Assimilation of ASCAT Sea Surface Wind Retrievals with Correlated Observation Errors

Boheng DUAN, Weimin ZHANG,2*, Xiaofeng YANG, and Mengbin ZHU

1 College of Meteorology and Oceanology, National University of Defense Technology, Changsha 410073

2 Laboratory of Software Engineering for Complex Systems, Changsha 410073

3 State Key Laboratory of Remote Sensing Science, Chinese Academy of Sciences, Beijing 100101

4 Hainan Key Laboratory of Earth Observation, Sanya 572029

5 Beijing Institute of Applied Meteorology, Beijing 100029

ABSTRACT Data assimilation systems usually assume that the observation errors of wind components, i.e., u (the longitudinal component) and v (the latitudinal component), are uncorrelated. However, since wind components are derived from observations in the form of wind speed and direction (spd and dir), the observation errors of u and v are correlated. In this paper, an explicit expression of the observation errors and correlation for each pair of wind components are derived based on the law of error propagation. The new data assimilation scheme considering the correlated error of wind components is implemented in the Weather Research and Forecasting Data Assimilation (WRFDA) system. Besides, adaptive quality control (QC) is introduced to retain the information of high wind-speed observations. Results from real data experiments assimilating the Advanced Scatterometer (ASCAT) sea surface winds suggest that analyses from the new data assimilation scheme are more reasonable compared to those from the conventional one, and could improve the forecasting of Typhoon Noru.

Key words: data assimilation, Advanced Scatterometer (ASCAT), wind components, correlated observation error

1. Introduction

Wind observations, especially those derived from satellites, are of great importance for improving the numerical prediction of tropical cyclones. Observation products from scatterometers constitute one of the most important sources of sea surface wind data. The ECMWF was the first numerical weather forecasting center in the world to successfully incorporate the first European Remote Sensing Satellite (ERS-1) scatterometer wind data into a global three-dimensional variational system(Andersson et al., 1998). Previous studies have shown that wind data derived from scatterometers are of great importance for weather forecasting and climate monitoring (e.g., Stoffelen and Cats, 1991; Stoffelen and van Beukering, 1997; Atlas and Hoffman, 2000; Isaksen and Stoffelen, 2000; Atlas et al., 2001; Candy, 2001; Lin et al., 2017). In particular, the usefulness of such data has been demonstrated in the prediction of tropical cyclones(Isaksen and Stoffelen, 2000) and extratropical cyclones(Stoffelen and van Beukering, 1997).

Before assimilation, wind observations are usually first transformed into longitudinal and latitudinal components,uandv, although most wind observations are observed or derived in the form of wind speed (spd) and wind direction (dir). Since theuandvcomponents are derived from observations of spd and dir, the observation errors of theuandvcomponents are actually correlated and directly impacted by the errors of spd and dir.However, most current data assimilation systems—for example, the Weather Research and Forecasting Data Assimilation (WRFDA) system (Barker et al., 2012)—assume for the sake of simplicity that the observation errors of wind components are uncorrelated (Huang et al.,2013). This simplification impairs the results of analysis since correlated observations bring in redundant information to the data assimilation. Huang et al. (2013) proposed a method for assimilating wind observations in their observed form, and thus the observation errors of spd and dir could be directly taken into account in the assimilation process. Although this method can help avoid the introduction of correlation into the wind components in the process of transformation, it may also introduce propagation error due to the observation operator that transforms background wind components to spd and dir.This also impacts the process of quality control (QC),which is based on the departure between the observation and background. In light of these problems, it is necessary to find a way to estimate the errors and correlation of the wind components derived from spd and dir, so as to make it possible to assimilate the correlated wind components directly into the data assimilation system.

The rest of this paper is organized as follows. In Section 2, we discuss the error characteristics of wind observation from different viewpoints, and give a definite expression of the observation errors and correlation for each pair of wind components based on the law of error propagation. In Section 3, we describe the possibility of directly assimilating the correlated wind components.The results from real data experiments with Advanced Scatterometer (ASCAT) wind observations are given in Section 4. Finally, conclusions are drawn in Section 5.

