere. This estimate is valid only when the observation operator is linear. Values of the estimate of the IWV analysis error for the three types of 1DVAR are illustrated in Fig. 11 for the Zoe case and Fig. 12 for the Chaba case. The error is in the 10% range for the 1DVAR Tb LOE and 1DVAR SRR cases. There is a larger spread in the error for the latter case. For the 1DVAR Tb SOE, the errors are much smaller and appear to be too low. As in Marécal and Mahfouf (2002), a polynomial curve will be fitted to the data to parameterize the IWV analysis error as a function of analyzed IWV. This approach will be chosen because it is too costly to compute the Hessian within the 1DVAR.

4. Sensitivity studies

a. Impact of observation error correlation between channels

The effect of including a correlation (r) of the observation error between channels is investigated in this section (i.e., off-diagonal elements of 𝗥 are nonzero). In Moreau et al. (2004), it was found that introducing and varying the correlation (i.e., r = 0.0, 0.5, 0.8) had almost no impact on the retrievals. However, these experiments were performed for a strong assimilation case with the same observation errors as the SOE experiment. Here, the correlation is varied for the LOE2 (section 3b, last paragraph) experiment, which has much higher observation errors. Table 4 lists OP and OA bias and SD for 1DVAR Tb experiments (Zoe case, the results are similar for the Chaba case) with r equal to 0.0, 0.4, and 0.8. The statistics are only listed for the five-channel dataset since no significant differences are found when changing r for the three-channel dataset. About 68% of the profiles in both study cases belong to the five-channel dataset. The OA bias and SD vary considerably as a function of r. For the 37H-GHz channel, OA SD takes on values of 7.98, 11.24, and 19.76 K as r increases (Table 4). The fact that r affects the five-channel dataset results and not the three-channel dataset is because the number of independent pieces of information required to solve the problem accurately should be higher with the five-channel dataset since the associated atmospheric profiles are less opaque. For the three-channel dataset, the precipitation signal dominates the problem.

For the Zoe case, the shape of the histogram of IWV increments (not shown) remains quite similar as r varies. For the Chaba case, the SD of IWV increments decreases slightly as r increases (Fig. 6), which is consistent with a narrower histogram as r increases. Table 5 lists the evaluation statistics [where O is either the PATER SRR or the Weng and Grody (1994) LWP, or the Alishouse–Petty IWV retrievals as described above] for the five-channel dataset. It is difficult to draw conclusions since sometimes the bias is reduced at the expense of an increased SD (e.g., for LWP). Nevertheless, a correlation of 0.4 seems to give the best overall results.

b. Impact of the size of the background temperature error

As mentioned in section 2b, background error statistics for temperature and natural logarithm of specific humidity were obtained with the NMC method. These statistics are obtained from a 3-month average and are computed for a spatial resolution of 1.5° × 1.5° (T108 or 240 × 120 grid points over the globe) even for the higher-resolution GEM mesoglobal. New statistics will be computed at higher resolution as in Buehner (2005) (C. Charette 2005, personal communication). Moreover, these error statistics are only an approximate of the true statistics since for example they are not flow dependent.

A 1DVAR Tb SOE experiment is performed where the temperature background error SD is reduced by a factor of 2 (over the TC Zoe area shown in Fig. 3). As a result, normalized T increments are smaller and normalized lnQ increments are somewhat larger than those of the control case (section 3a) while the IWV increments have hardly changed (not shown). The ratio between normalized T and normalized lnQ increments is increased from around 2 (control case) to around 2.7 for the SD and from about 1.4 (control case) to 1.9 for the mean. Thus, decreasing the T background error leads to a higher ratio of normalized T to lnQ increments, which is needed to satisfy the assumptions that underlay the two-step assimilation method.

c. Impact of moist physical schemes

Experiments with the 1DVAR SRR and 1DVAR Tb SOE are performed (over the smaller Zoe area) to evaluate the impact of specific moist physical schemes. The control experiment (experiment 0) has the three schemes described in section 2b (i.e., shallow- and deep-convection schemes, and a nonconvective scheme). Experiment 1 does not include the shallow-convection scheme while in experiment 2, only the nonconvective scheme is activated. Evaluation statistics for the 1DVAR SRR and 1DVAR Tb SOE similar to those found in Table 2 (where the observations O are SSM/I retrievals of SRR, nonrainy IWV, and LWP) are summarized in Figs. 13a–f. In this figure, the axis labeled “moist physical scheme” represents the different experiments (i.e., 0, 1, and 2). All experiments include a posteriori QC.

