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--> --> --> -->2.1. Desroziers method
Observation errors play a vital role in a data assimilation system. According to the best linear unbiased estimate, the analysis x a derived by the assimilation system can be expressed as the combination of the information from the background x b and observations y: \begin{equation} {x}_{\rm a}={x}_{\rm b}+{BH}^{\rm T}({R}+{HBH}^{\rm T})^{-1}[{y}-H({x}_{\rm b})] , \ \ (1)\end{equation} where R is the observation-error covariance matrix, B is the background-error covariance matrix, H is the observation operator that projects the background to observation space, and H is the linearized observation operator. It is obvious that R and B are used to determine the weighting of the observation and background in the analysis.In practice, the observation errors include systematic errors and random errors. Systematic errors refer to the biases in the data, and there are usually significant biases in satellite observations. To assimilate biased observations, a bias correction procedure is applied to correct them. Random errors are introduced by both the observations themselves and the observation operators. The main sources of random errors associated with the observation operator can be categorized as the forward model error, representativeness error, and preprocessing error. Detailed discussions of these error types can be found in (Daley, 1991) and (Janji? et al., 2017). All of these error sources may result in a correlation between the different observation positions and channels (Weston et al., 2014). Therefore, we need to study the possible structure of the off-diagonal observation-error covariance matrix to ensure that the observations are assimilated in an optimal way.
Because the true state is impossible to obtain, the covariance matrix of the observation errors is usually obtained by statistical methods. Posterior diagnostics can be used to estimate the observation-error covariance matrix (Desroziers et al., 2005). The diagnosed posterior observation-error covariance matrix $\tilde{R}$ can be expressed as \begin{equation} \tilde{{R}}=E({d}_{\rm o,a}^{\rm T}{d}_{\rm o,b}^{\rm T}) , \ \ (2)\end{equation} where do,a=y-H(x a) is the difference between the observation and analysis, d o,b=y-H(x b) is the difference between observations and background. When the observation and background errors are independent and R and B in Eq. (1) are exactly correct, the posterior $\tilde{R}$ is equal to the prior R.
The diagnostic results cannot be an exact representation of the true error characteristics, leading to an inaccurate observation- and background-error covariance matrix. (Waller et al., 2016b) reported that the diagnostics will underestimate the correlation length scale when the correlated errors are treated as uncorrelated. Additionally, the estimated correlation length scale will be overestimated when the assumed observation-error standard deviations are inflated. Because uncorrelated and inflated observation errors are used, the diagnostic deficiency will be partly offset. (Bormann, 2015) demonstrated that background-error dependence was relatively weak in estimating correlated errors. We also conducted experiments with different background-error specifications and found that the differences in the diagnosed results were rather small. Therefore, although the diagnostic has some limitations, it can still reflect the characteristics of observation-error correlation when the results are carefully interpreted.
In Eq. (2), d o,a and d o,b are assumed to be unbiased. Because a satellite data bias correction was performed in the assimilation system, this assumption was considered reasonable. However, we still subtracted the mean of the difference [as in Eq. (4.3) of (Stewart, 2010)] to ensure an unbiased result. (Waller et al., 2016a) suggested that the diagnostics were unaffected by biases in the observations when the mean residual values were subtracted. In addition, the matrix obtained by the diagnostic does not guarantee symmetry, and therefore we used $\tilde{R}=1/2(\tilde{R}+\tilde{R}^\rm T)$ to ensure a symmetrical result.
