Multiscale feature analysis of forecast errors of 500 hPa geopotential height for the CMA‐GFS model

Using ERA5 reanalysis data from March 2021 to February 2022 and the China Meteorological Administration Global Forecasting System (CMA‐GFS) operational forecast dataset of 500 hPa geopotential height in the Northern Hemisphere in the same period, the multiscale features of forecast errors are analyzed. The results indicate that the anomaly correlation coefficient (ACC) of 500 hPa geopotential height and its multiscale components in the Northern Hemisphere keep decreasing with the extension of forecast lead time, and there are no seasonal differences in the evolution of the ACC. The effective forecast skills by season for the CMA‐GFS model are above 6 days at multiscale, with the highest skills in winter and the planetary‐scale components. In space, significant seasonal differences are observed in the locations of the extreme values of multiscale forecast errors for 500 hPa geopotential height, and the spatial distribution of forecast errors reflects the inadequate prediction of the intensity of large‐scale trough and ridge systems at middle and high latitudes and the phase‐shift prediction of small troughs and ridges at middle latitudes. Generally, the forecast errors of the original field and planetary‐scale component show wavelike or banded distribution, and the synoptic‐scale forecast errors are always distributed in latitudinal wavelike patterns alternating between positive and negative, without significant differences in the distribution of land, sea, and terrain. The first empirical orthogonal function modes of multiscale forecast errors almost retain their respective feature. In temporal, the spring, summer, and autumn time series all have quasi‐biweekly positive and negative phase transitions within the monthly scale, and the significant phase transition in winter only occurs around January 1st. These results deepen the understanding of the distribution and possible causes of forecast errors of the CMA‐GFS model and provide ideas for the improvement and revision of the model.


| INTRODUCTION
As the core of the operational numerical prediction system, the development and improvement of global medium-range numerical weather prediction systems has been attached great importance by meteorological authorities around the world (Rennie et al., 2021;Xue & Liu, 2007).However, the atmosphere is one of the most complex nonlinear systems; hence, numerical models are limited in their ability to describe the dynamical and physical processes in the atmosphere and the multiscale interactions of the atmosphere, while the model's initial conditions uncertainty and data assimilation process also produce certain errors, so the existence of model forecast errors is inevitable (James, 2002;Xue et al., 2013).Normally, the bias between the time-averaged model forecast field and the observed/reanalysis field is stable and reflects the deficiencies in the model dynamics and physical processes, that is, systematic errors (Harr et al., 1983;Rodwell & Palmer, 2007).Systematic errors constrain the improvement of model prediction, and diagnosing systematic errors in numerical models can help to understand model performance and provide clues and directions on how to improve numerical models (Jung et al., 2005).
The middle troposphere at 500 hPa, which is usually called the level of nondivergence, is less influenced by terrain and is the most applicable height for forecasting weather system movement; therefore, 500 hPa geopotential height is the traditional meteorological element field for international verification and diagnosis.The China Meteorological Administration Global Forecasting System (CMA-GFS, formerly known as GRAPES-GFS) is a multiscale universal assimilation and medium-range numerical weather prediction system (Yu et al., 2019) developed independently by the CMA Earth System Modeling and Prediction Centre (CEMC), for which a new nonhydrostatic dynamic core is established and four-dimensional variational (4DVar) data assimilation is developed using a large number of satellite applications, the radar radial wind and wind profiler data, conventional observations from aircraft, oceanic/land surface stations and radiosondes, and so on (Shen et al., 2011;Xiao et al., 2020).Zhang et al. (2018) found that systematic errors for 500 hPa geopotential height forecast field from GRAPES-GFS V2.0 are extremely concentrated at middle and high latitudes compared with FNL reanalysis data, the errors increase slightly less than linearly with forecast lead time, and there are significant seasonal differences.However, the research from Tong et al. (2017) also showed that although 500 hPa geopotential height forecast field from the GRAPES-GFS model has large errors at middle and high latitudes, the forecast system configuration is closer to the reanalysis field; the magnitude of 500 hPa geopotential height forecast field at low latitudes is close to the reanalysis field, but the forecast system has some misalignment.In addition, when studying the medium-range forecast busts of GRAPES-GFS model in East Asia, it was also discovered that the sources of error differ from season to season, but in general are closely related to the main weather systems affecting that season (Shen et al., 2021).