1) Impact on SRR

Removing the deep-convection scheme (experiment 2) reduces the OP bias and SD somewhat. This is not surprising since the precipitation field generated by the deep-convection scheme has a high variance and is not well forecast. Removing the shallow-convection scheme (experiment 1) has no effect on OP bias and SD. This is expected since only liquid clouds are generated by this scheme and do not significantly affect the other schemes. The 1DVAR SRR has a smaller OA bias and SD than the 1DVAR Tb. This is also expected since SRR observations are assimilated in the 1DVAR SRR and not in the 1DVAR Tb.

2) Impact on IWV in nonrainy atmospheres

The OP IWV bias in nonrainy atmospheres is about −8 kg m−2. The forecast (which includes rainy areas) is too humid in “true” nonrainy areas as indicated by the observations. The OP IWV SD is ∼4.5 kg m−2. The 1DVAR Tb has a smaller IWV OA bias and SD than the 1DVAR SRR. This can be explained by the fact that unlike SRR, Tb contains direct information on IWV amount.

3) Impact on LWP

The OP bias is considerably less without the shallow-convection scheme (experiment 1) indicating that this scheme needs tuning in terms of liquid water production. The inclusion of the shallow-convection scheme (experiment 0) leads to a considerably larger OA bias and SD for the 1DVAR SRR than for the 1DVAR Tb. This results from the less direct dependence of SRR on cloud water than Tb.

In the 1DVAR, the GEM mesoglobal nonconvective prognostic scheme can optionally be replaced by the ECMWF nonconvective diagnostic cloud and precipitation scheme of Tompkins and Janisková (2004) named CLOUDST (presented in section 2b). This scheme produces much more cloud water and larger areas of precipitation than the GEM nonconvective scheme. CLOUDST has only a 22% occurrence of nonrainy or very low rain-rate situations (<2 mm h−1) compared to 37% for the GEM mesoglobal scheme. LWP evaluation statistics when the GEM nonconvective scheme is replaced with the CLOUDST scheme are displayed in Fig. 13b. The LWP OP has a much larger bias and SD when CLOUDST is used. The OA SDs are not that different whether CLOUDST is used or not and reflects to a certain degree the independence of the first-guess state as a result of a strong assimilation. However, when CLOUDST is used, the O–A bias is much larger whether Tb or SRR is assimilated.

5. Discussion and conclusions

A 1DVAR system has been developed for the GEM mesoglobal NWP model. This system constitutes the first component of the two-step method for the assimilation of rain/cloud information (Marécal and Mahfouf 2000, 2002). The control variables are temperature and natural logarithm of specific humidity profiles. The 1DVAR can assimilate either SRR or Tb (Moreau et al. 2004) and produces IWV increments to be assimilated in a 4DVAR assimilation system in the second step. The observation operator is limited to moist physical schemes when SRR is assimilated whereas when Tb is assimilated, a radiative transfer model is also needed to simulate Tb at the top of the atmosphere.

This paper addresses several issues raised while developing the 1DVAR system at MSC. First, consideration is given to the size of the observation error when Tb is assimilated. Second, the assimilation of Tb and SRR is performed for the same two case studies to compare the resulting analyses. Third, the question is addressed as to whether 1DVAR temperature increments can be neglected in the two-step method. Fourth, the impact of an interchannel observation error correlation is considered. Fifth, the impact of the different moist physical schemes as included in GEM mesoglobal is evaluated. Finally, the portability of moist physical schemes specifically developed for data assimilation (which also comes with their adjoints) is evaluated.

