## 1. Introduction

Although ensemble-based Kalman filter (EnKF) data assimilation schemes were first proposed more than a decade ago (Evensen 1994; Burgers et al. 1998; Houtekamer and Mitchell 1998) and several successful attempts at assimilating observations of the atmosphere have been reported in the last few years (e.g., Houtekamer et al. 2005; Whitaker et al. 2004, 2008; Szunyogh et al. 2008; Miyoshi and Sato 2007; Miyoshi and Yamane 2007; Torn and Hakim 2008; Bonavita et al. 2008), evidence has emerged only recently that EnKF schemes may be viable alternatives to the variational techniques in operational numerical weather prediction (e.g., Buehner et al. 2010a,b; Miyoshi et al. 2010).

In the present paper, we focus on the performance of one particular EnKF scheme, the local ensemble transform Kalman filter (LETKF), for assimilating satellite radiance observations. The LETKF algorithm was developed by Ott et al. (2004) and Hunt et al. (2004, 2007) and was tested on both simulated observations in the perfect-model scenario (Szunyogh et al. 2005) and on observations of the real atmosphere (Miyoshi and Sato 2007; Szunyogh et al. 2008; Whitaker et al. 2008). In particular, Szunyogh et al. (2008) and Whitaker et al. (2008) assimilated nonradiance observations in a reduced-resolution version of the model component of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) and found that the performance of the LETKF was superior to that of the Statistical Spectral Interpolation (SSI) of NCEP in data-sparse regions.^{1}

Our goal here is to extend the study of Szunyogh et al. (2008) by augmenting the observational dataset with satellite radiance observations. To assimilate these satellite observations, we employ techniques for the localization and bias correction of the satellite radiance observations, which were developed and tested in an idealized setting in Fertig et al. (2007, 2009). The observations we assimilate are the Advanced Microwave Sounding Unit-A (AMSU-A) level 1B brightness temperature data from an instrument flown on the Earth Observing System (EOS) *Aqua* spacecraft (Olsen 2007). Hereafter, we refer to brightness temperature and radiance observations collectively as radiance observations, as the assimilation of both of these types of data requires the use of a radiative transfer model. The performance of the LETKF in assimilating radiance observations is assessed by comparing the results to those obtained by assimilating only the nonradiance observations.

Ours is not the first attempt at assimilating satellite radiance observations with an implementation of an EnKF scheme on a model of operational complexity: some studies assimilated satellite radiance observations using an offline estimate of the observation bias, which was provided by a variational data assimilation system (e.g., Houtekamer et al. 2005; Miyoshi and Sato 2007; Buehner et al. 2010b), while Miyoshi et al. (2010) combined the ensemble based estimation of the state with an online deterministic estimation of the observation bias. The unique aspect of our study is that it uses the ensemble-based approach to estimate both the bias and the state.

The structure of the paper is as follows. Section 2 provides a summary of our implementation of the LETKF on the model component of the NCEP GFS, while section 3 is a brief description of the AMSU-A observational datasets. Section 4 explains the design of our numerical experiments, whose results are reported in section 5. Section 6 offers a summary of our conclusions.

## 2. The LETKF for the NCEP GFS model

In what follows, we explain our implementation of the LETKF algorithm using the model component of the NCEP GFS. We introduce the major components of the data assimilation algorithm and summarize the data assimilation procedure for the conventional nonradiance observations. Then, we explain the modifications required to assimilate satellite radiance observations.

### a. Definitions

*t*, observations are assimilated from the observation time window:The observations from

_{n}*τ*form the vector of observations

_{n}*t*= 6 h.

*γ**for the state space trajectory of the model in*

_{n}*τ*:where the vector

_{n}**x**(

*t*) is the finite-dimensional representation of the atmospheric state on the model grid. Similar to other ensemble-based data assimilation schemes, the LETKF algorithm prepares an ensemble of analyses,

*K*analyses,

*t*

_{n−}_{1}=

*t*− Δ

_{n}*t*. The associated computations consist of two main steps: the forecast step and the state-update step.

