It is widely accepted that the quality of background error covariance matrix (B Matrix) is crucial to every DA system, for its high relevance to NWP model’s systematic error. Before conducting experiments, we calculate B Matrix from 12-hour and 36-hour lead-time forecasts valid at 00:00 UTC and 12:00 UTC from Global Forecast System (GFS) operated by National Centers for Environmental Prediction (NCEP) [24] in January and June from 2017 to 2019 using NMC method [25] [26]. With a horizontal resolution of 1.25˚ × 1.25˚ and a vertical resolution of 31 levels from 1000 hPa to 1 hPa, this B Matrix can properly represent the error in initial condition generated by forecast model. In this study, B Matrices for January and June are generated respectively, as a series of monthly-separated B Matrix can better describe the sub-seasonal systematic error from NWP model than a unified one. Unlike B Matrix, the DA system inherits observational error covariance matrix (R Matrix) from open-accessed 1D-Var assimilation package released by Numerical Weather Prediction (NWP) Satellite Application Facility (SAF) in European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) [27] - [33]. This decision is made up for the reasons listed below: 1) observational error covariance matrix only comprises instrumental error and fast forward radiative transfer model error, and fast radiative transfer model error is relatively static compared to the error in NWP model; 2) instrumental error is supposed to be steady, as it is only related to instrument’s operating and healthy status.

Even though several Data Assimilation (DA) systems have the capability of assimilating infrared channel observations if the weighting function peak pressure level is above diagnostic cloud height over overcast areas [32] [34] [35] [36], the vertical resolution of GFS data used in this research is not sufficient enough to precisely diagnose cloud top height. Therefore, all experiments in this article will not assimilate any infrared observation over cloudy regions. To separate clear sky observation from cloudy ones, a cloud detection algorithm using equivalent cloud-top height and equivalent cloud cover calculated from the minimum residual method [37] is embedded in quality control process: if the equivalent cloud top height for an observation is above 700 hPa or the equivalent cloud cover is above 20%, cloud detection process will mark this observation as cloud-contaminated and reject it from data assimilation system.

The DA system used in the coming experiments runs a two-step quality control process to ensure the quality of observations before they are assimilated into initial condition. In the first step, Degree of information Freedom for Signal (DFS) [38] is calculated from following function:

D F S = H ′ ( x ) ⋅ B ⋅ ( H ′ ( x ) ) T ⋅ [ H ′ ( x ) ⋅ B ⋅ ( H ′ ( x ) ) T + R ] − 1 (1)

where H ′ is the Jacobian of fast forward radiative transfer model regarding initial condition x, R (B) is the background (observational) error covariance matrix, operator T stands for matrix transpose and operator ⋅ refers to dot production. Once finishing Equation (1) calculation, DA system initiates a channel screening process: if a channel observation’s value is smaller than 0.2, system will mark this observation as unqualified and discard it from data ingestion list. Based on the results from first step, the second step starts by calculating absolute brightness temperature difference between observation and simulation from initial condition. In this step, DA system eliminates the channel observation with an absolute difference larger than 1.0 K.

The primary function of a DA system is to determine the best estimation of initial condition (x) by minimizing the cost function based on quality-controlled observation [39] [40] [41] [42] listed below:

J ( x ) = 1 2 ( x − x b ) T ⋅ B − 1 ⋅ ( x − x b ) + 1 2 ( y − H ( x ) ) T ⋅ R − 1 ⋅ ( y − H ( x ) ) (2)

where H is a PC-based fast forward radiative transfer model which calculates simulated brightness temperature (BT) from initial condition. To find the minimum of J, DA system updates initial condition at each iteration ( n + 1 ) using the equations below (Rodgers 1976, 2000):

x n + 1 = x b + U − 1 ⋅ V (3)

U = B − 1 + [ H ′ ( x n ) ] T ⋅ R − 1 ⋅ H ′ ( x n ) (4)

V = [ H ′ ( x n ) ] T ⋅ R − 1 ⋅ [ ( y − H ( x n ) ) + H ′ ( x n ) ⋅ ( x n − x b ) ] (5)

where n stands for iteration steps. In this system, Havemann-Taylor Fast Radiative Transfer Code (HT-FRTC) [43] [44] is used to calculate H and H ′ , for that Principal-Component-based fast forward model is more computationally effective than other fast forward radiative transfer models, e.g. Radiative Transfer for TOVS (RTTOV) [45] [46] and Community Radiative Transfer Model (CRTM) [47]. Besides, the precision of HTFRTC is comparable to RTTOV and CRTM [48] in simulating full-spectrum hyperspectral infrared radiance or brightness temperature.