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2.2. FY-3A/B microwave sounders
The radiance data from the MWTS and MWHS onboard FY-3A/B were used in this study. The MWTS and MWHS instruments were placed on FY-3A with an equatorial local crossing time of 1015 (descending), and on FY-3B with an equatorial local crossing time of 1340 (ascending). Figure 1 displays the weighting functions of the channels of the two sensors. These instruments can obtain the vertical atmospheric temperature and humidity in different layers. The MWTS channels are similar to those of the MSU channels and channels 3, 5, 7 and 9 of the AMSU-A. Table 1 lists the channel characteristics of the MTWS, including the frequency, main absorber, and peak weighting function height. Channel 1 is the window channel used to obtain the surface temperature and emissivity. The other channels are used to detect atmospheric temperature in the troposphere and lower stratosphere. The MWHS and MHS have five similar channels, with the MWHS also including a dual-polarization channel at 150 GHz (channels 1 and 2), while channels 1 and 2 of the MHS are at 89 and 150 GHz. Channels 3-5 of the MWHS can provide information on mid- to upper-tropospheric humidity. The MWTS has 15 fields of view (FOVs) per scan line, a swath width of 2250 km, and a spatial resolution of 50 km at the nadir view. The MWHS has 98 FOVs, each with a swath width of 2700 km, and a spatial resolution of 15 km at the nadir view. The channel characteristics of the MWHS are shown in Table 2. The satellite data used in this paper were derived from the China Meteorological Data Service Center (CMDC, http://data.cma.cnhttp://data.cma.cn).Figure1. Weighting functions of MWTS channels 1-4 (red lines) and MWHS channels 1-5 (blue lines). The pressure levels (gray horizontal lines) for the background are shown.
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2.3. WRFDA three-dimensional variational assimilation system
The three-dimensional variational assimilation system (3DVar) provided by the WRFDA (version 3.8.1) (Barker et al., 2012) was used to assimilate all non-window channels of the MWTS and MWHS (channels 2-4 for MWTS and channels 3-5 for MWHS). The radiative transfer model used to simulate brightness temperature was RTTOV (version 11.3) (Hocking et al., 2013). The assumed observation errors of the MWTS and MWHS were determined empirically by referring to the AMSU-A and MHS assumed observation errors.The background at each time was interpolated using NCEP final (FNL) analysis data with a horizontal resolution of 1°× 1° for the same time. Because FNL data are observational data collected from many sources (but not FY-3A/B), only the MWTS and MWHS non-window channels were assimilated in the experiments. The background-error covariance was estimated by the National Meteorological Center method (Parrish and Derber, 1992) using the differences of the 12 and 24 h forecasts valid at 0000 and 1200 UTC over a period of one month. All assimilated observations passed through quality control procedures, including a limb check (reject observations at limb scan positions), cloud check, gross check, first-guess check, and others. Observations strongly affected by cloud and precipitation were eliminated by rejecting observations for which the absolute value of the window channel innovation exceeded a threshold (3 and 5 K for channel 1 of the MWTS and MWHS, respectively). Only observations over sea were assimilated (as shown in Fig. 2). A variational bias correction was applied to remove systematic errors before the radiance data were assimilated, in which a constant component and seven state-dependent predictors (1000-300 hPa thickness, 200-50 hPa thickness, surface skin temperature, total column precipitable water, scan position, and the square and cube of scan position) and their coefficients were included. The WRFDA started running two days before the experiment was initiated. The updated bias correction coefficients were used as the input coefficients for the next cycle. Twenty cycles per analysis time were conducted to spin up these coefficients. The initial coefficients at the beginning of the cycles were provided by the statistics of the initial MWTS and MWHS data. Figure 3 shows that the overall bias was close to zero after variational bias correction. Scan-dependent biases and geographical biases were also effectively removed (not shown). It has been shown that the assimilation of the MWTS and MWHS in the WRFDA with a similar set-up can have a positive impact on a forecast (Dong et al., 2014; Xu et al., 2016).
Figure2. Observation departures after bias correction for assimilated MWHS channel 4 observations onboard (a) FY3-A and (b) FY3-B located in the statistical domain on 10 February 2012. Histograms means show the distribution of sample numbers in the range of observation departures after bias correction. The binning width is 0.25 K.
Figure3. Distribution of sample number in the range of observation departures before and after bias correction for the FY-3A (a-c) MWTS and (d-f) MWHS. The binning width is 0.25 K.