In the forecast step, a *K*-member ensemble of forecasts from time *t _{n}*

_{−1}to time

*t*+ Δ

_{n}*t*/2 is prepared using the analysis ensemble members

*t*/2 long time period between

*t*

_{n}_{−1}and

*t*+ Δ

_{n}*t*/2. The Δ

*t*-long section of the trajectories, which falls into

*τ*provides the ensemble of background forecast trajectories,

_{n}*t*

_{n}_{−1}. To obtain

*t*

_{1}, we use a random set of operational NCEP analyses valid at different times as the initial conditions

**h**(

*γ**) that satisfiesHere,*

_{n}

*ε**is a vector of Gaussian random observation noise with zero mean and error covariance matrix*

_{n}*t*; thus, we can drop the subscript

_{n}*n*from the notation without sacrificing clarity. In particular, we introduce the notation

**x**

^{b}^{(k)}for the state at time

*t*along the trajectory

_{n}**x**

_{ℓ}that is composed of the model variables at model grid point ℓ. LETKF generates a

*K*-member ensemble of local analyses,

*k*th column of the matrix of ensemble perturbations

*k*th local background ensemble perturbation defined by

### b. Conventional observations

For the conventional (nonradiance) observations, the time interpolation component of the observation operator **h**(** γ**) is defined by the linear interpolation procedure described in section 2a, while in the two horizontal spatial dimensions,

**h**(

**) is a bilinear interpolation. Since the vertical coordinate in the NCEP GFS model is**

*γ**σ*(defined as the ratio of the pressure and the surface pressure) and the vertical position of the observations is given in pressure, the vertical interpolation for a given observation is carried out in three steps:

- We calculate the pressure at each
*σ*level at the horizontal location of the observation. This calculation requires the horizontal interpolation of the background surface pressure to the horizontal location of the observation and the multiplication of the results of the interpolation with*σ*at the given model level. - We define 28
*σ*layers, each bounded by a pair of*σ*levels (the lowest layer is defined by the model surface and the lowest*σ*level). - We find the
*σ*layer that contains the observation and linearly interpolate the logarithm of the pressure to the vertical location of the observation based on the pressure values at the two*σ*levels that bound the layer.

- The observation operator
**h**() is applied to each member of the ensemble of background trajectories, {*γ***γ**^{b}^{(k)}:*k*= 1, 2, … ,*K*}, to obtain an ensemble, {**y**^{b}^{(k)}:*k*= 1, 2, … ,*K*}, of the model-predicted values of the observables at the observation locations. The ensemble averageof the ensemble { **y**^{b}^{(k)}:*k*= 1, 2, … ,*K*} is computed and the matrixis constructed by taking its columns to be the vectors . - The localization is performed. For each location (grid point) ℓ, the observations that are thought to have useful information about the atmospheric state at grid point ℓ are selected for assimilation. The selected observations form the local observation vector
. The vector and the matrices and are formed by selecting those vector components and matrix elements that are associated with the selected set of observations at ℓ. The sensitivity of the LETKF scheme to the localization parameters and the number of ensemble members was investigated in detail in Szunyogh et al. (2005). The issue was further investigated in Szunyogh et al. (2008), where it was found that, within a reasonable range, the accuracy of the analysis and the short-term forecasts was only weakly sensitive to the localization parameters. Here, we use the same localization parameters as in Szunyogh et al. (2008): - In the horizontal direction, observations are considered from an 800-km radius neighborhood of the location (grid point) ℓ. The influence of observations is tapered as a function of the radius
*r*from the grid point. Formally, the tapering is achieved by multiplyingby a factor *μ*(*r*):*μ*(*r*) = 1 for*r*≤ 500 km and*μ*(*r*) = (800 −*r*)/300 for 500 km ≤*r*≤ 800 km.^{2} - In the vertical direction, observations are considered from a layer around ℓ. The depth of the layer is 0.35 scale height between model levels 1 and 15 (below
*σ*= 0.372), and, starting with level 15, the depth gradually increases with height to reach 2 scale heights at the top of the model atmosphere (defined by*σ*= 0.003, which is equivalent to about 3 hPa). (The scale height is defined by the vertical distance in which the pressure drops by a factor of*e*≈ 2.718.) Surface pressure observations are also considered from the local horizontal region when the state is analyzed at a model grid point, which is at or below model level 15. - The surface pressure components of the state vector are treated differently from the other components. To obtain the surface pressure analysis at a location ℓ, we use all surface pressure observations from an 800-km radius of ℓ and all temperature and wind observations from an 800-km radius of ℓ between model levels 2 (
*σ*= 0.982) and 5 (*σ*= 0.916). As for all other observation types, the influence of the surface pressure observations is tapered beyond a 500-km radius.