Mean Bias (MB) and Standard Deviation (SD) are the major variables in evaluating the quality of initial condition before and after assimilating hyperspectral infrared observation, which are widely used among various subjects [49]:

M B = ( ∑ n = 1 N ( x − O B S ) ) / N (6)

S D = s q r t ( ∑ n = 1 N [ ( x − O B S ) − M B ] 2 / N ) (7)

where OBS refers to conventional observations gathered by aircraft, ships, weather balloons, and surface stations. Statistically, mean bias is good at representing systematic bias and standard deviation is highly related to random bias.

By far, there are four types of hyperspectral infrared Michelson interferometer operating on different polar-orbiting satellites: Atmospheric Sounding Interferometer (IASI) [50], Cross-track Infrared Sounder (CrIS) [51], Russian advanced infrared atmospheric sounder (IKFS-2) [52], Hyperspectral Infrared Atmospheric Sounder (HIRAS) [53]. Among these instruments, observations from CrIS and IASI are more widely accepted and used by operational NWP centers until a recent successful attempt on assimilating IKFS-2 observation into ECMWF’s operational NWP system [54]. For this reason, experiments in this article only accept observations from IASI and CrIS instruments, which can be downloaded from NOAA Comprehensive Large Array-data Stewardship System (CLASS). Before initializing the DA system, the radiance observations are converted from radiance space to brightness temperature space: although the public has already accepted the concept that using radiance can improve data assimilation system’s accuracy, as instrumental noise is more constant in radiance space, using brightness temperature can help DA system overcome slow convergence problem while minimizing the cost function and thus increase computational efficiency because brightness temperature space is more linear than radiance space. 12-hour lead time forecast products from GFS in January and June 2020 valid at 00:00 UTC and 12:00 UTC will provide initial condition for the DA system since B Matrix is derived from historical GFS data. In performance evaluation, temperature and water vapor observation from radiosonde, Aircraft Meteorological DAta Reports (AMDAR), ship, and surface station [55] will help to quantify performance differences among upcoming experiments.

Before starting the experiments, we need to establish comprehensions of the spatial intersection between CrIS data coverage and conventional observation. As can be seen in Figure 3, around 00:00 (23:30-00:30) UTC and 12:00 (11:30-12:30) UTC, CrIS observation mainly covers the area from 45˚W to 45˚E and from 135˚E to 135˚W. In the CrIS data coverage area, most conventional observations locate in Europe, South Africa, Central Pacific, and Alaska. In marine region, the amount of conventional observation is far less than that over land in January (Figure 3(a)), this situation intensifies significantly in June (Figure 3(b)).

By comparing temperature mean bias departure profile from FullChannel experiment (green line) and SelectedChannel (red line) experiment in January (Figure 4(a)), full-channel assimilation method holds a better skill in reducing systematic error (mean bias) in initial condition than selected-channel assimilation in middle troposphere (700 - 400 hPa) and upper troposphere (400 - 100 hPa). On average, temperature mean bias reduction in FullChannel experiment against SelectedChannel experiment is 0.046 K in middle troposphere and 0.048 K in upper troposphere, but this improvement shrinks to 0.005 K in low troposphere (1000 - 700 hPa). Situations in June (Figure 4(b)) are fairly the same as January, except for a few changes: FullChannel experiment’s improvement against SelectedChannel experiment drops to 0.028 K in middle troposphere and 0.026 K in upper troposphere, while an additional 0.008 k performance increase can be found in low troposphere compared to January. From the results shown in Figure 4(c) and Figure 4(d), both full-channel and selected-channel assimilation methods can reduce the temperature field’s random error (standard deviation) in initial condition, but the magnitude of improvement is different. In low troposphere, FullChannel experiment is ahead of SelectedChannel experiment by 0.012 K in January and 0.014 K in June, this precedence continues to exist in middle troposphere with a slight magnitude change: 0.011 K in January and 0.025 K in June. Performance of FullChannel experiment is still in the lead (0.015 K) in January over upper troposphere, but get surpassed by SelectedChannel experiment in June with a 0.0064 K decrease. Results from Figure 4(e) and Figure 4(f) illustrate that most of the improvements that water vapor field receives from hyperspectral IR data assimilation locate in low troposphere and full-channel assimilation method eliminates another 13.17 mg/kg (27.02 mg/kg) from mean bias in January (June) regarding selected-channel assimilation method, while neutral and negative impact can be detected in middle and upper troposphere. Although no positive impact arises after assimilating hyperspectral IR observation in middle and upper troposphere, uncertainty in full-channel assimilation method’s performance enhancement makes the conclusion more unpredictable: in January, FullChannel experiment lags SelectedChannel experiment by 5.56 mg/kg in middle troposphere and 3.25 mg/kg in upper troposphere; but in June, FullChannel experiment is in the lead with 1.98 mg/kg in middle troposphere and 0.95 mg/kg in upper troposphere. Both FullChannel and SelectedChannel experiment contributes positive impact on random error cancellation to water vapor field, according to Figure 4(g) and Figure 4(h). With an extra 40.29 mg/kg (14.32 mg/kg) improvement in January (June) within low troposphere, full-channel assimilation method seizes a higher performance compared to selected-channel assimilation method, and so is in middle troposphere, where FullChannel experiment’s improvement over SelectedChannel experiment is 16.96 mg/kg in January and 10.77 mg/kg in June in an average scale. Different from the results in low and middle troposphere, full-channel assimilation method experiences an upper-troposphere performance degradation in June, as SelectedChannel experiment exceeds FullChannel experiment by 2.37 mg/kg.