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2.4. Experimental design
To diagnose the inter-channel and spatial correlation of the MWTS and MWHS observations, we selected observations and background data from 0000 UTC 10 February 2012 to 1800 UTC 24 February 2012 (four times per day). The domain used for the experiments (as shown in Fig. 2) consisted of 360× 300 horizontal grids, with a grid spacing of 30 km and 40 vertical levels up to 10 hPa (Fig. 1).Figure4. Diagnosed inter-channel correlation for assimilated MWTS channels.
In the WRFDA, the default thinning distance is 120 km for all satellite radiance data. Because the horizontal resolutions of the two instruments used in this study were both greater than 120 km, the thinning process prevented the estimation of correlations at distances of less than 120 km. This process had little effect on the diagnosed observation-error characteristics, although it resulted in a sub-optimal analysis (Waller et al., 2016a). As a result, the experiments were performed using unthinned data to obtain a complete understanding of the correlation structure. We also conducted the experiments using thinned data, and similar statistics were obtained.
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3.1. MWTS
Figure 4 shows the diagnosed observation-error standard deviations along with the assumed error standard deviations in the WRFDA. The statistical results show that the error standard deviations of the MWTS were all smaller than those of the WRFDA and were assigned as values greater than 0.25 K for all assimilated channels. Because an artificial inflation was conducted for the observation-error standard deviations to account for the unconsidered correlated errors, it was expected that the standard deviations of the observation-error used in the assimilation system would be much larger than the statistical results. The same sensor onboard different satellites may still have different error characteristics. Although FY-3A and FY-3B use the same microwave temperature sensor, the error standard deviations of the FY-3A MWTS differed from that of the FY-3B MWTS. It was apparent that the observation-error standard deviations of the FY-3B MWTS were greater than that of the FY-3A MWTS, especially for channel 3.Figure 5 shows the diagnosed inter-channel correlation for the FY-3A and FY-3B MWTS assimilated channels. The absolute values of all the correlation coefficients were less than 0.2, except for the correlation between channels 2 and 3 on FY-3B, which had a coefficient of -0.28. The inter-channel correlations may be caused by a combination of the radiative transfer model, representativeness error, and quality control procedures. Because the same sensor had the same parameters in the radiative transfer model and also the same representativeness error, the difference in the inter-channel correlations for different satellites may be related to the quality control procedures.
Figure5. Diagnosed and original observation-error standard deviations for assimilated MWTS channels.
Figure 6 shows the spatial correlation of the MWTS channels and the number of observations used to calculate them. The amount of assimilated radiance data varied because of the omissions during the quality control process, with channel 4 having more observations assimilated than others. To better reflect the spatial correlation, the observation samples were binned with an interval of 60 km, which was close to the horizontal resolution of the MWTS. Although the spatial correlation of FY-3A and FY-3B was clearly different, all channels had a strong correlation (>0.2) within 120 km. The correlation length scale of some channels reached 180 km (FY-3A channel 3 and FY3B channel 2). The spatial correlation length scale was greatest for channel 3, with a value up to 300 km. This was very different from the spatial error correlation for AMSU-A reported by (Bormann and Bauer, 2010), which was less than 0.2 even at the least distance. There were two likely reasons for this. One was the difference between the AMSU-A and MWTS observations and the other was the use of different assimilation systems, which would lead to differences in the quality control process and radiative transfer model. The correlation length scales of the MWTS were consistent with the default thinning distance of 120 km. This suggests that it is reasonable to use this thinning distance to offset the spatial observation-error correlation.
Figure6. Diagnosed spatial observation-error correlation (lines) and the number of observation samples (bars) for the MWTS onboard (a-c) FY-3A and (d-f) FY-3B.
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3.2. MWHS
The standard deviations of the MWHS observation-error are shown in Fig. 7. The observation-error standard deviations of channels 3 and 5 for FY-3A/B were similar at about 1.3 and 1.1 K, respectively. The difference for channel 4 was large, with a standard deviation of 1.35 K for FY-3A, while it was less than 1.05 K for FY-3B. As with the results for the MWTS, due to the use of the inflated error standard deviations in the assimilation system, the statistical error standard deviations were less than the assumed errors in the assimilation system. Because the MWHS channels had larger instrumental noises than those of the MWTS, the error standard deviations of the MWHS were much larger.Figure7. As in Fig. 4 but for the MWHS.