2. Error characteristics of wind components

2.1 Wind observation errors

In most data assimilation systems, wind observations are assimilated in the form ofuandvcomponents. For one wind vector collected in the form of spd and dir, the wind components can be acquired by using the equation

It is certain from Eq. (1) that both the spd error and the direrror can impact the uncertainties ofuandv.However, data assimilation systems (Barker et al., 2012)usually use the observation errors ofuandvestimated only from the spd error. It is obvious that the errors ofuandvare correlated to each other, but they are assumed to be independent in data assimilation systems.

Next, we discuss the impact of considering the difference in the wind observation errors between the windvector (spd,dir) form and the wind-components (u,v)form. Figure 1 shows the region where the wind “truth”might be located when given the wind observation with uncertainties. Figure 1a is from the wind-vector perspective where the uncertainties (δspd, δdir) of one wind-vector pair (spdo, diro) are given. Figure 1b, meanwhile, is from the wind-components perspective, which assumes that the uncertainties (δu, δv) of the (uo,vo) are fixed and uncorrelated. For most observing systems acquiring wind information in the form of spd and dir, Fig. 1a is more reasonable for the real world. Under this circumstance,the uncertainties of wind components are no longer fixed values, but vary along with the values of spd anddir when their uncertainties are given. As shown in Fig. 1a,the possible region where the wind “truth” might be located will change in different (spdo,diro) pairs, as will the range ofutandvt(where the superscript t means “truth”).In other words, the real errors of wind components do not only depend on the values of spd error and dir error, but also on the observed values of spd and dir, and the errors of each (uo,vo) pair should be correlated.

Fig. 1. Region where the wind “truth” might be located when given the wind observation with uncertainties. The red arrow is the wind observation, while the blue ones are the corresponding wind components. The red shadow region is the place where the wind “truth” might be located given the uncertainties of the wind observation. Panel (a) is from the wind-vector perspective, in which the uncertainties of spd and dir (δspd and δdir) are known. Panel (b) is from the wind-components perspective, in which the uncertainties of u and v (δu and δv) are known.

2.2 Correlated errors of observed wind components

In this subsection, we calculate the correlated errors of observed wind components based on the law of error propagation (Ochoa and Belongie, 2006). The law of error propagation is widely used in the area of mapping and surveying. It maps the error (or errors) from one variable (or variables) to another. A brief introduction to this theory is given as follows.

Letx∈Rnbe a random vector with mean μxand covariance matrix ?x, and letf:Rn→Rmbe a nonlinear function. Assume thaty=f(x), andJ∈Rm×nis the Jacobian matrix ?f/?xevaluated at μx. Then, the random vectory∈Rmhas mean μy≈f(μx) and covariance?y≈J?xJT[see Ochoa and Belongie (2006) for further details], where the superscript T denotes the transpose.

For the wind vector transformation equation from(spd,dir) to (u,v) [see Eq. (1)], we havex=(spd,dir)Tandy=(u,v)T, and the Jacobian matrix is

The covariance matrix ofycan thus be written as

The correlation coefficient ρu,vcan also be calculated by using the equation

To better demonstrate how the error characteristics of wind components would vary along with the observed values of wind, we calculate the standard deviations of the wind components within a certain range of the observed values. Furthermore, in this study, an assumption is made that the standard deviations of spd and dir are unchanged and are set to 2 m s?1and 20°, respectively.The standard deviations ofuandvalong with their correlationareshown inFig. 2;theobserved wind vectors vary withinthe rangeof ?25to25 m s?1withan interval of 0.05ms?1. Eachpoint inthe(u,v) coordinate space corresponds to a (spd,dir) vector. We can see from Fig.2a that the standard error ofudoes not remain constant but grows rapidly in its vertical (latitudinal) direction.This is because the magnitude of the standard deviation is mainly determined by its second term, which changes rapidly with the spd if the value ofvis nonzero. The same is true forv(see Fig. 2b). Furthermore, the errors of the wind components are highly correlated, especially in the diagonal direction, as shown in Fig. 2c. Errors of the(u,v)pair are negatively correlated; while if the signs are opposite, the correlation is positive. The correlation is zero only when the value of one component is zero.However, the absolute value of the correlation would always be less than one.

