# FuXi-DA: A generalized deep Learning data assimilation framework for
assimilating satellite observations

Xiaoze Xu1,2 , Xiuyu Sun222footnotemark: 2, Wei Han3, Xiaohui Zhong4,2, Lei
Chen2, Hao Li4,233footnotemark: 3

1School of Atmospheric Physics, Nanjing University of Information Science and
Technology
2Shanghai Academy of Artificial Intelligence for Science (SAIS)
3CMA Earth System Modeling and Prediction Centre (CEMC), China Meteorological
Administration
4Artificial Intelligence Innovation and Incubation Institute, Fudan University
This work was completed while Xiaoze Xu was interning at SAIS.These authors
contributed equally to this work<EMAIL_ADDRESS><EMAIL_ADDRESS>authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Data assimilation (DA), as an indispensable component within contemporary
Numerical Weather Prediction (NWP) systems, plays a crucial role in generating
the analysis that significantly impacts forecast performance. Nevertheless,
the development of an efficient DA system poses significant challenges,
particularly in establishing intricate relationships between the background
data and the vast amount of multi-source observation data within limited time
windows in operational settings. To address these challenges, researchers
design complex pre-processing methods for each observation type, leveraging
approximate modeling and the power of super-computing clusters to expedite
solutions. The emergence of deep learning (DL) models has been a game-changer,
offering unified multi-modal modeling, enhanced nonlinear representation
capabilities, and superior parallelization. These advantages have spurred
efforts to integrate DL models into various domains of weather modeling.
Remarkably, DL models have shown promise in matching, even surpassing, the
forecast accuracy of leading operational NWP models worldwide. This success
motivates the exploration of DL-based DA frameworks tailored for weather
forecasting models. In this study, we introduces Fuxi-DA, a generalized DL-
based DA framework for assimilating satellite observations. By assimilating
data from Advanced Geosynchronous Radiation Imager (AGRI) aboard Fengyun-4B,
FuXi-DA consistently mitigates analysis errors and significantly improves
forecast performance. Furthermore, through a series of single-observation
experiments, Fuxi-DA has been validated against established atmospheric
physics, demonstrating its consistency and reliability.

_K_ eywords Weather Prediction $\cdot$ Data Assimilation $\cdot$ Deep Learning
$\cdot$ Satellite Data Assimilation

##  1 Introduction

Accurate weather forecasts are crucial for saving lives, emergency management,
mitigating disaster impacts, and preventing economic losses due to severe
weather events. At present, weather forecasts primarily rely on numerical
weather prediction (NWP) models [1]. These traditional models generate weather
forecasts by solving the governing partial differential equations based on the
current state (called the analysis) of atmosphere and surface as initial
conditions. Since 2022, a new generation of deep learning (DL) models for
medium-range weather forecasting has been recognized for their potential in
medium-range weather forecasting, demonstrating forecast performance
comparable to leading NWP models [2, 3, 4, 5]. Weather forecasting is
essentially an "initial value" problem, where the greatest source of
uncertainty is the initial conditions [6]. Therefore, the accuracy of these
initial conditions is of paramount importance, regardless of the forecasting
method used [7, 8]. Currently, both NWP and DL-based weather forecasting
models depend on the analysis, which is derived from a combination of NWP
output and observation through data assimilation (DA) systems, ensuring the
highest possible accuracy of the initial conditions. The DA systems establish
a foundation for reliable weather forecasting and are equally important as
weather forecasting models.

Over the past few decades, advancements in NWP models and DA systems have
significantly improved the accuracy of weather forecasts [1]. Modern
operational DA systems need to process a massive number of observations,
including satellite data, radiosondes, aircraft and surface observations
within a one-hour time window to produce the analysis for forecasting models
[9]. Despite these advances, the development of DA systems has faced several
challenges. One primary challenge is the significant volume of observations,
many of which are compromised by their inadequate quality [10, 11]. This
issue, coupled with the extensive computational resources and time required
for data processing, forces operational DA systems to find a balance between
increasing the volume of assimilated observation and maintaining the
timeliness of forecasts. For instance, while the European Centre for Medium-
Range Weather Forecasts (ECMWF) receives 800 million observations daily, only
60 million of these pass the quality control for integration into the
Integrated Forecasting System (IFS) [12]. Moreover, satellite observations are
predominantly assimilated under clear-sky conditions, neglecting valuable
cloud and precipitation data, resulting in the use of only 5–10% of total
satellite data in global weather forecasts [1, 13]. Remarkably, this small
fraction contributes to the majority of relevant information for weather
forecasts [1, 14]. Another challenge lies in the complexity of traditional DA
workflows, which results in substantial delays from the acquisition of new
observations to their operational application. The ever-increasing volume of
available observations amplifies this challenge, posing greater scientific
challenges for DA systems. Notably, satellites measure variables such as
radiances, brightness temperatures, or reflectivities, which do not directly
correspond to model variables like temperature, humidity or ozone.
Consequently, observation operators must be developed to convert model
variables into model equivalents of the observed data for comparison, a
process that not only must be tailored for each new type of observation [14],
but also introduces additional errors [15]. Lastly, widely adopted DA methods,
such as the four-dimensional variational (4D-Var) [16] and ensemble-
variational (En-Var) method [17] are computationally expensive. These
challenges underscore the need for continuous innovation in DA techniques to
enhance the efficiency and accuracy of weather forecasting in the face of
growing data volumes and computational demands.