The inter-channel correlation of the MWHS is shown in Fig. 8. The figure shows that the channels of the MWHS had significantly correlated errors, especially between adjacent channels (i.e., channels 3 and 4, and channels 4 and 5). The correlations between the two satellites were also different. For adjacent channels, the correlations for FY-3B were greater than those for FY-3A, which were 0.6 and 0.4, respectively. Figure 9 shows the MWHS spatial observation-error correlations calculated with an interval of 15 km and the observation samples used. Because the MWHS had a higher spatial resolution and a larger scanning angle than the MWTS, it had many more assimilated observations than the MWTS. The MWHS had a significant spatial error correlation when the distance of separation was within 60 km; the distance was about four times the length of the observing resolution. The default thinning distance of the MWHS in the WRFDA system is 120 km, which is greater than the spatial observation-error correlation length scales of the MWHS. This thinning distance will reject many uncorrelated observations; thus, the amount of data used was reduced and the information contained in the observations was lost. This suggests that it may well improve the present data thinning scheme. For example, using a smaller thinning distance or combining the present thinning scale with a suitable observation-error inflation may be a more reasonable approach.
Figure8. As in Fig. 5 but for the MWHS.
Figure9. As in Fig. 6 but for the MWHS.
Figure10. Statistical results at different scan positions for the FY-3B (a, b) MWTS and (c, d) MWHS. Panels (a, c) are the observation-error standard deviations for different channels, and (b, d) are the inter-channel observation-error correlations.
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3.3. Scan position and latitudinal dependence of error statistics
(Bormann and Bauer, 2010) showed that the error statistics obtained by the Desroziers method had an anisotropic characteristic at different scan positions and scan lines. (Waller et al., 2016a) found that the inter-channel observation-error statistics varied spatially, but their research domain was so small that it could not reveal the relationship between error statistics and the geographical locations of the observations. The anisotropy on different scan lines identified by (Bormann and Bauer, 2010) may also be due to the different latitude of the observing location. Therefore, it is necessary to study the difference in the statistical results in scanning position and geographical location.Figure11. Statistical results at different latitudinal bands for the FY-3B (a-c) MWTS and (d-f) MWHS. Panels (a, d) are the observation-error standard deviations, (b, e) are the inter-channel observation-error correlations, and (c, f) are the spatial observation-error correlation of MWTS channel 4 and MWHS channel 5, respectively.
Figure 10 shows the differences in the statistical results of the instruments in different scan positions. Except for some small changes in parts of the diagnosed results (error standard deviation of MWTS channel 2, and the inter-channel correlation of MWHS channels 3 and 4), the other error statistics, including the observation-error standard deviations, and inter-channel correlations, displayed a clear scan position dependence, especially for the MWTS. The error standard deviations in the limb position were larger than those near the nadir point. The inter-channel correlations showed a pronounced asymmetry and the correlations on the left of the nadir point differed from those on the right. MWTS channels 3 and 4 exhibited strong inter-channel error correlations at the beginning of several scan positions, but only a very small correlation in other positions. This oscillatory distribution of the diagnostic results was similar to the striped distribution of the statistics found by (Bormann and Bauer, 2010). It should be noted that the limb observations of each scan line were removed during the quality control process, and the limb observations in this study were not the observations made at the scan boundary. The scan position dependence of error standard deviations was mainly attributed to the design of the instruments, but the scan position dependence of error correlations may be due to the different performance of the radiative transfer model at different scan positions. The deficiency in the scan bias correction may also be the reason for this phenomenon. In addition, the results for FY-3A were slightly different from the results for FY-3B, which indicates that the scan position dependence of the error features was satellite-specific. To check the comparability, similar experiments using data from 10-25 August 2012 were conducted and similar characteristics were found.