Fig. 2. Standard errors and corresponding correlation of u and v within a certain range of the observed values: (a) standard errors of u, denoted as σu; (b) standard errors of v, denoted as σv; and (c) error correlation of u and v, denoted as ρu,v.

Based on the above analysis, we can see that the new scheme improves the wind assimilation process from two aspects. On the one hand, it changes the fixed weight assigned to the wind observation based on the observation error in the assimilation process. The observation error of theuwind component can reach 7 m s?1when the spd exceeds 20 m s?1, as can be seen from Fig. 2a. Under strong convective weather conditions, such as in the area of the typhoon center, the spd is much higher, and so too is the uncertainty of the observed wind, meaning that reducing the weight of such wind observations is appropriate. In addition, consideration of the correlation between the observation errors of wind components can be seen as another correction to the observation weight. On the other hand, the increase in the magnitude of the observation error is equivalent to appropriately relaxing the threshold condition of QC. This ensures that more observations can be incorporated into the assimilation system, especially for strong convective weather conditions.

3. Assimilation of correlated wind components

After the new observation errors and corresponding correlation of the pair of (u,v) components are obtained,the QC scheme must first be adjusted. The threshold for the criterionis no longer a fixed value, but varies with the observation error. Another important factor is the accuracy of the background error covariance, since it plays an important role in the QC procedure and assigning weight to the minimization of the cost function. For extreme weather conditions, background variances are typically more than 10 times larger than the local climatology (Bonavita et al., 2017). Thus,we use a flow-dependent background error variance based on the hybrid ensemble transform Kalman filter–three-dimensional variational data assimilation (ETKF–3DVAR) scheme developed for WRF (Wang et al.,2008), which will also make a difference to the QC. The cost function of the assimilation also needs to be adjusted since the error correlation between the pair of(u,v)components is considered.

3.1 Cost function of correlated wind components

The mathematical formulation of the hybrid method in WRFDA is described in Wang et al. (2008). Therefore,we only give a brief introduction here. The final analysis increment δxof the hybrid is a sum of two terms,

where the first term δx1is the analysis increment associated with the 3DVAR static background covariance and the second term is the increment associated with the flow-dependent ensemble covariance. The vectorakdenotes the extended control variable for thekth ensemble member. The symbol ? denotes the Schur product of the vectors √akandandis thekth perturbation normalized by, whereKis the ensemble size:

Here,xkis thekth ensemble forecast andis the mean of theK-member ensemble forecasts.

The analysis increment δxis obtained by minimizing the following cost function:

Compared to the normal 3DVAR cost function, the weighted sum of the termsJsandJereplaces the usual background term. Here,Jsis the traditional 3DVAR background term associated with the static covarianceB,andJeis the background term associated with the flowdependent ensemble covariance. Coefficients β1andβ2are the weights assigned to the static background-error covariance and the ensemble covariance and are constrained by

MatrixBis the background error covariance matrix,and the superscripts ?1 and T denote the inverse and adjoint, respectively, of a matrix or linear operator. Here,ais formed byKvectorsak, in whichAis a block-diagonal matrix that defines the spatial covariance ofa.

Here, we mainly concentrate on the observation termJo, of whichd=y?H(xb) is the innovation vector andyis the observation vector. In a hybrid data assimilation scheme, the backgroundxbis the ensemble mean forecast,His the nonlinear observation operator,His the linearizedH, andRois the observation error covariance matrix.

We only consider the correlation between the pair of wind components within one wind vector observation,and assume that errors between different wind vector observations are independent. Therefore, the minimization of the cost function of the observation can be calculated independently. For wind components, the cost function of one observation can be written as

where δu=ua?uband δv=va?vbare the analysis increments ofuandv, respectively; andduo=uo?ubanddvo=vo?vbare the innovations. Assuming thatX=(x1,x2)T,withx1=δu?duo andx2=δv?dvo , then the cost functionJo(δx)is a quadratic function of vectorX, which is

whereAis the inverse of the error covariance matrix.The condition that the quadratic functionf(X) has a minimum is that the matrixAis a positive definite matrix,which is the same as saying that the inverse ofAis a positive definite matrix. From the equation above, we have

such that

Then, for the error covariance matrix, we have

Therefore, the error covariance matrix is a positive definite matrix; that is, the quadratic functionf(X) has its minimum.