In recent years, DL models have demonstrated remarkable potential across
various aspects of weather and climate modelling, catalyzing numerous efforts
to integrate DL models with DA techniques [18]. These efforts have largely
focused on conducting proof-of-concept studies within simplified dynamical
systems, applying DL models to address specific challenges within the
traditional DA framework. Notable examples include the Lorenz 63 and Lorenz 96
systems [7, 19], as well as the 2-dimensional (2D) shallow water equations
[20]. The scope of these applications encompasses the development of inverse
observation operators [21] for mapping observational data to physical states,
the estimation of error covariance matrices [22, 23], the joint training of
surrogate models for the dynamic system along with DA problem solver [24, 25,
26], and the derivation of tangent-linear and adjoint models [27]. Despite
these achievements, a considerable gap exists between the simplified models
used in these studies and actual weather systems in terms of the complexity
and data dimensions. This discrepancy underscores the urgent need for direct
application of DL models to real-world weather systems. Recent studies have
shown that DL models can match the forecasting prowess of NWP models with
significantly lower computational costs [28], opening new avenues for DA
development. However, the operational deployment of such DL models still rely
on the analysis generated using traditional NWP models and DA systems. The
optimization of this analysis data is for NWP model forecasts make it less
ideal for DL-based weather forecasting models. Therefore, there is a crtical
need to design a DA system specifically tailored for these novel DL-based
weather forecasting models. This systems should integrate real-world
observations with the ultimate goal of establishing an operational framework
that operates independently of traditional NWP and DA systems.

Given the pivotal role of satellite DA in improving the skill of medium-range
weather forecasts [29, 14], this study introduces FuXi-DA, a pioneering,
generalized DL-based DA framework. This framework is designed for the
assimilation of satellite observations to produce the analysis that optimize
the forecast performance of DL-based weather forecasting models.Figure 1 shows
the comparison between FuXi-DA and the traditional DA process, taking the
variational method as an example. FuXi-DA advances beyond traditional DA
systems by introducing several innovative features:

  * •

To address the discrepancy in information content between background and
observational data, FuXi-DA employs separate encoders for processing each type
of data. This approach ensures the assimilation process fully leverages the
extensive information contained in background data, while accurately
integrating observational data. Notably, background data often surpasses
observational data in spatial coverage and has more complete information on
weather systems.

  * •

The pre-processing of observations involves a significant amount of laborious
work, including data thinning, cloud detection, bias correction, and more
[14]. Such data pre-processing steps are not included in the FuXi-DA
framework, or are implicitly integrated into the model architecture.
Furthermore, it has been shown that FuXi-DA possesses the capability to
implicitly learn from historical data to mitigate the impact of clouds on the
assimilation of satellite data.

  * •

Traditional DA methods require the estimation of observation and background
error covariance matrices, crucial for adjusting the influence of background
and observational data within the DA system [30]. FuXi-DA obviates this
requirement through a innovative unified fusion neural network. This network
effectively adeptly adjusts the weight of observations and background data
without the need to estimate their error covariance matrices. This advancement
addresses the challenges associated with the large size and impractical
storage demands of the background error matrix, as well as the difficulty of
accurate estimation due to the absence of a "true state."

  * •

Furthermore, FuXi-DA facilitates the joint training of DA with any DL-based
weather forecasting model, specifically FuXi in this study, to not only refine
analysis accuracy but also enhance medium-range forecast performance. This
approach extends the forecast optimization window beyond the traditional
assimilation window, a significant advancement over the traditional 4D-Var
approach, which limits optimization to the observation availability window.
Importantly, by leveraging supervised information from medium-range forecasts
during training, FuXi-DA enables the simultaneous optimization of both
assimilation and forecasting processes. This dual optimization significantly
improves forecast accuracy.

In this study, we assess the efficacy of Fuxi-DA by integrating it with the
Advanced Geosynchronous Radiation Imager (AGRI) aboard Fengyun-4B satellite.
To illustrate the advantages and effectiveness of incorporating AGRI data, we
developed a comparative correction model (refers to EXP_CORR). This model
replicates the architecture and training process of FuXi-DA (refers to
EXP_ASSI), with the only difference being its exclusion of AGRI data
assimilation. Our results reveal that the assimilation of AGRI data
consistently enhances forecast accuracy, demonstrating FuXi-DA’s capability to
leverage satellite data for forecast improvement. Further, multiple single-
observation experiments confirm that FuXi-DA’s consistency with established
atmospheric physics. It is important to note that while FuXi-DA was developed
to refine the analysis for enhancing the forecast performance of FuXi, its
framework is sufficiently flexible to be adapted for use with other DL-based
weather forecasting models.

Figure 1: The comparison between the Fuxi-DA and the variational data
assimilation.

##  2 Results

Fengyun-4B/AGRI data, as the first selected satellite observation, have been
successfully assimilated into FuXi-DA. Three experiments are designed to
evaluate the impact of assimilating AGRI data in FuXi-DA, including performing
a 6-hour forecast (called the background) using FuXi, correcting the 6-hour
forecast fields using a correction model, and assimilating AGRI data into the
6-hour forecast fields using FuXi-DA (named EXP_CTRL, EXP_CORR and EXP_ASSI,
respectively). The EXP_CORR experiment is designed for a fairer comparison,
since the improvements in EXP_ASSI experiment can be attributed to both the
correction of the background data and the assimilation of AGRI data. The
comparison between EXP_CORR and EXP_ASSI is the primary focus of the results.

After assimilating AGRI data in FuXi-DA, a significant reduction in analysis
error is observed, particularly for relative humidity and geopotential.
Furthermore, the impact of these observations is primarily evident in the
middle and upper troposphere within the observed area. Specifically, the
regionally-averaged latitude-weighted root mean square errors (RMSEs) of R300
and Z500 decrease by approximately 4.47% and 2.02%, respectively, within the
coordinates of 52.5°E to 147.5°W and 80°S to 80°N. Additionally, the skill of
global forecasts is improved after assimilating AGRI data. Within the forecast
lead time of 7 days, there are statistically significant improvements, with
the RMSE of Z500 decreasing by approximately 0.67% at the forecast lead time
of 1 days and 0.34% at the forecast lead time of 7 days. We also demonstrated
that the introduction of forecast fields loss improves the skill of long-term
forecasting. Finally, FuXi-DA model exhibits excellent physical consistency
and possesses the capability to automatically distinguish between observations
under cloudy and clear conditions.