- In the horizontal direction, observations are considered from an 800-km radius neighborhood of the location (grid point) ℓ. The influence of observations is tapered as a function of the radius
- The weight vector
is computed byIn Eq. (5), is the local analysis error covariance matrix, which is computed byHere, *ρ*≥ 1 is a multiplicative covariance inflation factor andis the identity matrix. In our implementation, *ρ*is a smoothly varying three-dimensional scalar field:*ρ*tapers from 1.25 at the surface to 1.2 at the top of the model atmosphere in the SH extratropics and from 1.35 to 1.25 in the NH extratropics, while*ρ*changes smoothly throughout the tropics (between 25°S and 25°N) from the values of the SH extratropics to the values of the NH extratropics. - The matrix
, whose columns are the local weight vectors for the ensemble perturbations, is computed by . - The weight vector
is added to each row of . The columns of the resulting matrix are the members of the ensemble of weight vectors .

### c. Satellite radiance observations

The assimilation procedure is more complicated for the radiance observations than for the conventional observations. The primary source of the added complexity is the observation operator **h**(** γ**), whose components for the radiance observations involve the use of a complex radiative transfer model,

**h**(

**) = [**

*γ***h**

^{(nr)}(

**),**

*γ***h**

^{(r)}(

**)]**

*γ*^{T}. The first component

**h**

^{(nr)}(

**) maps the state trajectory for the nonradiance observations and the component**

*γ***h**

^{(r)}(

**) maps the trajectory for the radiance observations.**

*γ***b**is the bias-correction term. The implementation of the observation operator defined by Eq. (7) requires a computational procedure to estimate

**b**. Following the general practice of numerical weather prediction (e.g., Eyre 1992; Derber and Wu 1998; Harris and Kelly 2001; Dee 2005), we make the estimation of

**b**part of the data assimilation process. In particular, we assume that the

*O*

^{(r)}components of

**b**can be estimated by the linear combinationof a set of scalar predictors {

*p*(

^{i}*t*):

*i*= 1, … ,

*I*} [

*O*

^{(r)}is the number of radiance observations]. Our task is to estimate the bias-correction parameters:

*I*+ 1) ×

*O*

^{(r)}bias parameters is computationally not feasible, because in a typical NWP application

*O*

^{(r)}is of order 10

^{6}–10

^{8}. To reduce the number of bias-correction parameters, we use the same set of predictors for all radiance observations, and assume that the bias-correction parameters are the same for all observations taken by a given instrument in a given channel. Thus, when the total number of channels we assimilate is

*J*, the total number of bias-correction coefficients is

*Q*= (

*I*+ 1) ×

*J*. These coefficients,

*β*. When all predictors are zero, the bias correction for the

*j*th channel is equal to

*intercept*for the

*j*th channel. In addition, because the bias-correction procedure introduces a dependence of the observation operator on

**, for the radiance observations we write Eq. (7) asWith this notation Eq. (2) becomeswhere**

*β***y**

^{o(nr)}is the vector of nonradiance observation and

**y**

^{o(r)}is the vector of radiance observations, while

*ε*^{nr}is the vector of random observation errors for the nonradiance observations and