In short, although some studies have carried out error diagnostic and forecast performance assessments for 500 hPa geopotential height of the CMA-GFS model, the multiscale features of the weather system have not been separated, so the spatial and temporal distribution features of the multiscale errors of the CMA-GFS model need to be clarified.Therefore, in this paper, a comprehensive analysis of forecast error multiscale features of 500 hPa geopotential height for the CMA-GFS model is analyzed to further reveal the model forecast performance and error sources.

| Dataset
In this paper, we focus on the error analysis of 500 hPa geopotential height in nontropical latitudes of the Northern Hemisphere (20 -90 N, 0-360 ), and the study period is from March 2021 to February 2022, where March to May is spring (MAM), June to August is summer (JJA), September to November is autumn (SON), and December to February is winter (DJF).
The data used include (1) the CMA-GFS operational forecast dataset starting from 00 UTC (Shen et al., 2020), with a model forecast length of 240 h, a horizontal spatial resolution of 0.25 Â 0.25 , and a geographic range of (89.875 N-89.875 S, 0-359.75); (2) hourly reanalysis data from ECMWF, the fifth-generation global atmospheric reanalysis dataset ERA5 (Hersbach et al., 2020) with a horizontal spatial resolution of 0.25 Â 0.25 and a geographical range of (90 N-90 S, 0-359.75 ).In this paper, the forecast error is calculated by referring to the ERA5 reanalysis data as the "true value."

| Methods
Fast Fourier transform (FFT) as a fast algorithm of discrete Fourier transform is a signal analysis method commonly applied in spectroscopy, geophysics, numerical signal, and so on, and its positive inverse transform can process the obtained high and low-frequency signals for scale separation (Wang et al., 2008).It has been pointed out that the first 8 waves can mostly represent the basic features of atmospheric motions (Chu, 1964).Therefore, in this paper, 500 hPa geopotential height from the CMA-GFS model is filtered in the specific latitudinal wavenumber range of 0-3 and 4-8 waves based on the FFT to obtain the corresponding planetary-scale and synoptic-scale waves.
The main statistical methods used for model forecast error analysis are mean error (ME) and anomaly correlation coefficient (ACC), with the following expressions: where the subscript c indicates the climate state.ME is used to diagnose the bias between the forecast F and the analysis A, and as the absolute value of ME is closer to 0, it indicates that the forecast is better.To analyze the spatial and temporal distribution features of forecast error from the CMA-GFS model, the empirical orthogonal function (EOF) method is subsequently used (Molteni et al., 1988).ACC is usually taken to diagnose the consistency of spatial distribution between F and A, which can reflect the overall forecast performance of the model (WMO, 2006;Yu et al., 2014), and internationally, the forecast is considered valid if ACC of 500 hPa geopotential height is above 0.6 (Zhang et al., 2019).

| Seasonal statistics of space distribution of model forecast fields and forecast errors
Using 500 hPa geopotential height in the nontropical Northern Hemisphere (NNH) as an example, this paper compares the ERA5 reanalysis data and the CMA-GFS model forecast field in different seasons and analyzes the model forecast errors, thereby making a preliminary evaluation of the performance of the CMA-GFS model.First, the statistical analysis of seasonal mean ACC variation with forecast lead time of 500 hPa geopotential height and its planetary-scale and synoptic-scale components in NNH during 2021 can directly reflect the forecast effectiveness of the CMA-GFS model.In Figure 1, the ACC, or the forecast skill, of 500 hPa geopotential height and its multiscale components decrease with the extension of forecast lead time, regardless of the season.The ACC of the planetary-scale components are generally larger than the original field, and the ACC of the synoptic-scale components are generally smaller than the original field, with effective forecast skill of about 7-8 days for the original field, 7-9 days for the planetary-scale components, and 6-7 days for the synoptic-scale components.It is noteworthy that forecast skills of 500 hPa geopotential height and its multiscale components are comparable in JJA (Figure 1b), and the effective forecast skill is highest in DJF (Figure 1d); this may be due to the weather systems being more stable in winter, while systems are active in summer with high convective activity and more small and medium-scale weather processes.In addition, the variation of forecast skills throughout 2021 is similar to that of the seasons, and overall, the effective forecast skills are above 7 days at multiscale (figure omitted).