Table 1: The reduction rates of the regionally-average latitude-weighted RMSE for 65 upper-air atmospheric variables in the EXP_ASSI experiment compared to the EXP_CTRL experiment (left) and compared to the EXP_CORR experiment (right).  Pressure (hPa) | EXP_ASSI compared to the EXP_CTRL | EXP_ASSI compared to the EXP_CORR
---|---|---
R |  T |  Z |  U |  V |  R |  T |  Z |  U |  V
1000 |  -2.10 |  -2.24 |  -1.63 |  -1.37 |  -1.54 |  -0.88 |  -1.03 |  -0.82 |  0.29 |  0.36
925 |  -1.28 |  -1.78 |  -1.54 |  -0.73 |  -0.65 |  -0.39 |  -0.66 |  -0.87 |  0.33 |  0.32
850 |  -0.84 |  -1.19 |  -2.68 |  -0.59 |  -0.45 |  -0.40 |  -0.43 |  -1.08 |  0.23 |  0.21
700 |  -1.65 |  -1.51 |  -2.03 |  -0.61 |  -0.59 |  -1.04 |  -0.61 |  -1.33 |  -0.17 |  -0.20
600 |  -2.36 |  -1.73 |  -2.80 |  -0.66 |  -0.76 |  -1.69 |  -0.87 |  -1.62 |  -0.28 |  -0.37
500 |  -3.55 |  -1.41 |  -2.82 |  -0.86 |  -1.02 |  -2.77 |  -0.77 |  -2.02 |  -0.51 |  -0.60
400 |  -4.81 |  -1.50 |  -3.24 |  -1.16 |  -1.32 |  -3.86 |  -0.81 |  -2.31 |  -0.70 |  -0.87
300 |  -5.49 |  -1.76 |  -4.11 |  -1.40 |  -1.50 |  -4.47 |  -1.07 |  -3.01 |  -0.90 |  -1.03
250 |  -5.26 |  -1.49 |  -5.08 |  -1.47 |  -1.61 |  -4.02 |  -0.98 |  -3.58 |  -0.00 |  -1.06
200 |  -3.31 |  -1.00 |  -5.31 |  -1.57 |  -1.55 |  -2.21 |  -0.64 |  -3.95 |  -1.20 |  -1.07
150 |  -1.58 |  -0.83 |  -5.32 |  -1.21 |  -1.57 |  -0.39 |  -0.52 |  -4.09 |  -1.05 |  -1.31
100 |  -1.33 |  -0.42 |  -4.56 |  -0.36 |  -0.67 |  -0.36 |  -0.30 |  -2.89 |  -0.25 |  -0.52
50 |  -0.21 |  -0.07 |  -1.69 |  -0.57 |  -0.09 |  -0.09 |  -0.10 |  -1.50 |  -0.40 |  -0.01
Figure 2: Comparison of analysis after a single assimilation. The time series
of the globally-averaged latitude-weighted RMSE (a, b, c, d) and the time
series of the normalized RMSE difference (e, f, g, h) of EXP_CTRL (black
lines), EXP_CORR (blue lines) and EXP_ASSI (red lines), and the spatial
distribution of normalized RMSE difference (i, j, k, l) for EXP_ASSI relative
to EXP_CORR for R300, T300, U300 and V300. Sub-figure in (e) displays the
original normalized RMSE difference at certain moments. The original data is
represented by solid lines with high transparency, while solid lines with low
transparency indicate the smoothed values.

###  2.1 Impact on the background

This section compares the differences in background and analysis errors
resulting from a single assimilation. The differences, within the observation
region, between EXP_CTRL and EXP_ASSI are discussed first. The left side of
Table 1 displays the reduction rates of the regionally-average latitude-
weighted RMSE in the EXP_ASSI experiment compared to the EXP_CTRL experiment.
As illustrated in the table, the assimilation of AGRI observations leads to a
significant decrease in RMSEs across all variables, most notably in relative
humidity and geopotential. To facilitate a more equitable comparison, the
right side of Table 1 shows the comparison between the EXP_ASSI and EXP_CORR
experiments. Significant improvements in relative humidity and geopotential
are also observed in the middle and upper troposphere, with the RMSE of R300,
R500, Z300, and Z500 decreasing by approximately 4.47%, 2.77%, 3.01%, and
2.02%, respectively. The notable improvement in humidity can be attributed to
the three crucial water vapor channels of AGRI (channels 9-11). Additionally,
there is a decrease in the RMSE of low-level temperatures, notably the 1000
hPa temperature, associated with the three window channels (channels 12-14).

Figure 2 shows the time series of RMSE and the normalized difference of
globally averaged latitude-weighted RMSE for R300, T300, U300 and V300, as
well as the horizontal distribution of the reduction rate of RMSE in the
EXP_ASSI experiment compared to EXP_CORR experiment. The EXP_CTRL, EXP_CORR
and EXP_ASSI experiments are represented by black, blue and red colors,
respectively. In Figure 2a-2h, the original data is represented by solid lines
with high transparency, while solid lines with low transparency indicate a
moving average with a window size of 4. The spatial distribution of the
normalized RMSE difference was calculated from regionally-averaged latitude-
weighted RMSE over a range of 5° × 5°. As shown in Figure 2a-2d, the RMSE
values for EXP_CTRL, EXP_CORR and EXP_ASSI experiments have consistently
remained low. At almost all times, the EXP_CORR and EXP_ASSI experiments
outperform the EXP_CTRL experiment, exhibiting negative values in the
normalized RMSE differences. Furthermore, throughout the long-term evaluation,
all variable in the EXP_ASSI experiment consistently perform better. Compared
to the EXP_CORR experiment, the reduction in RMSE within the observation area
is more significant in the EXP_ASSI experiment, which is consistent with our
expectations. It is worth noting that the reduction in the RMSEs is more
pronounced at 00 and 12 h (6-hour forecasts from 06 and 18), while the
improvement is relatively small when assimilating at 06 and 18 h (6-hour
forecasts from 00 and 12). At 00 and 12 h, there are typically more
observations used in the 4D-Var component of ERA5 (for example, radiosonde
soundings are often available at 00 and 12 h), resulting in higher analysis
accuracy. Therefore, the background obtained from the 6-hour forecasts of FuXi
at 00 and 12 h are more accurate. In this case, the assimilation model trusts
the background more, leading to less impact from observations.