*ε**is the vector composed of the random part of the observation errors for the radiance observations.*

^{r}The predictors {*p ^{i}*(

*t*):

*i*= 1, … ,

*I*} can be defined by any scalar physical parameters from the model or from the information provided with the observations (e.g., Eyre 1992; Derber and Wu 1998; Harris and Kelly 2001). Typical examples for model-based predictors are the skin temperature and the thickness of different atmospheric layers, while an example for an observation-related predictor is the scan angle at which the radiance observation is taken by the satellite-based observing instrument.

**x**by the

*Q*components of

**to obtainand apply the LETKF algorithm to the augmented state vector**

*β***z**instead of the state vector

**x**.

**from one analysis time to the next. In our current implementation of the method, we evolve the bias parameters between two analysis times by assuming persistence of the bias parameters:Using Eq. (12) in the forecast step of the analysis does not mean that the estimates of the bias-correction parameters cannot change with time. In fact, the state-update step typically changes the value of the bias parameters, that is,which leads toThe second important issue is the nonlocal nature of the observation operator for radiance: in contrast to the case of the conventional observations, where the observation operator for a given observation depends on the model state only at the nearby grid points, the output of**

*β**ω*, which is computed by

_{m}*m*= 1, 2, … ,

*M*. For a given observation,

*T*is the temperature at model level

_{m}*m*,

*B*(

*T*) is the Planck function, and the weights {

_{m}*ω*:

_{m}*m*=1, 2, … ,

*M*} satisfy the following condition:We apply the cutoff-based observation strategy of Fertig et al. (2007) to select the model levels where a given observation is assimilated. In particular, we choose a cutoff parameter

*η*(0 <

*η*≤ 1), which we will use in step 3 of the data selection procedure below to determine the depth of the layer where the observation will be assimilated. (We use the same value of

*η*for all channels.) We determine the layer for each observation in the following steps:

- We apply
to all members of the ensemble of background trajectories { **γ**^{b}^{(k)}:*k*= 1, 2, … ,*K*} to obtain an ensembleof the weight *ω*for the given observation._{m} - We find the model level
, where takes its maximum value for each ensemble member. - We search for the top,
, and the bottom, , of the deepest layer around level , in which the weighting function satisfies the condition . - We compute the ensemble mean of the index of the top layers
and the ensemble mean of the index of the bottom layers, . - We assimilate the observations at model levels that fall into the layer bounded by
and .

*l*. The potential to incorrectly eliminate correlations is higher for the radiance than the conventional observations, because the neighboring satellite channels typically have broad overlapping weighting functions; thus,

*η*: the smaller the value of

*η*, the more channels are assimilated simultaneously and the lower the chance that correlations are eliminated incorrectly. This argument is also supported by the results of Fertig et al. (2007), who showed that assimilating each radiance observation at multiple model levels, instead of the single level where

*η*, of course, increases the computational cost, because more observations are assimilated at each location. Thus, we determine the value of

*η*by numerical experimentation, choosing a value of

*η*, which is slightly smaller than the value at which the analysis and forecast accuracy starts to degrade noticeably.

- The ensemble of model-predicted radiance values at the observation locations is obtained by applying
**h**^{(r)}to the background trajectories {**γ**^{b}^{(k)}:*k*= 1, … ,*K*}. - The horizontal localization is done the same way as for the conventional observations, while the vertical localization is done by the cutoff-based strategy. The components of the local augmented state vector
**z**_{ℓ}at location ℓ are the components of the local state vector**x**_{ℓ}, and the local vector of bias-correction parameters*β*_{ℓ}, which is composed of the bias-correction parameters for the channels that are assimilated at location ℓ.