The spatial distribution features of the CMA-GFS model forecast errors are analyzed in detail below.Figure 2 shows the multiscale components of the seasonal mean forecast field and corresponding forecast errors for 500 hPa geopotential height in NNH during MAM 2021 for different lead times (120,192, and 240 h) from the CMA-GFS model.The forecast errors for 500 hPa geopotential height during MAM are mainly concentrated at middle and high latitudes, and the error extremes are distributed in a wavelike pattern with alternating positive and negative distribution and banded pattern, among which the centers of maximum forecast error are mainly located in Western Europe, near Lake Baikal and Baffin Island in the region between the troughs and ridges, and the centers of minimum error are mainly located near the Ural Mountains-Caspian Sea, Arctic Ocean, North Pacific (NP) and Greenland in the ridges (Figure 2a2,a3,a4).The planetary-scale forecast errors for 500 hPa geopotential height are also concentrated in the middle and high latitudes, and the distribution feature of the error centers is approximately the same as the original field, but the location of the extremum centers are slightly different compared to the original field, in which the centers of maximum error are mainly located near the Barents Sea and Nunavut Territory in front of the ridge, and the minimum error centers are mainly located in Arctic Ocean, NP and Greenland in the ridge area (Figure 2b2,b3,b4).The synoptic-scale forecast errors are mainly concentrated between small troughs and ridges at midlatitudes, with a phase difference of about 1/4 wavelength from the forecast F I G U R E 2 Seasonal mean 500 hPa geopotential height (left column) and its planetary-scale (middle column) and synoptic-scale components (right column) in NNH during MAM 2021, from ERA5 reanalysis fields (a1, b1, c1) and the CMA-GFS model forecast fields (contours, units: gpm) with forecast errors (shading, units: gpm) for forecast lead times of 120 h (a2, b2, c2), 192 h (a3, b3, c3), 240 h (a4, b4, c4), respectively.fields, which do not have obvious differences in the distribution of land, sea, and terrain, and the axis of the extremum centers around 50 N also in a wave train of alternating positive and negative values (Figure 2c2,c3,c4).
In addition, although forecast errors for 500 hPa geopotential height and its multiscale components in NNH during MAM gradually increase with the extension of forecast lead time, the spatial distribution of errors is similar, that is, the location of extremum centers which hardly changes with lead time, and such feature is also found in other seasons, which is consistent with the findings of previous studies (Jung, 2005;Zhuang et al., 2010).Therefore, the spatial distribution of the multiscale components of the forecast field and corresponding forecast errors in the other seasons are only given for 240-h forecast (Figure 3).
From Figure 3, it can be noticed that forecast errors for 500 hPa geopotential height and its planetary-scale components in other seasons are also mainly located at middle and high latitudes.The extreme centers of forecast error in JJA show a band-like spread, with negative deviations mainly located in the area south of 70 N and positive deviations mainly located in the Arctic marginal sea (Norwegian Sea and Beaufort Sea) north of 70 N (Figure 3a1,b1).The error extremum centers during SON are banded in the area south of 50 N with mainly negative deviations and the area north of 50 N shows alternating positive and negative wavelike distribution with the centers of maximum error mainly located in the Western Siberia and Alaska regions between troughs and ridges, and the centers of minimum error are mainly located in the regions near Western Europe, Central Siberia, and Labrador Peninsula (LP; Figure 3a2,b2).The centers of forecast errors with maxima and minima in DJF are also characterized by wavelike and banded patterns of distribution, but the negative deviations are predominant which are mainly located in and around the Arctic Ocean, near the western coast of North America, the eastern coast of the United States, East European Plain (EEP) and the Iranian Plateau, with only partial positive deviations in Northwest Pacific and the region near Hudson Bay (Figure 3a3,b3).The spatial distribution features of synoptic-scale forecast errors for 500 hPa geopotential height in NNH in other seasons are similar to those in MAM, but the locations of the axes of error extreme centers are slightly different: the axis is about 50 -60 N in JJA (Figure 3c1), near 50 N in SON (Figure 3c2), and near 30 N and 55 N in DJF (Figure 3c3).