To further analyze the impact of assimilating AGRI observations, Figure 3
shows error distributions and analysis increments at 12:00 on October 15 2023.
Figure 3a-3f display the horizontal distributions of R300 and Z500 for ERA5,
EXP_CORR, and EXP_ASSI, and there are no significant differences between them.
Figure 3g and 3j show the errors between EXP_CORR and ERA5, while Figure 3h
and 3k display the errors between EXP_ASSI and EXP_CORR. As shown in Figure 3h
and 3k, the analysis increments generated by assimilating AGRI data are
concentrated primarily in the observation area (Figure 3i). Furthermore,
assimilating AGRI data leads to a reduction in analysis errors. For instance,
in Figure 3g, there is a significant negative deviation to the north of
Australia. After assimilating AGRI data, a positive increment is generated (as
seen in Figure 3h). The improvement in geopotential is clearer. There are
positive deviations to the east of Australia and negative deviations to the
west of Australia (as seen in Figure 3j). After assimilating AGRI, a negative
increment was generated on the east side, and a positive increment on the west
side (as seen in Figure 3k).

Figure 3: Visualization of spatial distributions, error distributions and
analysis increments of R300 and Z500, as well as visualization of
observations. Spatial maps of R300 for ERA5 (a), EXP_CORR (b), and EXP_ASSI
(c); spatial maps of Z500 for ERA5 (d), EXP_CORR (e), and EXP_ASSI (f);
spatial maps of error distribution for R300 (g) and Z500 (j); spatial maps of
analysis increments for R300 (h) and Z500 (k); and a spatial map of AGRI
channel 9 brightness temperature (i).

###  2.2 Impact on the forecast

Examining longer forecast ranges after a single assimilation, Figure 4
presents the normalized RMSE difference among EXP_CTRL, EXP_CORR and EXP_ASSI
for global 10-day forecasts across three variables at three pressure levels.
The forecast skill is enhanced through correction and assimilation, and
EXP_ASSI performs better compared to EXP_CORR. Similar to analysis, the
improvement in relative humidity is most significant after assimilating AGRI
observations. Observations’ impact on forecasts is observed mainly in the
middle and upper troposphere, exhibiting larger differences at 300 and 500
hPa, and smaller differences at 850 hPa between EXP_CORR and EXP_ASSI. With
increasing forecast lead times, the improvements caused by assimilating AGRI
data decrease gradually. Nevertheless, when compared to EXP_CORR, EXP_ASSI
exhibits significant improvement in the first 7 days, as evidenced by the blue
solid line extending beyond the red shading, and the reduction of Z500 errors
ranges from 0.67% at the forecast lead time of 1 day to 0.34% at the forecast
lead time of 7 days. It is worth mentioning that the accuracy of the forecast
has been significantly enhanced through the joint optimization of forecasting
and analysis, as illustrated in Figure5. This improvement is likely
attributable to the improved dynamic consistency of the initial fields
resulting from the addition of multi-time-step forecast loss.

Figure 6 depicts the spatial distribution of the impact from assimilating AGRI
data on relative humidity forecasts. At the forecast lead time of 1 day,
improvements in the humidity forecast are observed primarily within the
observation areas. With increasing forecast lead times, the influence of
assimilation expands rapidly. Driven by the mid-latitude westerlies,
improvements in forecasting primarily propagate eastward. At the forecast lead
time of seven days, the impact of assimilating AGRI data extends globally.
Although there are negative effects in some regions for long-term forecasting,
assimilating AGRI data demonstrates positive impacts in a greater number of
regions.

Figure 4: Comparison of global 10-day forecasts after a single assimilation.
The normalized RMSE difference of EXP_CTRL (black lines), EXP_CORR (blue
lines) and EXP_ASSI (red lines) for R300, R500 and R850 (a, b, c), T300, T500
and T850 (d, e, f), as well as Z300, Z500 and Z850 (g, h, i). The shaded areas
represent 95% confidence intervals. Figure 5: Comparison of global 10-day
forecasts after a single assimilation. The normalized RMSE difference of CTRL
(black lines), EXP_ASSI using a single-time-step loss (blue lines) and
EXP_ASSI using a multi-time-step loss (red lines) for R300, R500 and R850 (a,
b, c), T300, T500 and T850 (d, e, f), as well as Z300, Z500 and Z850 (g, h,
i). The shaded areas indicate a 95% statistical confidence interval. The
shaded areas represent 95% confidence intervals. Figure 6: Visualization of
spatial distribution of normalized RMSE difference for humidity forecasts:
EXP_ASSI compared with EXP_CORR. The first, second and third rows represent
forecast lead times of 1 day, 3 days, 5 days and 7 days, respectively. The
first, second and third columns represent pressures at 300, 500 and 850 hPa
respectively. Statistics of latitude-weighted RMSE within a 5-degree by
5-degree range.

###  2.3 Interpretation

Unlike previous works that utilized simulated observations for DL-based DA
studies, we investigated whether physics-free DA networks are consistent with
prior knowledge of atmospheric physics. Traditional DA methods frequently
employ single-observation experiments to ascertain the validity of the impact
generated by observations, which depends on the observation operator and the
estimation of error in the background and observation. Here, we employed a
similar approach. Given that the DA network lacks explicit observation
operators and error statistics, and since using a single observation as input
is not feasible in the trained FuXi-DA, we implemented some adjustments. The
specific practice involved two inferences using the trained FuXi-DA. The first
inference utilized the 6-hour forecast and original observations as inputs,
while the second introduced a perturbation at a single point location to the
original observations as model input. The difference between the analyses
obtained from the two inferences is determined to characterize the analysis
increment resulting from the perturbation. We begin with the traditional
incremental analysis formula and provide a proof of this method in Appendix A.