**x**

^{a}^{(k)}:

*k*= 1, 2, … ,

*K*} of {

**z**

^{a}^{(k)}:

*k*= 1, 2, … ,

*K*} are obtained as before, collecting the state vector components

*β*

^{a}^{(k)}:

*k*= 1, … ,

*K*} from the bias-correction component,

*Q*bias parameters. We achieve this goal by averaging the local estimates of each bias parameter over all locations ℓ where it is estimated by the following formula:Here,

*q*th components of

*β**and*

^{a(k)}*ϕ*

_{ℓ}is the latitude at location ℓ; and the factor cos(

*ϕ*

_{ℓ}) accounts for the dependence on the latitude of the area represented by a grid point. The factor

*q*th component of the bias parameter vector

**at location ℓ. (Here,**

*β**β*

_{q}_{,ℓ}at location ℓ.) Weighting with the inverse of the variance ensures that locations where the uncertainty in the estimate of a given bias parameter is larger contribute with a smaller weight to the global estimate of that bias parameter.

## 3. The observations

Following the convention of operational numerical weather prediction for global models, we use a 6-h window and prepare analyses 4 times a day: at 0000, 0600, 1200, and 1800 UTC. A typical example for the number of observations we assimilate is shown in Table 1.^{3} On any given day, we assimilate about 1 million observations, of which about 15%–20% are radiance observations. These radiance observations fill important data voids in the coverage by the conventional data (see Figs. 1 and 2). We process many more observations than indicated by Table 1, but the number of observations is reduced by selecting only a subset of the radiance observations for assimilation and by rejecting observations that do not pass quality control. The data selection strategy and the quality control procedure are explained in section 4.

Number of assimilated observations on a typical day (31 Jan 2004).

### a. Conventional observations

We assimilate all conventional observations that were assimilated operationally at NCEP between 0000 UTC 1 January 2004 and 1800 UTC 29 February 2004. This dataset includes observations of the surface pressure by synoptic land stations; virtual temperature and surface pressure by surface marine observing platforms; splash-level virtual temperature by dropsondes; virtual temperature and wind by rawinsondes; sensible temperature and wind by commercial airliners; flight-level virtual temperature and wind by reconnaissance planes; cloud-drift wind by the *Meteorological Satellite-5* and *-7* (*Meteosat-5*) and (*Meteosat-7*), the *Geostationary Operational Environmental Satellite-8* and *-10* (*GOES-8*), and (*GOES-10*); and the Quick Scatterometer (QuikSCAT) surface wind by scatterometers. Figure 1 shows the spatial distribution of the assimilated temperature observations for a typical 6-h observation time window.

### b. AMSU-A level 1B brightness temperature data

AMSU-A is primarily a temperature sounder that provides atmospheric information in the presence of nonprecipitating clouds. We assimilate a subset of the AMSU-A level 1B brightness temperature dataset, which contains calibrated and geolocated brightness temperatures in kelvin for 15 microwave channels. We assimilate only 8 of the 15 channels, since the observations from channels 1, 2, 3, and 15 have a strong surface signal component, while channels 12, 13, and 14 are strongly influenced by the atmospheric conditions at altitudes that are higher than the top of our model atmosphere. Figure 2 shows the spatial distribution of the assimilated AMSU-A observations for a typical 6-h observation time window.

The number of vertical levels used in the computation of the radiative transfer is one of the input parameters of the CRTM. After consulting colleagues with extensive experience with the CRTM, we decided to use 101 levels. Thus, in our implementation, the **h**^{(r)}(** γ**) observation operator for the AMSU-A observations involves an interpolation of the background fields from the 28 model levels to the 101 levels used in the computation of the radiative transfer. Since the radiative transfer depends on the full atmospheric state, the maximum value of the weighting function,

AMSU-A channels selected for assimilation. The pressure level of peak sensitivity for each channel is shown for a randomly selected analysis time (1800 UTC 18 Feb 2004) at 2 particular locations (45°N and 45°S) along the date line.