Moreau et al. (2004) specified an observation error where the contribution of moist physical schemes (forward operator) was ignored (as explicitly stated in section 3b of their paper), leading to strong underestimations of these quantities. Consequently, strong assimilations are performed by giving a much larger weight to the observations than to the background field. The large resulting IWV increments could actually have a detrimental impact on the model hydrological cycle. Bauer et al. (2006a) have addressed this issue by having a more severe a priori quality control and also by introducing a bias correction scheme on Tb. In this study, a more balanced contribution between background and observations is obtained by assigning an observation error estimated in rainy areas from the average background error mapped to observation space (i.e., H𝗕HT). As expected, the resulting assimilation is weaker and produces smaller IWV increments. Meanwhile, analyzed fields of SRR and LWP show that a considerable amount of information is still extracted from the observations. In particular, the analyzed LWP fields are rather similar when either a small or a large observation error (SOE or LOE) is used. However, the IWV analysis in the clear sky (in the PMW sense) of the weaker assimilation is not as good with a large bias remaining (Zoe case: −3.71 kg m−2) and higher SD (Zoe case: 3.81 compared to 2.83 kg m−2). Profiles with observations that are clear sky (in the PMW sense) could be excluded from the 1DVAR or selectively given a lower observation error. This would correspond to providing a priori information on the profile. Instead of taking the average of the background error in observation space (for rainy profiles) as was done here, it would be possible to specify an observation error that varies with the background SRR. However, when the deep-convection scheme is dominant (i.e., a highly nonlinear scheme), the equivalent error is shown to have a very large spread. This raises the question of the usefulness of this scheme for the assimilation of cloud/rain observations. Over the oceans (limited to 40°S–N) and for the same time as TC Zoe, deep convection is only dominant (convective precipitation greater than stratiform) for half of the grid points where SRR is between 1 and 10 mm h−1 (25% of all grid points with SRR greater than 0.01 mm h−1). This is understandable because part of the liquid water generated in convective towers is detrained in the large-scale environment.

The assimilation of Tb and SRR is performed for the same case study to compare the resulting analyses. As in Mahfouf et al. (2005), the SRR observation error is specified according to Bauer et al. (2002) and does not include the contribution of the moist physical schemes. The SD of the IWV increments for the 1DVAR SRR lies in between that of the 1DVAR Tb SOE and LOE. The range of IWV increments is similar for the 1DVAR Tb SOE and the 1DVAR SRR. However, the histogram of the IWV increments is considerably more peaked for the latter. By comparing the analyzed LWP with Weng and Grody (1994) retrievals for the 1DVAR SRR, only a slight improvement with respect to the background is noticed. For the 1DVAR SRR, no direct information on clouds is contained in the observation, which is the case for 1DVAR Tb. Moreover, analyzed IWV in nonrainy areas still has a large bias (Zoe case: −3.18 kg m−2) and SD (Zoe case: 4.84 kg m−2) with respect to the Alishouse–Petty retrievals. Unlike Tb, an observed SRR value of zero (i.e., nonrainy profile) does not include direct information on IWV.

As discussed above, the assimilation case studies for the 1DVAR Tb with SOE produces a large range of increments. It is also shown that reducing the temperature background error reduces the temperature increments while increasing the humidity increments. Using higher observation errors in the 1DVAR Tb system, which would be more realistic, will also ensure that the ratio of normalized temperature to normalized humidity error is large and thus justifying the two-step assimilation method. The current GEM mesoglobal background errors from the “NMC” method are a rather crude estimate of the true background errors and will be improved (Buehner 2005).

The impact of adding an interchannel observation error correlation, contrary to Moreau et al.’s (2004) findings, is significant for the five-channel dataset. This is because the number of independent pieces of information required to solve the problem accurately needs to be higher with the five-channel dataset since the associated profiles are less opaque. For the three-channel dataset, the precipitation signal dominates the problem. The moist physical schemes used in 1DVAR are the same as those of the GEM mesoglobal model and consist of a shallow- and deep-convection schemes, and a prognostic nonconvective scheme. The shallow- and deep-convection schemes can optionally be excluded. The shallow-convection scheme, designed to transport moisture through the cloudy boundary layer, only generates cloud water content. It is shown that when this scheme is excluded, the agreement between LWP from the Weng and Grody (1994) retrievals and that analyzed is actually improved. This is also the case in brightness temperature space. These findings highlight the fact that the production of liquid water by this scheme needs to be calibrated. Furthermore, it was shown that for TC Zoe the deep-convective scheme deteriorated the fit between modeled and observed Tb, due to a misplacement of the precipitating areas by the GEM model.

A nonconvective cloud and precipitation scheme developed by Tompkins and Janisková (2004) at ECMWF for data assimilation has been included in the 1DVAR. This scheme has the advantage of being diagnostic in terms of cloud condensate. It is rather linear and simple enough for the adjoint scheme to be available. However, this scheme is found to produce too much clouds and too extensive areas of low precipitation and would need tuning to provide comparable background fields to those of GEM mesoglobal. Moist physical schemes used in NWP centers around the world can be very different. It is well known that precipitation forecasts from different NWP centers are rarely in very good agreement between each other and agreement with observations is still fairly poor (Ebert et al. 2003). Hence, moist physical schemes are far from being readily portable. This is not the case for the radiative transfer model (i.e., RTTOV) and its adjoint, which are readily portable.