Profiles in Figure 4 demonstrate hyperspectral IR data assimilation’s impact on improving temperature and water vapor field’s accuracy in initial condition as well as full-channel assimilation’s performance enhancement over selected-channel assimilation in spatial average scale. Still, regional performance difference between full-channel and selected-channel assimilation methods remains uncertain. To disclose this, we calculate the standard deviation in a 7.5˚ × 7.5˚ horizontal resolution by organizing conventional observation and initial condition (before and after data assimilation) to the same projection. Figure 5 shows the spatial distribution standard deviation departure between FullChannel and SelectedChannel experiment in integral (a), low (b), middle (c), and upper (d) troposphere. In Figure 5(a), FullChannel experiment suffers from performance decrease in high latitude region (60˚N - 90˚N and 60˚S - 90˚S) comparing to SelectedChannel experiment, this symptom may result from underestimation of surface skin temperature in initial condition [56]. Due to a lack of conventional observation, FullChannel experiment suffers from a performance decrease in central and northern Africa. Beneath 400 hPa in mid-latitude region, standard deviation in FullChannel experiment is approximately 0.01 K higher than that in SelectedChannel experiment, comparing to SelectedChannel experiment. On account of conventional observation shortage in upper troposphere, the spatial variation of standard deviation departure is relatively higher than the other parts in the troposphere. But in southern and western Europe, where the observation

amount is the highest, marginal enhancement from FullChannel experiment in reducing standard deviation continues persisting. In Figure 6, the spatial distribution of water vapor standard deviation departure is calculated using the same approach as is used in Figure 5. The result in Figure 6(a) shows that performance deficiency in full-channel assimilation over high latitudes (compared to selected-channel assimilation) still exist in the water vapor field, but in low (Figure 6(b)) and middle troposphere (Figure 6(c)), symptom alleviates remarkably compared to the situation in upper troposphere (Figure 6(d)). Meanwhile, in areas where a sizeable amount of conventional observation locates, e.g. Europe, the performance of FullChannel experiment is always in the lead.

From the results listed above, we can conclude that: 1) both full-channel and selected-channel assimilation method can improve the accuracy of initial condition by reducing temperature field’s systematic and random error, together with the random error in water vapor field; 2) generally, the performance of full-channel assimilation overpasses selected-channel assimilation in correcting low and middle troposphere temperature and water vapor field. To find the reason for the conclusions, we calculate the weighting function and DFS for each channel used in FullChannel experiment based on the average atmospheric variable profile from ECMWF [57]. By comparing the DFS for temperature (Figure 7(a)) and water vapor (Figure 7(b)) of each channel used in FullChannel experiment (dots) and SelectedChannel experiment (the dots that shares the same

wavenumber with blue triangles), we could find out that the DFS for channels used in selected-channel assimilation is always higher than 0.1 and most of these channels are assimilated in both FullChannel and SelectedChannel experiment, while some of these channels will be discarded by quality control process in the DA system. In addition to the channels used in SelectedChannel experiment, FullChannel experiment also assimilates other channels rejected by channel selection scheme whose DFS is normally higher than 0.38, and a considerably large amount of them are located in LWIR region, especially atmospheric window bands. Plus, weighting function peak pressure level in Figure 7(c) and Figure 7(d) indicates that these additional channels assimilated in FullChannel experiment are partial to modify temperature and water vapor field in low and middle troposphere. This could explain why improvement from full-channel assimilation locate in middle and low troposphere: as so many high-DFS channels carrying useful temperature and water vapor information in middle and low troposphere are assimilated by DA system, modified initial condition from FullChannel experiment is supposed to have a better performance than SelectedChannel experiment in middle and low troposphere.