## 4. Numerical experiments

The primary goal of our numerical experiments is to determine how much improvement is achieved in the analyses when, in addition to the conventional observations, we assimilate the AMSU-A observations with the proposed strategy. We assess the performance of the data assimilation system when the AMSU-A observations are included by comparing the analysis and short-term (48 h) forecast errors with those from two reference experiments. In one of these reference experiments, we assimilate the AMSU-A observations but do not apply bias correction to the radiance observations, while in the other reference experiment, we assimilate only the conventional observations.

### a. Experiment design

For the sake of computational efficiency, in the two experiments that assimilate radiance observations, we do not assimilate more than one radiance observation per channel at a given grid point. Instead, we assimilate the first observation from the dataset that satisfies all quality control criteria. In particular, we do not assimilate observations from mixed-surface footprints (e.g., from areas where seawater is mixed with ice), observations from channels 4 and 5 over land, and observations for which the scan angle is larger than 35°. We also reject observations for which the difference between the observed value and **h**(** γ**) is more than 5 times larger than both the ensemble spread (standard deviation of the ensemble) and the presumed standard error of the observations.

The model used in this study is the 2004 model component of the operational NCEP GSF truncated to T62L28 resolution. This model is identical to the one that was used in Szunyogh et al. (2008) and Whitaker et al. (2008). The only important improvement in our LETKF data assimilation system, compared to the one we evaluated in Szunyogh et al. (2008), is the correction of a coding error that led to the rejection of most scatterometer observations in the former implementation of the system. This correction leads to an improvement of the analyses and short-term forecasts in the Southern Hemisphere extratropics near the surface. We use this improved set of analyses as the baseline for the evaluation of the results obtained with the augmented observational dataset. Despite the aforementioned coding error, the former version of the LETKF provided analyses and short-term forecasts that in the SH, on average, were more accurate at the 99% significance level than those obtained with the then-operational SSI of NCEP at the same T62L28 resolution. Consequently, our baseline dataset consists of reasonably high quality analyses.

### b. Verification methods

*t*(

_{i}*i*= 1, 2, … ,

*T*). Since we verify forecasts started from the 0000 and 1200 UTC analyses between 10 January and 27 February 2004,

*T*= 2 × 49 = 98. We measure the error in the state estimate at pressure level

*t*with the root-mean-square error:In Eq. (20),

_{i}*t*onto the verification grid. Since the AMSU-A observations from the

_{i}*Aqua*satellite were not assimilated by the operational NCEP analysis system,

^{4}and the algorithm used by the then-operational SSI data assimilation system of NCEP is substantively different from our LETKF algorithm, we have reason to believe that most of the changes we detect in the quality of the analyses are not due to correlation between the errors in

*t*test for correlated data described in Example 5.2 of Wilks (2006). In particular, we define the time series,of the difference between the pairs of the root-mean-square-errors for the two experiments. The sample mean,is typically different from zero. The test computes the likelihood that the true mean of the random variable sampled by

- The effective sample sizeis computed based on the assumption that
describes a first-order autoregressive process. The autocorrelation coefficient is computed byHere, and . If were zero, would equal *T*, but as the autocorrelation increases,decreases. [The sample size *T*has to be replaced by the effective sample sizebecause forecast errors at verification times separated only by 12-h tend to be strongly correlated.] - The test statisticis computed, where
is the sample variance for the time series , *t=*1, 2, … ,*T*). Under the assumption that both time series of root-mean-square errors sample normally distributed random processes, when the true mean of the random process sampled byis zero, the random variable *z*is normally distributed with standardized statistics. - The likelihood
*l*that the particular value of*z*we obtain for a given set of analysis or forecast errors is from a standardized normal distribution is determined (e.g., with the help of a table of cumulative probabilities for the standardized normal distribution). - The difference between the accuracy of the forecasts for the two configurations is deemed statistically significant at the (1 −
*L*) level, if*l*≤*L*. For instance, the difference between the two time series of root-mean-square errors is statistically significant at the 99% level if ‖*z*‖ ≥ 2.58 and only at the 90% level if ‖*z*‖ ≥ 1.65.