3.2 Variance inflation

The observation error matrixRois a diagonal matrix under the independent error assumption, so the minimization of the cost function of wind components can be calculated separately. Unlike the traditional scheme, the minimization for the new scheme should be considered simultaneously since the wind components are correlated. The same is true for the gradient of the cost function, in which the gradient for the new scheme is a vector rather than a scalar, as shown by

whereg1andg2denote the gradients ofuandv, respectively. The inverse of the error covariance matrix can be derived as follows:

Here, ρ is short for ρu,v. We can see from the equation above that when ρ=0, the gradient vector becomes

which is equal to the scheme with independent error assumption. However, when the absolute value of ρ is close to one, the magnitude of the gradient value will be extremely large and slows down the convergence of the minimization process. To avoid this problem, we must limit the value of ρ to a certain range. In this study, we adopt the following constraint:

where the subscriptxdenotesuorv. The reason for inflation is to reduce the weight of the observation in the analysis as a compensation for the reduction of correlation.The variance inflation guarantees that the matrixAwill still be a positive definite matrix.

The new variances and correlation are shown in Fig. 3.The inflation has no impact on the low spd observations,because wind components with absolute values of correlation larger than 0.9 are mainly distributed in the region where both the absolute values ofuandvare larger than 13 m s?1. Therefore, this strategy will make a difference to the wind observations in high-wind conditions, especially for tropical cyclones.

Fig. 3. As in Fig. 2, but after variance inflation.

3.3 Adaptive QC

Conventional QC procedures based on selective blacklisting and first-guess rejections are too blunt to deal with the task of whether to retain an observation with a large initial departure from the model background (Bonavita et al., 2017). Observations with a large departure from the background do not mean that they have large errors.Sometimes, the background also has large errors, especially in extreme weather conditions. The strategy of QC should be considered carefully to assign a reasonable weight to the observation, since the observation error is varying with the observed value.

For traditional QC using an error cutoff method, an observation that satisfiescan be retained by the QC step, whereyis the observation andhis the observation (or forward model) operator linking the model statexto the observation; σoand σbrepresent the standard deviation of the observation error and background equivalent error, respectively; and α is the cutoff factor. For the WRFDA system, α is usually set to 5,which means an observation with departure more than five times the value ofwill be rejected by QC. However, in extreme weather conditions like those of a typhoon, it is often the case that high spd observations near the typhoon center are screened out by such a QC scheme. This is usually because of the incorrect expression of background error covariance in the meteorological environment of the typhoon.

In situations in which we can retain the high-windspeed observational information, as well as the incremental 3DVAR within the bounds of validity of the tangent linear hypothesis (see Section 3.1), we use an adaptive QC scheme that was proposed by ECMWF (Bonavita et al., 2017). Here, we give the new QC scheme directly as

4. Real data experiments

4.1 ASCAT wind data

The wind observation data used in this study were produced by Remote Sensing Systems (Ricciardulli and Wentz, 2016), which were derived from ASCAT, which is one of the instruments carried on board the MetOp polar satellites operated by the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) (Bi et al., 2011). The data files contain the measurements for one orbit of the satellite around the earth and are organized by observation cells that are perpendicular to the direction of the satellite travels.

Scatterometer wind products have been used in many operations, such as those of ECMWF (Isaksen and Stoffelen, 2000), the UK Met Office (Candy, 2001), and the NCEP (Bi et al., 2011). ASCAT obtains three independent backscatter measurements using three different viewing directions separated by a short time delay. Then, the surface spd and dir are calculated using these “triplets”within the C-2015 geophysical model function (GMF;Ricciardulli and Wentz, 2012).

The ASCAT wind product used in this study is from theMetOp-Asatellite, with a grid size of 25 km. The spd error and dir error are assumed uncorrelated in this study and are set to 2 m s?1and 20° for experimental use, respectively. Figure 4a shows the ASCAT wind field of the center of Typhoon Noru (2017) at 2100 UTC 25 July 2017.