Table 2: Observation perturbations test Channel |  Condition |  Brightness Temperature Perturbation
---|---|---
9 |  clear |  +1 K
9 |  clear |  +5 K
9 |  clear |  -1 K
11 |  clear |  +1 K
11 |  cloudy |  +1 K

We randomly selected two observation points at 00:00 on October 26, 2023, one
under clear conditions and the other under cloudy conditions (as shown in
Figure A.1.1). The distinction between clear and cloudy conditions was made
using the AGRI Cloud Mask (CLM) product [31]. Considering that relative
humidity showed the most significant improvement after assimilating AGRI data,
channels 9 (a high-level water vapor channel) and 11 (a low-level water vapor
channel) were used for single-observation experiments. Table 2 lists the five
perturbation experiments conducted in this section. To further analyze the
impact of introducing perturbations at different times within the assimilation
window, perturbations were introduced at the beginning, middle, and end of the
assimilation window, respectively. Figure 7 shows the results of introducing a
1 K perturbation at the clear point. Figures 7a-7c display the analysis
increments of R300 resulting from perturbing channel 9. The decrease in
relative humidity is consistent with the theory of radiative transfer in the
atmosphere, indicating a negative correlation between brightness temperature
and humidity. Specifically, a smaller amount of water vapor results in less
absorption of surface-emitted radiation, thus the brightness temperature is
warmer [32]. For a clear illustration, the humidity Jacobian functions for
channels 9-11 are shown in Figure B.1 (a detailed description related to the
Jacobian is provided in Appendix B). We also observed the performance of the
increments under different perturbation values. By introducing a perturbation
of -1 K, positive increments occurred, and the magnitude of these increments
was consistent with those caused by adding a perturbation of 1 K (as shown in
Figure A.1.2). However, introducing a perturbation of 5 K resulted in the
increments of relative humidity being approximately five times larger than
those from a perturbation of 1 K. These phenomena align with the results of
traditional DA methods. Figures 7h-7m show the analysis increments of R500
resulting from introducing a perturbation to channel 11, displaying results
similar to those of channel 9. It is important to note that the characteristic
of the analysis increments is the dispersion from the observation point
towards the surroundings, which is caused by the spatial correlations of the
background.

To further demonstrate physical consistency, Figure 7 also shows the vertical
distribution of the relative humidity increment. The analysis increment of
channel 9 reveals the vertical correlation of the background, with the peak
increment occurring around 300 hPa (as shown in Figures 7d-7f). Figure 7g
shows the normalized weighting functions for channel 9 (a detailed description
of the weighting function is provided in Appendix B). It can be noted that the
peak of the analysis increment aligns with the peak of the weighting function,
both located around 300 hPa. Additionally, the findings for channel 11 are
similar to those for channel 9, except that the peak increment for channel 11
occurs at a lower altitude, around 500 hPa (as shown in Figures 7k-7m). These
results demonstrate that the impact of observations on the assimilation
results is correct, further demonstrating the stability of FuXi-DA. Notably,
the analysis increments are very small after introducing a 1 K perturbation.
One reason is that the observations were not sparsified, so adjacent
observations can be correlated. When multiple observations jointly impact the
same location in the background, the expected analysis increment will be
larger. Another possible reason is the high accuracy of the 6-hour forecast
field of FuXi, leading the assimilation model to trust the background more.
Additionally, all perturbation experiments show an interesting phenomenon: the
closer to the end of the assimilation window, the greater the impact of
observations. The same phenomenon appears in the traditional 4D-Var
assimilation system [33]. In fact, the assimilation method of FuXi-DA is more
similar to the the three-dimensional variational (3D-Var) method. However, by
incorporating the temporal information of the observational data, FuXi-DA has
demonstrated phenomena consistent with the 4D-Var method, which is
encouraging.

Figure 7: The analysis increment of relative humidity resulting from
introducing a perturbation of 1 K for the clear observation (pentagram) at the
beginning (the first column), middle (the second column), and end (the third
column) of the assimilation time window, respectively. Horizontal distribution
of R300 increments resulting from introducing a perturbation of 1 K to channel
9 (a, b, c); vertical distribution of relative humidity increments resulting
from introducing a perturbation of 1 K to channel 9 (d, e, f), obtained from a
cross-section delineated by the dashed lines in (a); horizontal distribution
of R500 increments resulting from introducing a perturbation of 1 K to channel
11 (h, i, j); vertical distribution of relative humidity increments resulting
from introducing a perturbation of 1 K to channel 11 (k, l, m), obtained from
a cross-section delineated by the dashed lines in (a); normalized weighting
functions for channel 9 (g) and channel 11 (n). The atmospheric profile comes
from the US standard atmosphere. The rapid radiative transfer model adopted is
RTTOV version 13.2.

Traditional satellite DA methods require complex modeling for assimilating
observations under cloudy conditions [10, 14, 11]. Additionally, many cloud-
affected observations from instruments that have not achieved all-sky
assimilation are discarded. In this study, the FuXi-DA model assimilates
observations under both cloudy and clear conditions. To demonstrate the
adaptive adjustment ability of the DA network under different conditions, a
cloudy single-observation experiment was conducted. Figure 8 shows the
analysis increment of relative humidity resulting from introducing a 1 K
perturbation to channels 9 and 11 at the cloudy observation point (as shown in
Figure A.1.1). There is no significant analysis increment around the cloudy
observation point, which indicates that FuXi-DA possesses autonomous cloud
detection capabilities. The limited impact of cloudy observations on the
background is due to clouds having a strong ability to absorb infrared
radiation, making it difficult to obtain information below the clouds.
Additionally, the lack of hydrometeors (such as liquid, ice, snow, hail,
graupel) and other particles (including mixed phase) in the background makes
it difficult for the network to establish a connection between the observation
and the background. The varied performance of FuXi-DA under clear and cloudy
conditions demonstrates the potential of the DL model to achieve all-sky
assimilation.

Figure 8: Similar to Figure 7 but for cloudy observation (pentagram).

##  3 Discussion

Both numerical weather prediction (NWP) systems and the recently developed
deep learning-based weather forecasting models rely on accurate initial
conditions. Modern NWP systems utilize data assimilation (DA) to determine the
current state of the atmosphere. This process is very complex and requires
significant computational resources. Currently, satellite observations have
become the dominant source of observational information in DA systems, but
assimilating these data is more complex and consumes more computational
resources. Therefore, we have developed a scalable satellite DA framework
based on deep learning (DL) models, named FuXi-DA. FuXi-DA employs different
encoders for various types of data, resolving the inconsistency between
observational and background variables by facilitating interactions among
diverse data in latent space. Starting from the general form of traditional
variational methods [8], FuXi-DA automatically learns the weights for
observations and backgrounds in a unified fusion neural network to facilitate
the assimilation process. Using the FuXi-DA assimilation framework, we have
successfully implemented the assimilation of Fengyun-4B/AGRI data.