*t*. Then, we computeat each grid point for both experiments. We denote the value of

_{i}### c. LETKF parameters

Most of our choices of the LETKF parameters, which define the localization for the conventional observations and the variance inflation for the state vector components, are discussed in section 2. Since observation density has a large influence on the optimal level of variance inflation (e.g., Satterfield and Szunyogh 2011), retuning the variance inflation factor, *ρ*, for the configurations of the data assimilation system, which assimilate the AMSU-A observations, would likely lead to a further increase of the accuracy of the analyses and the ensuing forecasts. Notwithstanding the potential positive effects of retuning *ρ* on the accuracy of the analyses in the experiment that assimilates the AMSU-A observations, for the sake of a conservative comparison to the results of the reference experiments, we opt not to retune *ρ*.

We find that the ensemble of bias-correction parameters collapses for some of the bias parameters unless we apply an additional inflation, with coefficient *ρ _{β}* > 1, to the

*ρ*= 1.07 to the ensemble perturbations of all

_{β}*Q*bias parameters, we can avoid a collapse of the ensemble for all

*Q*bias parameters.

*p*

_{1}) and the scan angle (

*p*

_{2}), that is, the bias-correction term is estimated bySince we estimate all bias parameters adaptively

^{5}and the number of bias parameters for each channel is (

*I*+ 1) = 3, the total number of bias parameters that we estimate is

*Q*= (

*I*+ 1) ×

*J*= 24. The areal average values of the bias-correction parameters are obtained from the local values by averaging them over all observation locations in three zonal latitude bands (90°–30°S, 30°S–30°N, 30°–90°N) using Eqs. (18) and (19). We chose this particular set of predictors and averaging regions based on a large number of numerical experiments with different predictors suggested in the literature. We define the initial value of the estimates of the

*Q*bias parameters by a set of random samples from a standardized normal distribution.

We find that a 60-member ensemble provides a sufficiently large number of degrees of freedom to obtain accurate estimates of the bias parameters and the atmospheric state. We also find that a cutoff value of *η* = 0.8 provides a performance that is similarly good to that for lower values, but at a lower computational cost.

## 5. Results

### a. Analysis and forecast verification results

Figures 3 and 4 show the time evolution of the root-mean-square error

We show the time series of root-mean-square error only for the SH extratropics because this is the region where the difference between the time series from the different experiments is statistically highly significant (at the 99% at most pressure levels). The full vertical profile of

The geographical distribution of the improvement in the 48-h forecasts is shown in Figs. 7 and 8. The only difference between these two figures is that, in Fig. 8, the difference between the forecast errors is not shown at locations where it is not statistically significant at the 90% level. (We include the figure showing unfiltered results to illustrate the effect of filtering based on statistical significance.) This pair of figures indicates that the analyses are improved over the oceans, with the largest improvement between and east of Cape Horn and the Antarctic Peninsula, while the analyses are degraded over Antarctica. The statistically significant improvement in the surface pressure forecasts indicates that the ensemble-based estimate of the background error covariance matrix provides useful information about the cross correlation between the surface pressure and the atmospheric state variables that directly affect the radiative transfer.

In summary, we can conclude that the assimilation of radiance observations with our proposed strategy is a source of analysis improvement that leads to significant forecast improvement in the SH midlatitudes, which are especially large in the upper troposphere and the stratosphere.

### b. The behavior of the bias parameters

To illustrate the behavior of the bias-correction terms, we choose two channels: one that has the average peak sensitivity in the lower troposphere (channel 4) and one that is most sensitive, on average, to the atmospheric conditions in the stratosphere (channel 11). (See Table 2 for typical pressure levels of peak sensitivity for the different channels.) We investigate the time evolution of the bias-correction terms for these two channels in the extratropical SH region (Fig. 9).