This paper emphasizes the fact that the 1DVAR system must account for errors in the moist physical schemes (part of the forward operator), and for interchannel observation error correlation in the case where the five lowest-frequency SSM/I channels (rather than the three lowest-frequency channels), and finally that the assimilation of Tb (rather than SRR) should be favored due to the direct dependence of Tb on LWP and IWV.

The 1DVAR system will be generalized (but limited to 40°S–N or tropical belt where the dynamic range of precipitation is high) to provide IWV increments for a 4DVAR assimilation cycle using the two-step method. Avenues will have to be found to speed up the 1DVAR system since the intended application is operational assimilation. So far, one option has been investigated and consists of maintaining the transpose of the Jacobian (H) fixed after a preset number of iterations. Results show that doing this after five iterations leads to very similar analyzed fields. The fact that the 1DVAR system uses the same moist physical schemes as in the next version of the operational forecast model (GEM mesoglobal) will allow a complementary evaluation of these schemes by simulating Tb and comparing these with observations as in Chevallier and Bauer (2003) over several weeks.

Acknowledgments

The authors thank Adrian Tompkins and Martha Janiskova for making their code CLOUDST available for testing. The authors are also grateful for the constructive and helpful comments provided by the reviewers.

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APPENDIX

Specification of the Observation Error

The following formulation follows Daley (1991). Given a zero mean Gaussian-distributed random variable x and observation y, the optimal estimate xa of the true value xt is
i1520-0493-135-1-152-ea1
where σ2o and σ2b are estimates (not necessarily correct) of the observation and background error variance, respectively. Correct values of these quantities are denoted by σ2ot and σ2bt, respectively.

Then, the actual expected analysis error variance is σ2a = 〈(xaxt)2〉, where 〈·〉 denotes the ensemble average.

Using Eq. (A1), it can be shown that
i1520-0493-135-1-152-ea2
where OA = yxa and (σ2bt + σ2bt) is the true innovation variance.
If σ2bt = σ2b and σ2ot = σ2o, then 〈(OA)2〉 = (σ4o/σ2o + σ2b). In general however, from Eq. (A2) we have
i1520-0493-135-1-152-ea3
where OP = yxb.
From the OA and OP SD provided in Table 1 (TC Zoe with SOE observation error) and for the 19H-GHz channel of the five-channel dataset for example, Eq. (A3) can be rewritten as
i1520-0493-135-1-152-ea4
Therefore, in the analysis equation [Eq. (A1)], the relative weight is 0.19 for the background state and 0.79 for the observation (i.e., the observation is given much more weight than the background state).

It is desirable to have a more balanced contribution of each piece of information in the analysis. By choosing σ2o equal to σ2b, the ratio (σ2o/σ2o + σ2b) is equal to 0.5 and then the value of [〈(OA)2〉/〈(OP)2〉] is 0.25. This ratio increases from 0.19 to 0.31 when the observation errors are set to LOE.

Fig. 1.
Fig. 1.

Flowchart describing the simulation of the SSM/I brightness temperatures from the 1DVAR control variables, which are temperature and humidity.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 2.
Fig. 2.

Mean (solid line) and standard deviation (SD; dotted line) of (a) T and (b) lnQ background errors for GEM mesoglobal over the geographical area that extends from 40°S to 40°N for profiles for the TC Zoe case (0000 UTC 27 Dec 2002).

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 3.
Fig. 3.

Brightness temperature at 19 GHz V for TC Zoe 0000 UTC 27 Dec 2002. (a) F15 SSM/I observations and (b) GEM mesoglobal 12-h forecast.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 4.
Fig. 4.

1DVAR first-guess fields of (a) IWV in kg m−2, (b) cloud LWP in kg m−2, (c) SRR in mm h−1, and (d) ratio in percent of convective to total SRR for TC Zoe 0000 UTC 27 Dec 2002.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 5.
Fig. 5.

Analyzed SRR (mm h−1) for TC Zoe 0000 UTC 27 Dec 2002: (a) 1DVAR Tb SOE, (b) 1DVAR Tb LOE, (c) 1DVAR SRR, and (d) PATER retrievals from F15 SSM/I.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 6.
Fig. 6.

Mean and SD of increments (analysis minus first guess) for 1DVAR experiments SOE, LOE, LOE2 (r = 0.0, 0.4, and 0.8), and SRR. The interchannel error correlation coefficient (r) is zero for experiments 1DVAR SOE, LOE, and SRR. The 1DVAR experiment LOE2 is similar to LOE but with a different a posteriori quality control (see text).