Results indicate a significant improvement in analysis and forecast accuracy
after assimilating AGRI data. The successful assimilation of AGRI data in
FuXi-DA represent the first step towards a fundamental shift from traditional
DA methods to DL-based DA models. FuXi-DA’s strong performance in fusing
background and observation information, its design simplicity and scalability,
the rapidity of computation, and the unique approach to handling observation
data create a possibility for building an end-to-end DL-based weather
forecasting system. Additionally, FuXi-DA demonstrates consistency with prior
knowledge of atmospheric physics, enhancing confidence in the application of
multi-source observations in DL-based DA models.

Compared to traditional DA systems, FuXi-DA not only significantly simplifies
the data pre-processing steps but also substantially reduces the computational
resources required. In traditional DA systems, specific modules such as data
thinning, quality control, bias correction, error estimation, and observation
operators need to be customized for different observations, consuming
significant human and computational resources. As mentioned above, FuXi-DA
avoids the manual construction of observation operators and error estimation
by encoding each type of data separately and employing a unified fusion
module. Furthermore, FuXi-DA does not require data thinning and cloud
detection operations, thus maximizing the utilization of observations.
Although it has not demonstrated the capability to assimilate observational
data under cloudy conditions, FuXi-DA effectively distinguishes observations
between clear and cloudy conditions. Introducing hydrometeors and other
particles into the forecast model could further enhance the capability of all-
sky assimilation. Additionally, we incorporate satellite zenith angle and
spatio-temporal information into the inputs of FuXi-DA, and adjust
observations within the fusion module to achieve bias correction
functionality. In summary, FuXi-DA reduces pre-processing steps through its
strong learning capability and achieves the necessary processes through its
unique modeling approach. Most importantly, FuXi-DA greatly saves
computational resources. Compared to the extensive computational resources
consumed by traditional DA systems [1], FuXi-DA completes one assimilation on
a single A100 GPU in less than 10 seconds.

Looking to the future, our goal is to build an end-to-end DL-based weather
forecasting system. To achieve this objective, we are endeavoring to
incorporate more satellite observation data into FuXi-DA, including advanced
infrared sounders, microwave sounders, and imagers, among others [14].
Furthermore, we will continue to explore the application of more unevenly
distributed and sparse observation data in DL-based assimilation models, such
as radiosonde soundings and aircraft meteorological data reports [34, 35].
Additionally, considering that only one year of data were used for model
training, which is significantly less compared to the amount of data typically
used in training DL forecast models, combining real observations with
simulated observations for joint training is expected to yield better results
in future work.

##  4 Method

###  4.1 Data

ERA5 is the fifth generation of the ECMWF reanalysis dataset, generated by
assimilating high-quality and abundant global observations using ECMWF’s IFS
model. The ERA5 data is widely regarded as the most comprehensive and accurate
reanalysis archive. Therefore, we use the ERA5 reanalysis dataset as the
ground truth for model training. Moreover, ERA5 data is also used as the input
for generating the background in the FuXi model. In this work, we use 5 upper-
air atmospheric variables at 13 pressure levels (50, 100, 150, 200, 250, 300,
400, 500, 600, 700, 850, 925, and 1000 hPa), and 5 surface variables. The 5
upper-air atmospheric variables are geopotential (Z), temperature (T), u
component of wind (U), v component of wind (V), and relative humidity (R).
Additionally, 5 surface variables are T2M, 10-meter u wind component (U10),
10-meter v wind component (V10), mean sea level pressure (MSL), and total
precipitation (TP). In total, 70 variables are used in this study, with a
spatial resolution of 0.25° and a temporal resolution of 6 hours.

Fengyun-4B/AGRI has a total of 15 channels, including 3 visible (VIS)
channels, 3 near-infrared (NIR) channels, 2 mid-wave infrared (IR) channels, 3
water vapor channels, and 4 long-wave IR channels. The footprint at nadir for
the VIS channels is 1 km, for the NIR channels and one mid-wave IR channel it
is 2 km, and for the remaining channels, it is 4 km [36]. In our research,
observations from channels 8-15 are used for assimilation, with an
assimilation time window of plus or minus one hour. Since the footprint of
these channels is 4 km (2748 × 2748 latitude-longitude scattered points), and
the background fields have a spatial resolution of 0.25° (721 × 1440 latitude-
longitude grid points), achieving correspondence of the respective positions
within the network for these two types of data poses a challenge. We processed
the satellite observations into "super-observations" that matched the
resolution of the background fields. This approach makes the observations more
representative of the spatial scales in the model and effectively reduces the
uncertainty of the observations [37, 38]. The specific process includes: (1)
Cropping a regular grid area that includes the observation area based on the
background coordinates. For AGRI data, the corresponding background size is
approximately 640 × 640. (2) Averaging the satellite observations within
±0.125° around each grid point of the cropped area to represent the satellite
observations at that grid point. (3) Applying mask processing to positions
within the cropped area that do not have matched observations. After
processing, the satellite observations were processed into 640 × 640 latitude-
longitude grid point data with a spatial resolution of 0.25°.

The data for FuXi-DA is provided in the form of input-output pairs, where the
inputs include background fields and satellite observations, and the output is
ERA5 data. The available data spans a period from June 2022 to December 2023.
The data is divided into training, validation, and testing sets. The training
includes 1460 samples spanning from June 2022 to May 2023. The validation set
contains 244 samples, corresponding to the period from June 2023 to July 2023,
while out-of-sample testing is performed using 612 samples from August 2023 to
December 2023.