As a whole, the forecast fields of 500 hPa geopotential height and its multiscale components in NNH from the CMA-GFS model have obvious seasonal differences in the spatial distribution of errors compared with the ERA5 reanalysis data, and forecast errors for the original 500 hPa geopotential height and its planetary-scale component in NNH are the smallest in summer compared with other seasons, and synoptic-scale forecast errors are comparable in magnitude in all seasons.The main reason for the systematic configuration bias in the model forecast is the lack of depth in the description of the largescale trough-ridge system at middle and high latitudes, F I G U R E 3 Seasonal mean 500 hPa geopotential height (left column) and its planetary-scale (middle column) and synoptic-scale component (right column) in NNH during JJA (a1, b1, c1), SON (a2,b2,c2) and DJF (a3,b3,c3) in 2021, from ERA5 reanalysis fields (purple contours, units: gpm) and the CMA-GFS model forecast fields (black contours, units: gpm) with forecast errors (shading, units: gpm) for 240 h forecast.and some offset in the phase description of the unstable small troughs and ridges at middle latitudes.

| Spatial and temporal evolutionary feature of multiscale forecast errors
To deeply analyze the spatial and temporal evolution of multiscale forecast errors for 500 hPa geopotential height in NNH, this paper chooses to use the EOF method, which has the advantage of condensing the information on the variation of the meteorological element field into the first few modes and the resulting eigenvectors are orthogonal to each other (Déqué, 1988).Similarly, in this section, the EOF decomposition of multiscale forecast errors for 500 hPa geopotential height in all seasons with a forecast lead time of 240 h are shown only.The results show that, except for the original field and its planetaryscale component in JJA, the first EOF mode of multiscale forecast errors in the other seasons and the synoptic-scale forecast errors in summer are able to pass the North test.The first EOF mode, which is the most dominant component, is sufficient to capture most of the information about the spatial and temporal distribution of forecast errors.Therefore, the first EOF mode of multiscale forecast errors for 500 hPa geopotential height in NNH for each season at 240-h forecast is presented in Figure 4, and the corresponding time series are presented in Figure 5.
The first EOF mode of forecast errors for 500 hPa geopotential height original field in all seasons shows an alternating positive and negative wavelike pattern along the latitudinal direction, with obvious seasonal differences.In MAM, there are two axes, one around 50 N where the extremum centers in "negative-positive-negative-positive" distribution are located around 180 -50 W in the Western Hemisphere (WH) with two positive centers mainly in the Northeast Pacific (NEP) and Gulf of Saint Lawrence, and the other around 70 N with negative (center in the Canadian Arctic Archipelago, CAA) to positive (center near Scandinavia Peninsula, SP) spatial distribution located around 120 W-60 E (Figure 4a1); in JJA, the extremum centers are mainly located north of 60 N, where the positive centers are located near the Laptev Sea, the Bering Strait and LP, and the negative centers are located near EEP, NP and CAA (Figure 4a2); in SON, the central axis of the error extremes is around 60 N located in the distribution of "negative-positivenegative-positive" at the extreme centers around the southeast side of Newfoundland, Irminger Sea, SP, and the Ural Mountains, respectively (Figure 4a3); and in winter, the centers of error extremes in the range of 120 E-50 W show the distribution feature from northwest to southeast with positive centers near the Eastern F I G U R E 4 Spatial distribution of the first EOF mode of forecast error for 500 hPa geopotential height original field and its multiscale components in NNH from the CMA-GFS model with the forecast lead time of 240 h during all seasons in 2021.
Siberia, the Great Basin and Newfoundland, and negative centers mainly in NEP and Baffin Island (Figure 4a4).
The spatial distribution of the first EOF mode of the planetary-scale forecast errors is slightly different from the original field, but the range of extremes is similar.The forecast errors are distributed in a latitudinal bandlike in MAM, with the axis of the positive values being about 60 N and the negative central axis being about 70 N (Figure 4b1).Except for MAM, the forecast errors in other seasons are distributed in a latitudinal wavelike pattern, with the axes located around 60 -70 N as a whole.The extreme centers of errors in JJA are mainly located in the marginal sea of the Arctic Ocean and the surrounding land (Figure 4b2); the axes in SON and DJF are slightly more southerly than that in JJA (Figure 4b3, b4), the spatial distribution of "positive-negative-positive" in SON is located around 70 W-120 E (Figure 4b3), while the positive centers in DJF are mainly located in the Eastern Hemisphere (EH), and the negative centers are mainly located in WH (Figure 4b4).