Figure 4 also shows the new error characteristics of theuandvcomponents, in which Figs. 4b, c present the standard deviations of the observation errors ofuandv,respectively, which are derived from the standard deviations of the spd and dir; and Fig. 4d shows the corresponding correlation. We can see from Fig. 4 that the error standard variances of wind components are affected by both spd and dir, which is consistent with the theory presented in Section 2. However, the error correlation is highly direction-related, and it reaches its maximum when theuandvhave the same amplitude.

Fig. 4. ASCAT wind field of Typhoon Noru with a grid size of 25 km: (a) wind speed (spd) field, (b) standard deviation of u component, (c)standard deviation of v component, and (d) error correlation of wind components.

4.2 Experimental design

This study uses Typhoon Noru (2017) as a numerical example, which generated in the Northwest Pacific Ocean on 20 July 2017 and further intensified into a typhoon on 23 July 2017. Here, we implement the new data assimilation scheme that considers the correlation of theuandvcomponents and the adaptive QC in the WRFDA system. The WRFDA was developed by NCAR(Barker et al., 2004) and is a widely used operational system. It can produce a multivariate incremental analysis in the WRF model space (Zhang et al., 2009). The setup of the assimilation region is 80×80 with a resolution of 25 km, and the vertical discretization is 51 layers. The time of assimilation is 2100 UTC 25 July 2017. We use the FNL (final) global analysis data provided by NCEP as the initial field and boundary conditions, and take the ensemble mean of 40 forecasts [generated using the Data Assimilation Research Testbed (Anderson et al., 2009)]adjustment from 1800 UTC 23 to 2100 UTC 25 July 2017 as the background field of the assimilation system.After the assimilation, a 24-h forecast is made, which is a forecast to 2100 UTC 26 July 2017. A set of assimilation and comparison experiments are carried out, as shown in Table 1.

The control experiment is the natural run of the forecast without assimilation of any observations. Accordingly, we have three groups of assimilation experiments,including two wind component (u,v) schemes and one wind vector (spd,dir) scheme.

5. Results

Figure 5 gives the ensemble mean of the wind fields of forecasts and corresponding flow-dependent error of theuandvcomponents. The background error of the wind field shows very similar characteristics to that of the observation in the central area of the typhoon. As shown in Figs. 4b, c; 5b, c, the error distribution is highly correlated to the values of spd and dir.

The results of QC by using different assimilation schemes are shown in Table 2 and Fig. 6. The reason for the observation gap in the center of the typhoon area for the wind-component schemes is mainly because of the position of the typhoon center in the background wind field (27.5°N, 157.5°E) being different from that of the observed wind field (28°N, 158°E). The misalignment of the typhoon vortex structure will result in a large deviation between the wind components of the background and observation. However, the use of the adaptive QC loosens the observation error in the high-wind region,which is equivalent to relaxing the QC threshold, and thus more observations can be reserved in the QC step(as shown in Fig. 6). For the wind vector scheme, the adaptive QC in the (spd,dir) form results in a much larger QC threshold than that of the wind-component schemes.Reasonable observation error also means better weight assignment for the cost function of assimilation, which is of great significance for the minimization process of assimilation. As listed in Table 2, the number of iterations of minimization for the new scheme is much smaller than that of the traditional schemes.

Table 1. Different assimilation schemes for the ASCAT wind observation

Table 2. QC and number of iterations of minimization in different assimilation schemes

Fig. 5. Ensemble mean of the wind fields of forecasts and corresponding flow-dependent error of the u and v components: (a) ensemble average of the wind fields of forecasts, (b) standard deviation of the u component, and (c) standard deviation of the v component.

Fig. 6. QC using different assimilation schemes, in which the blue dots represent the wind observations that are successfully assimilated into the data assimilation system: (a) (u, v) with independent errors, (b) (spd, dir) with independent errors, and (c) (u, v) with correlated errors.