###  4.2 Model and Architecture

In the three-dimensional variational (3D-Var), the general form for the
analysis can be expressed as:

$x^{a}=x^{b}+BH^{T}(HBH^{T}+R)^{-1}[y^{o}-H(x^{b})]$ (1)

where $x^{a}$ is the state vector of analysis field, $x^{b}$ is the state
vector of background filed, $y^{o}$ is the observation vector, $H$ is the
observation operator, $B$ is the covariance matrix of background error and $R$
is the covariance matrix of observation errors, see e.g. [8] for this standard
result. The construction of FuXi-DA was inspired by this form. We depict DA
learning as an incremental learning process, where FuXi-DA learns analysis
increments from a fusion module and applies them to the background.

Due to the inconsistency between the background field variables and
observational variables, we encode each input separately and implement
interactions among various data within a unified fusion model. Compared to the
traditional DA system, which interpolates background fields to the position
and time of each observation and then processes them through an observation
operator for comparison with observations [14], FuXi-DA unifies and simplifies
this process by converting both observations and background into the latent
space. This approach significantly saves human and computational resources and
avoids errors introduced by the observation[15, 39]. As shown in Figure 9, the
model architecture of the FuXi-DA consists of three branches. The three
branches are the background information flow (yellow boxes), the mixed
information flow (gray boxes), and the observation information flow (violet
boxes), respectively. All branches adopt the same U-net architecture [40]. The
background flow has two down-sample and two up-sample stages, while the others
has two down-sample stages and one up-sample stages. The channels for each
U-net stage are 256, 512 and 256 respectively.

Firstly, the input data for FuXi-DA includes the background, observations, and
additional observation information. The background has dimensions of 70 × 721
× 1440, where 70, 721, and 1440 represent the total number of input variables,
latitude and longitude grid points, respectively. In FuXi-DA, the background
input is first reshaped into 70 × 720 × 1440 through bilinear interpolation.
The processed observations have dimensions of 8 × 8 × 640 × 640, where 8, 8,
640, and 640 represent the total number of time frames, channels, latitude and
longitude grid points, respectively. Considering that satellite observation
biases usually vary with scan angle and geographic location [41, 39, 42], we
encode the longitude, latitude, satellite zenith angle, and temporal
information of the satellite observations. The longitude, latitude, and
satellite zenith angle are taken as the cosine, sine, and cosine,
respectively. For time encoding, the observation time is first transformed
into the day of the year and the minute of the day. Then, we calculate both
the cosine and sine for the day of the year and the minute of the day. By
encoding longitude, latitude, satellite zenith angle, and time and adding them
as additional observation channels in the model, the size of observations
becomes 8 × 15 × 640 × 640. The observation input is first reshaped into 120 ×
640 × 640. Then, the background input is cropped to correspond to the
positions of the observations and concatenated with the reshaped observations
to obtain the input of the mixed information flow, with a size of 190 × 640 ×
640. Following the model architecture illustrated in Figure 9, the output is
initially reshaped to 70 × 720 × 1440, then restored to the original input
shape of 70 × 721 × 1440 by bilinear interpolation.

The fusion module receives information from the three branches and facilitates
information interaction through the U-net module with two down-sample and two
up-sample stages. Each fusion module has three outputs. The first output is
added to the background information flow. Similar to how traditional DA
systems adjust the weights of the background and observations through
background errors and observation errors, FuXi-DA automatically learns the
weights of the background and observations within this module and outputs
information that implicitly contains analysis increments. The second output is
propagated as mixed information to the subsequent network. The third output is
added to the observation information flow. This operation draws on the
traditional bias correction process, where predictions computed from the
background field are typically used for bias correction [43, 41, 39].

In Fuxi-DA, each down-sample stage consists of a 2 × 2 2-dimensional (2D)
convolution layer with a stride of 2, a layer normalization layer [44], a
sigmoid-weighted linear unit (SiLU) activation [45] and a 3 × 3 2D convolution
layer with a stride of 1. Each up-sample stages consists of a 3 × 3 2D
convolution layer with a stride of 1, a layer normalization layer, a sigmoid-
weighted linear unit (SiLU) activation, a 3 × 3 2D convolution layer with a
stride of 1 and a pixel-shuffle layer [46] with an upscaling factor of 2.
Furthermore, a skip connection is included that concatenates the outputs from
the down-sample stage with the output of up-sample stage before being fed into
the next up-sample stage.

Figure 9: Overall architecture of FuXi-DA model.

###  4.3 Model training

The FuXi-DA model is built using the PyTorch framework [47]. The model was
trained on 4 Nvidia A100 GPUs for a total of 6000 iterations, with a batch
size of 1 per GPU. The entire model training process lasted about 8 hours. The
AdamW [48] optimizer is used with parameters $\beta_{1}$ = 0.9, $\beta_{2}$ =
0.999 and a weight decay coefficient of $1e^{-5}$. The learning rate was set
using a warmup and a cosine annealing schedule [49]. First, the learning rate
increases during the training process according to the formula:

$lrate=[(step\\_num-1)/warmup\\_steps]\times(stop\\_lrate-
start\\_lrate)+start\\_lr$ (2)

This corresponds to increasing the learning rate linearly for the first
warmup_steps training steps. We used $warmup\\_steps=500$,
$start\\_lrate=1e^{-8}$, and $stop\\_lrate=2e^{-3}$. Second, the learning rate
decreases through a cosine annealing schedule with an initial learning rate of
$2e^{-3}$ and maximum number of iterations of 6000. We use the latitude-
weighted $L1$ loss to minimize the errors between Fuxi-DA’s outputs and the
ERA5, which is defined as follows:

$L1\\_Loss=\frac{1}{C\times{H}\times{W}}\displaystyle\sum_{c=1}^{C}\displaystyle\sum_{i=1}^{H}\displaystyle\sum_{j=1}^{W}\alpha_{i}|\widehat{X}_{c,i,j}-X_{c,i,j}|$
(3)

where $C$, $H$ and $W$ are the number of channels and the number of latitude
and longitude grid points, respectively. $\widehat{X}$ is the ground truth.
$\alpha_{i}=H\times\left.{cos\Phi_{i}}\middle/\right.{\displaystyle\sum_{i=1}^{H}cos\Phi_{i}}$
is the weight at latitude $\Phi_{i}$. In the actual training process, we
employed a multi-time-step loss. The output of FuXi-DA is used as the input
for FuXi for long-range forecasting, and the errors between the forecast
results and ERA5 need to be minimized. The final loss function used in Fuxi-DA
is defined as follows:

$Total\\_Loss=L1\\_Loss^{0}+\frac{1}{T}\displaystyle\sum_{t=1}^{T}L1\\_Loss^{t}$
(4)

$T$ represents the number of time steps, where at $t=0$, $X$ denotes the
current analysis field, and at $t>0$, $X$ represents the forecast field. In
this study, we set $T=10$, indicating the inclusion of the analysis field and
10 forecast fields spanning 6-60 hours.