To validate the conclusions drawn from CrIS experiments Section, supplementary experiments that assimilate IASI data are conducted in this section. From the spatial distribution of conventional observations amount in Figure 8, what we can confirm is most observations are located along the coastal area in far east Asia, eastern Australia, Brazil, and southern England. Amount-wise, observations used in evaluating initial condition’s accuracy after IASI assimilation are not as sufficient as are used in CrIS Experiments, as the maximum observation amount within a 2.5˚ × 2.5˚ grid is around 100, while this amount can be as high as 200 in CrIS experiment results evaluation. Thus, results from IASI data experiment and CrIS data assimilation may slightly differ from each other.

Profiles in Figure 9 presents the mean bias and standard deviation departure in initial condition between FullChannel and SelectedChannel experiment after assimilating IASI observation. In low troposphere, temperature field accuracy from FullChannel experiment is higher than SelectedChannel experiment, as the mean bias from FullChannel experiment is 0.005 K (0.009 K) smaller than SelectedChannel experiment in January (June) and the standard deviation is 0.012 K (0.014 K) smaller respectively. Below 850 hPa, full-channel assimilation experiences an accuracy loss on mean bias reduction (Figure 9(a) and Figure 9(b)), but it still keeps high proficiency on standard deviation reduction throughout low troposphere. FullChannel experiment’s mean bias (0.074 K in January and 0.077 K in June) and standard deviation (0.025 K in January and 0.018 K in June) improvements against SelectedChannel experiment in middle troposphere indicate full-channel assimilation is better at diminishing temperature error in initial condition than selected-channel assimilation method, so is in the upper troposphere, where the additional mean bias (standard deviation) reduction is 0.041 K (0.037 K) in January and 0.053 K (0.042 K) in June. In the water vapor field, most of the positive impact concentrates in low and middle troposphere. In January, FullChannel experiment obtains 5.73 mg/kg (12.06 mg/kg) mean bias

decrease in low (middle) troposphere over SelectedChannel experiment, but this number drops to −7.62 mg/kg in upper troposphere. The same situation happens in June as well, but with a subtle magnitude change: 45.93 mg/kg (48.13 mg/kg) decrease against SelectedChannel in low (middle) troposphere and 12.95 mg/kg increase in upper troposphere. Unlike the negative impact from IASI assimilation in mean bias reduction over upper troposphere, FullChannel experiments can diminish water vapor standard deviation more effectively than SelectedChannel experiment. Same as mean bias, most of the standard deviation improvement from full-channel assimilation concerning selected-channel assimilation stays in low and middle troposphere, while only marginal improvements can be detected in upper troposphere: in January, 40.39 mg/kg (16.94 mg/kg) improvements are found in low (middle) troposphere, but only 8.64 mg/kg improvement locates in upper troposphere; even though the magnitude levels up to 9.08 mg/kg in June in upper troposphere, it still falls behind low and middle-level improvement (32.58 mg/kg and 15.32 mg/kg).

Results from standard deviation departure between full-channel and selected-channel assimilation of IASI data share the same conclusion with CrIS experiments: 1) performance of full-channel assimilation over high latitude area is lower than selected-channel assimilation (Figure 10(a)); 2) most of the mid-latitude performance degradation in full-channel assimilation occurs in low troposphere over East Asia (Figure 10(b)), but this phenomenon alleviates in middle troposphere (Figure 10(c)) and upper troposphere (Figure 10(d)). Figure 11 shows the spatial distribution of water vapor standard deviation departure between FullChannel and SelectedChannel experiment, From which we can conclude: 1) standard deviation from FullChannel experiment is lower than SelectedChannel in the areas with accumulated observation amount larger than 20; 2) latitude-related performance variation is highly detectable in low troposphere than the other level in the troposphere.

Based on the results listed above, it is straightforward to conclude that both full-channel and selected-channel assimilation method can improve the accuracy of temperature and water vapor field in initial condition, and full-channel assimilation method is more skillful than selected-channel at reducing initial condition’s systematic and random error in low and middle troposphere. Furthermore, evidence from DFS value and weighting function peak pressure level of each channel used in IASI data assimilation showed in Figure 12 are in accordance with the ones disclosed in Figure 7, which provides effective supplementary proof that atmospheric window channels in hyperspectral IR observation can help data assimilation system improve temperature and water vapor field’s accuracy in initial condition.