For verification, we use the reanalysis data of ERA-Interim as the “truth.” ERA-Interim is the latest global atmospheric reanalysis, available from 1979, produced by ECMWF, and is continuously updated in real time (Dee et al., 2011). Figure 7 presents the analysis errors (analysis minus “truth”) of the spd field, and Fig. 8 gives the corresponding mean error and root-mean-square (RMS)value of the different experiments. Compared to the background, the wind-component assimilation schemes improve the spd field and the new scheme has the best fit to the reanalysis. As shown in Fig. 7, the assimilation of ASCAT wind components has improved the wind field structure in the typhoon area [the typhoon center is located at (28°N, 158°E) at the analysis time], which is critical to the track forecast. Although more observations are assimilated in the wind vector scheme, the improper observation error assigned for the high-wind observations may cause disharmony between the model states and the observations.

The sea surface pressure errors of different experiments are also compared in Figs. 9, 10. Assimilation of ASCAT wind observations using wind-component schemes improves the sea surface pressure field compared to the background. The scheme using wind components with correlated error shows the best result for the pressure field (see Fig. 10d for details). However, they all have a bias compared to the reanalysis.

Fig. 7. Wind speed (spd) analysis errors of different experiments: (a) background, (b) (u, v) components with independent error, (c) (spd, dir)with independent error, and (d) (u, v) components with correlated error.

Fig. 8. Mean error and root-mean-square (RMS) of spd analysis errors of different experiments: (a) background, (b) (u, v) components with independent error, (c) (spd, dir) with independent error, and (d) (u, v) components with correlated error.

Table 3 gives the analysis errors of the different assimilation schemes. It shows that in all cases the analysis errors of the spd field and pressure field improve when using the new assimilation scheme, which suggests that the new method is very promising for data assimilation.

Table 3. Mean error and RMS of analysis errors in different experiments

It is also instructive to examine how the forecast skill changes with time. Figure 11a shows the best track and forecast typhoon paths of the different experiments, and Fig. 11b gives the errors of the forecast typhoon paths.The best-track data are from the China Meteorological Administration’s tropical cyclone database (tcdata.typhoon.org.cn; Ying et al., 2014). The different assimilation schemes each show some improvement in terms of the forecast path compared to that of the control experiment. The new scheme shows very promising results in the initial stage of the forecast compared to the other schemes. The track forecast of the new scheme begins to diverge at 18 h; however, it is still effective for the 24-h forecast.

Fig. 9. As in Fig. 7, but for pressure analysis errors.

Fig. 10. As in Fig. 8, but for pressure analysis errors.

The intensity of the typhoon based on the different schemes is also compared (Fig. 12), from which we can see that the new scheme shows strong improvement in intensity forecasts compared to the traditional scheme.To better demonstrate the results of the different experiments, Table 4 gives the average forecast errors of the typhoon track, minimum pressure, and maximum spdof the typhoon eye. In each case the new scheme shows a promising result.

Table 4. Mean error of the typhoon forecast in different experiments

Fig. 11. The forecast (a) tracks and (b) track errors of Typhoon Noru (2017).

Fig. 12. (a) Minimum pressure and (b) maximum spd forecast errors of the typhoon eye.

6. Summary

This paper presents a new method for assimilating remotely sensed sea surface wind components by considering their correlation. We have theoretically demonstrated how different it can be when considering the error characteristics of wind observations in their original form and the wind-components form. Given the error variances of wind observations in the form of spd and dir, the error variances and correlation of corresponding wind components can be calculated by the law of error propagation. Results from real data experiments compared to the traditional (u,v) assimilation scheme and the (spd,dir)scheme verify the ability of this new method.

Since the main focus of this study is on the correlation of the wind components, other observation error sources, such as representativity error, have not been addressed, which may also affect the configuration of the observation error variance matrix. We also assume in this study that the observation errors of spd and dir are not correlated; however, large direction errors in wind observation are correlated to lighter spd for in situ observations (e.g., Dobson et al., 1980; Schwartz and Benjamin,1995; Gao et al., 2012), as well as remotely sensed wind observations (e.g., Plant, 2000; Ebuchi et al., 2002). We will deal with these aspects in future studies.