Figure12. As in Fig. 11, but for the statistics obtained using the data for August.
To determine the performance of the statistical results obtained at different geographical locations, the research domain was divided into three regions: Northern Hemisphere (>10°N); Southern Hemisphere (<10°S); and equatorial zone (<10°N and >10°S). Figure 11 shows the difference in the diagnostics at different latitudinal bands. In terms of the observation-error statistics, the two sensors performed differently in the different areas. For the MWTS, the error standard deviations and spatial error correlation were similar in each region, but the inter-channel errors were latitude- and channel-specific. For the MWHS, the error standard deviations and inter-channel correlation of the Northern Hemisphere were significantly different from those of the Southern Hemisphere and equatorial regions. This may be due to the differences in weather conditions at different latitudes, because the time period we used to calculate the error characteristics was in the winter of the Northern Hemisphere and summer of the Southern Hemisphere. (Waller et al., 2014b) also found that the representativeness error was sensitive to the synoptic situation. To further verify the relationship between the statistical results and the weather conditions, we used the data from 10-25 August 2012 to recalculate the error correlations. During this period, the seasons in the Northern Hemisphere and Southern Hemisphere were the opposite of those in the previous experiment. The statistical results for all domains obtained by the data for this time period were basically the same as previous results (not shown), but the performance of the error statistics in different latitudinal bands was very different from the original results (Fig. 12). The inter-channel correlation between MWTS channel 3 and the other channels was very different from that of the original experiment in the equatorial region and Southern Hemisphere. The MWHS results changed slightly in the equatorial region, but the distribution in the Northern Hemisphere and Southern Hemisphere was almost the opposite of that in the original experiment. This further suggested that the latitudinal dependence of the error statistics was related to the difference in weather conditions. The observation error was associated with the nonlinear observation operator, which indicated that the observation error could be attributed to the initial state. In different seasons, the dominant synoptic situation in specific latitudinal bands was different. For example, in the summer, the subtropical high over the Northwest Pacific was the main synoptic situation in the Northern Hemisphere area of the domain. In this situation, the field was more homogeneous, and the atmosphere was relatively stable. However, in the winter, the atmosphere over the Northwest Pacific was more varied and less homogeneous. This difference may influence the representativeness errors because different weather conditions may contain different scales or features and processes that were represented in the observations and not in the background. The different behaviors of the Desroziers diagnostic could be an alternative explanation. For example, the reliability of the assigned background errors may be situation-dependent.
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3.4. Forecast experiments using a modified thinning scheme
The above results suggested that an improved thinning scheme was needed to improve the assimilation of the MWTS and MWHS. To test the forecast impact of the improved thinning scheme, we performed the following two data assimilation experiments: (1) using a default thinning scheme (the thinning distances were 120 km for both the MWTS and MWHS); (2) using a modified thinning scheme (the thinning distances were 120 km for MWTS and 60 km for MWHS). A five-day forecast was made after data assimilation for both experiments using WRF V3.7.Figure 13 shows the forecast verification of the two experiments against the analysis. More than 94% of the verification metrics were improved, with reductions in the forecast root-mean-square error of between 0.3% and 1.5% compared to the analysis. This indicates that using a modified thinning scheme improved the forecast accuracy for most variables and levels at different forecast lead times. Some verification metrics, such as the temperature and relative humidity at 500 hPa for the equatorial zone, showed degradations for the modified thinning scheme compared with the default scheme. This may be because the smaller thinning distance meant some poor-quality observations were assimilated. Another aspect to note was that the contribution from analysis errors may be significant and the forecast impact evaluated against analyses may not be an accurate indicator of forecast accuracy.
Figure13. Normalized differences of forecast root-mean-square errors between cases using the modified thinning and default thinning schemes at different latitudinal bands. The verification metrics used are geopotential height (H), temperature (T), and relative humidity (R) at (a) 850 hPa, (b) 500 hPa and (c) 200 hPa for 1-5-day forecasts. Negative values mean an improvement using the modified thinning scheme compared with the default.