The time evolution of the bias-correction term is strikingly different for the two channels: while for the channel with peak sensitivity near the surface (channel 4), the time evolution of the bias correction is characterized by a diurnal oscillation around a nearly constant level, for the channel with peak sensitivity in the stratosphere (channel 11), the value of the bias correction shifts from a negative value (about −0.7 K) to a positive value (about 0.3 K). To better understand the behavior of the bias correction, in Fig. 10 we show the contribution of the three predictors

Finally, Fig. 11 shows the analysis and the spread of the analysis ensemble for the bias parameters *ρ* and *ρ _{β}*, the domain average of the ensemble spread is stable and sufficiently large to allow for continuous changes in the bias parameters. The temporal variability of the bias parameters is clearly larger for channel 11 than for channel 4, indicating that the larger variability in the contribution of the different bias-correction terms observed in Fig. 10 for this channel is the result of changes in the estimates of the bias parameters.

## 6. Conclusions

In this paper, we tested the techniques developed by Fertig et al. (2007, 2009) for the assimilation of satellite radiance observations in a realistic setting for the first time. The results suggest that the tested strategy can extract useful information about the atmospheric state, especially in regions where the satellite radiance observations are the dominant source of observational information. While our initial results with the ensemble-based bias corrections are promising, several important challenges remain to be addressed. Most importantly, augmenting the local state vector with the bias components significantly increases the dimension of the local state vector (e.g., we added *Q* = 24 extra components to the 4 or 5 components of the state vector in this paper). Since increasing the number of state vector components inevitably increases the dimensionality of the space of uncertainty, we expect that increasing the number of satellite channels will require increasing the number of ensemble members. The fact that we were able to obtain good results without increasing the ensemble size, while increasing the dimension of the local state vector by a factor of 7, is promising for the future, but does not guarantee that the ensemble size would remain manageable in case of a further massive increase of the number of bias parameters.

Our approach for bias correction, which is based on a simultaneous estimation of the state and bias parameters based on an ensemble, is not the only way to estimate and to correct for the bias in the radiance observations in an ensemble-based data assimilation system. Fertig et al. (2009) also introduced, in addition to the algorithm tested here, a two-step approach in which first the bias-correction parameters are estimated with an ensemble-based scheme and then the state is estimated in a subsequent step. Moreover, Miyoshi et al. (2010) uses a deterministic approach to obtain a single estimate of each bias parameter simultaneously with the ensemble-based estimate of the atmospheric state. A comparison of the different approaches for observation bias correction in ensemble-based data assimilation systems should be the subject of future research.

We thank Ross N. Hoffman of AER, Inc., for his helpful comments on the paper. The thorough reviews by two anonymous reviewers significantly helped to improve the presentation of our results. The work of J. A. on this project was partially funded by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, National Council for Scientific and Technological Development of Brazil) under Grants PDE 201185/2005-9 and PU 484245/2006-6. Further funding for this research was provided by NASA (Grants NNX08AD40G, NNX07AV45G, and NNX08AD37G) and NSF (Grants ATM0722721 and ATM0935538).

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^{1}

The SSI was the operational three-dimensional variational (3D-Var) system of NCEP until April 2007.

^{2}

This tapering function was introduced in Hunt et al. (2007) and tested in Szunyogh et al. (2008), where it was found that the tapering (i) had no effect on the accuracy of the analyses and the short-term forecasts in densely observed regions, but (ii) improved the accuracy in sparsely observed regions by making the spatial changes in the weight vector

^{3}

The small (less than 0.5%) difference between the number of observations of the zonal and meridional components of the wind is a result of our approach of treating the two components of the wind as independent scalar variables in the data assimilation process (e.g., we perform quality control of the two wind components independently).

^{4}

In 2004 NCEP assimilated AMSU-A observations from the *NOAA-15* and *NOAA-16* satellites.

^{5}

We note that some organizations (e.g., NCEP) estimate the scan angle bias predictor by a separate offline procedure.