###  4.4 Experience setup

Three experiments were designed to assess the impact of assimilating AGRI data
on analysis and forecasts. The control experiment, referred to as “EXP_CTRL,”
pertains to the 6-hour forecast results from FuXi. Given that FuXi was trained
with data from 1979 to 2015, biases are present in the 6-hour forecasts during
the study period of this work. Consequently, the output of FuXi-DA is
influenced by both assimilation and correction. To render the results more
rigorous, we trained a correction model (the corresponding experiment is named
"EXP_CORR"). The training dataset for the correction model is consistent with
that used to train FuXi-DA, except that satellite observations were excluded.
Additionally, the model structure adopts a similar architecture to FuXi-DA,
but with the fusion module removed, while the training approach is completely
consistent with FuXi-DA. The assimilation experiment is named EXP_ASSI.

###  4.5 Evaluation method

The Latitude- weighted root mean square error (RMSE) are used to evaluate
analysis and forecast performance, which is calculated as follows:

$RMSE(c)=\sqrt{\frac{1}{H\times{W}}\displaystyle\sum_{i=1}^{H}\displaystyle\sum_{j=1}^{W}\alpha_{i}(\widehat{X}_{c,i,j}-X_{c,i,j}})^{2}$
(5)

Additionally, The normalized RMSE difference between experiment A and
experiment B calculated as $(RMSE_{A}-RMSE_{B})/RMSE_{B}$. Similarly, the
regionally-average latitude-weighted RMSE is calculated as follows:

$RMSE(c)=\sqrt{\frac{1}{(H_{max}-H_{min})\times({W_{max}-W_{min}})}\displaystyle\sum_{i=H_{min}}^{H_{max}}\displaystyle\sum_{j=W_{min}}^{W_{max}}\widehat{\alpha}_{i}(\widehat{X}_{c,i,j}-X_{c,i,j}})^{2}$
(6)

where $H_{min}$, $H_{max}$, $W_{min}$ and $W_{max}$ represent the range of
grid point indices corresponding to the selected region.
$\widehat{\alpha}_{i}=(H_{max}-H_{min})\times\left.{cos\Phi_{i}}\middle/\right.{\displaystyle\sum_{i=H_{min}}^{H_{max}}cos\Phi_{i}}$
is the weight at latitude $\Phi_{i}$.

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## Appendix

##  Appendix A Single-observation experiments

According to Equation 1, the analysis can be simplified as:

$x^{a}=x^{b}+K[y^{o}-H(x^{b})]$ (A.1)

Here, $K=BH^{T}(HBH^{T}+R)^{-1}$ is Kalman gain matrix. If a perturbation
$\Delta y$ is added to $y^{o}$, the new analysis $x^{a^{\prime}}$ can be
expressed as:

$x^{a^{\prime}}=x^{b}+K[y^{o}+\Delta y-H(x^{b})]$ (A.2)

From equations A.1 and A.2, the analysis increment $\Delta x^{a^{\prime}}$
resulting from the perturbation can be expressed as:

$\Delta x^{a^{\prime}}=K[\Delta y]$ (A.3)

Figure A.1: Visualization of spatial distributions of AGRI CLM product and the
selected observation locations (pentagram and triangle) at 00:00 on October
26, 2023. Blue solid lines represent the 500 hPa geopotential height contours
of the background at intervals of 80, and black wind shafts indicate the 850
hPa wind vectors of the background (a half barb represents a wind speed
increment of 2 m/s, a full barb represents a wind speed increment of 4 m/s and
a flag represents a wind speed increment of 20 m/s). Figure A.2: The analysis
increment of relative humidity resulting from introducing a perturbation of 1
K (a, b, c) and 5 K (d, e, f) to channel 9 for the clear observation
(pentagram) at the beginning (the first column), middle (the second column),
and end (the third column) of the assimilation window, respectively.

##  Appendix B Weight Function and Jacobian

The radiative transfer model (RTM) RTTOV version 13.2 is used as the
observation operator to simulate the clear-sky radiances and the atmospheric
profile comes from the US standard atmosphere. RTTOV is developed by the
Satellite Application Facility on Numerical Weather Prediction (NWP SAF) of
EUMET- SAT. The weighting function (WF) of a channel with central wavenumber
$v$ in a satellite is defined as:

$WF=\frac{{\partial}\tau(v,\theta,p)}{{\partial}lnp}$ (B.1)

$\tau(v,\theta,p)$ is the transmittance of the channel with central wavenumber
$v$, which depends on the absorption coefficient of absorbed gas in the
atmosphere and their vertical distribution of density. $p$ is the pressure and
$\theta$ is the satellite zenith angle. Clearly, the WF represents the
contributions of the atmosphere at different altitudes to the radiance
observed by the satellite. The Jacobian is used to evaluate the sensitivity of
a radiance to a physico-chemical parameter. For a channel with central
wavenumber $v$, it represents the sensitivity of the brightness temperature
with respect to a change in a geophysical parameter ($X$) such as humidity in
our case. It is defined as:

$J_{v}(X)=\frac{{\partial}BT(v)}{{\partial}X}$ (B.2)

Figure B.1: Normalized weighting functions (a) and humidity Jacobian functions
(b) for channel 9-11. The atmospheric profile comes from the US standard
atmosphere. The rapid radiative transfer model adopted is RTTOV version 13.2.