The first EOF mode variance contributions of synoptic-scale forecast errors are around 15%, and all pass the North test.The first mode spatially shows a latitudinal wavelike distribution without obvious differences in land, sea, and terrain distribution, and the axis is generally located near 50 -60 N. The axis is to the north in JJA with the extremum centers surrounding the latitudinal circle (Figure 4c2), the axis in DJF is southerly and the errors are more prominent in WH (Figure 4c4), and MAM and SON are transitional seasons (Figure 4c1,c3).In general, it can be found from the distribution features of the first mode for synoptic-scale forecast errors by season that the CMA-GFS model is relatively weaker in the forecast of small troughs and ridges at midlatitudes in WH than in EH.
In the time series corresponding to the first EOF mode of multiscale forecast errors for 500 hPa geopotential height in NNH for all seasons given in Figure 5, it can be seen that except for DJF, there are quasi-biweekly positive and negative phase transitions within the monthly scale F I G U R E 5 Corresponding standardized time series of the first EOF mode of forecast error for 500 hPa geopotential height original field and its multiscale components in NNH from the CMA-GFS model with the forecast lead time of 240 h during all seasons in 2021, the solid red lines are the 9-point running average. in the time series for all seasons, while the obvious positive and negative phase transitions in DJF are around January 1st.In summary, the daily variation in the evolution of the time coefficients is less pronounced and mainly shows cyclical variations on longer time scales, while the multiscale components are highly correlated with the time series of the original fields.

| CONCLUSION AND DISCUSSION
In this paper, the multiscale features of the CMA-GFS model forecast errors for 500 hPa geopotential height in NNH from March 2021 to February 2022 are diagnostically analyzed together with ERA5 reanalysis data, and the following conclusions are obtained.
There are no significant seasonal or scale differences in the ACC evolution of 500 hPa geopotential height and its multiscale components, and they all decrease with the extension of forecast lead time; the effective forecast skill is above 6 days at multiscale in general, and the effective forecast skills are the highest in winter and the planetary-scale components.In addition, the forecast skills for 500 hPa geopotential height original field and its planetary-scale components are much smaller in summer than in other seasons, because the circulation situation is more complex in summer.
The forecast error distribution of 500 hPa geopotential height and its multiscale components show significant seasonal differences in the location of the error extremum centers.The forecast errors for the original field and planetary-scale components mainly reflect the inadequate prediction of the intensity of large-scale trough and ridge systems at middle and high latitudes, with the extremum centers of errors in the form of wavelike of alternating positive and negative and banded patterns, while the synoptic-scale forecast errors indicate some deviation in the forecast of the phase of small troughs and ridges at middle latitudes, with the centers of error extremes only showing a latitudinal wavelike distribution of alternating positive and negative values, and there is no obvious difference in the distribution of land, sea and terrain.In addition, the magnitude of errors increases gradually with forecast lead time, while the position of maxima or minima centers hardly changes.The first EOF mode of forecast errors for 500 hPa geopotential height and its multiscale components remains most of the above spatial features, where the extreme centers of the planetary-scale forecast errors are mainly around the north of 60 N and the synoptic-scale component are more prominent in WH.Temporally, the time series of all seasons, except winter, have quasi-biweekly positive and negative phase transitions within the monthly scale, and the obvious positive and negative phase transition in winter is around January 1st.
In general, basic spatial and temporal features of the forecast field and forecast errors for 500 hPa geopotential height in NNH are discussed from the perspective of scale separation in this paper.In terms of error revision, some scholars have used the Anomaly Numerical-correction with Observations method based on historical data and the leastsquares approach based on MEs in past intervals to revise the forecasts of GRAPES model, effectively reducing the model systematic errors (Tong et al., 2017;Xue et al., 2015).However, the causes of forecast errors are complex, and more in-depth studies on the multiscale features of forecast errors of the CMA-GFS model will be conducted in the future, focusing on the model dynamical framework and parameterization scheme, with a view to providing more clues for model improvement and error revision.

F
I G U R E 1 The variation of ACC of the seasonal mean 500 hPa geopotential height and its planetary-scale and synoptic-scale components in NNH with forecast periods during (a) MAM, (b) JJA, (c) SON, and (d) DJF in 2021.