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 7.
Fig. 7.

The OP (solid line) and OA (dashed line) histograms for TC Zoe 0000 UTC 27 Dec 2002 for the 1DVAR Tb SOE. Observations (O) are in (a) PATER SRR retrievals (mm h−1), (b) Alishouse–Petty IWV retrievals in nonrainy areas (kg m−2), and (c) Weng and Grody (1994) LWP retrievals (kg m−2). All retrievals are for F15 SSM/I.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 8.
Fig. 8.

(a) Mean and (b) SD of T (solid lines) and lnQ (dotted lines) increments (analysis minus first guess) normalized by SD of background errors (Fig. 2) for TC Zoe 0000 UTC 27 Dec 2002 and experiment 1DVAR Tb SOE.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 9.
Fig. 9.

Background error in brightness temperature space for the SSM/I channels as a function of first guess SRR (mm h−1) for TC Zoe 0000 UTC 27 Dec 2002: (a) 19V GHz, (b) 19H GHz, (c) 22V GHz, (d) 37V GHz, and (e) 37H GHz.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 10.
Fig. 10.

Background error in SRR space as a function of first-guess SRR (mm h−1) (small Zoe area). (a) 1DVAR SRR with all moist physical schemes (dotted line indicates 500% error and solid line 100%), (b) same as in (a) but with a linear scale instead of a logarithmic scale for the background error, and (c) same as in (b) but without deep convection.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 11.
Fig. 11.

Estimate of IWV analysis error from inverse of Hessian of cost function for TC Zoe: (a) 1DVAR SOE, (b) 1DVAR LOE, and (c) 1DVAR SRR. The 10% error line is shown.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 12.
Fig. 12.

Same as in Fig. 11, but for Typhoon Chaba.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Fig. 13.
Fig. 13.

The (left) OP and (right) OA bias (dotted lines) and SD (solid lines) statistics for TC Zoe 0000 UTC 27 Dec 2002 (small area). Stars denote 1DVAR SRR and diamonds denotes 1DVAR Tb. Observations (O) for (a), (b) SRR, (c), (d) IWV in nonrainy areas, and (e)–(h) LWP are, respectively, PATER SRR retrievals (mm h−1), Alishouse–Petty IWV retrievals in nonrainy areas (kg m−2), and Weng and Grody (1994) LWP retrievals (kg m−2). All retrievals are for F15 SSM/I. For the moist physical scheme on the horizontal axis, 0 includes all three schemes, 1 does not include shallow-convection scheme, and 2 only has nonconvective scheme. The nonconvective scheme is (a)–(f) “Sundqvist” and (g), (h) CLOUDST.

Citation: Monthly Weather Review 135, 1; 10.1175/MWR3265.1

Table 1.

Observed brightness temperature departures (K) from first-guess OP and analyses OA for three- and five-channel datasets for the 1DVAR Tb SOE. The departures are expressed in terms of bias and SD. Observation errors are 3 K for vertically polarized channels and 6 K for horizontally polarized channels. Here, No. is the number of profiles.

Table 1.
Table 2.

The OP and OA bias and SD evaluation statistics where the observations (O) are SSM/I retrievals of SRR, IWV in nonrainy areas, and LWP as described in section 3a. Results are presented for the three 1DVAR cases described in section 3.

Table 2.
Table 3.

Mean background error in observation space or equivalent error used in the 1DVAR Tb LOE experiment. Statistics shown are for TC Zoe 0000 UTC 27 Dec 2002. LOE observation errors listed in the last column of the table are the same for both the Zoe and Chaba cases. Profiles for the five-channel dataset are defined as rainy if background SRR is greater than 0.0001 mm h−1.

Table 3.
Table 4.

Bias and SD of observed brightness temperature (SSM/I F15) departures (K) from first-guess OP and analyses OA for the five-channel dataset and for correlation (r) between observation errors for different channels of 0.0, 0.4 and 0.8 for TC Zoe 0000 UTC 27 Dec 2002. Observation errors are as specified in Table 3 (i.e., LOE).

Table 4.
Table 5.

Bias and SD of observation departures from first-guess OP and analyses OA for the five-channel dataset and for correlation (r) between observation errors for different channels of 0.0, 0.4, and 0.8 for TC Zoe 0000 UTC 27 Dec 2002 for the 1DVAR Tb LOE2. The observations (O) are SSM/I F15 retrievals of SRR, IWV in nonrainy areas, and LWP as described in section 3a.

Table 5.
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