The Chinese Meteorological Society
Article Information
- WANG, Ting, Jie XIANG, Jianfang FEI, et al., 2017.
- Evaluation of the Impact of Observations on Blended Sea Surface Winds in a Two-Dimensional Variational Scheme Using Degrees of Freedom. 2017.
- J. Meteor. Res., 31(6): 1123-1132
- http://dx.doi.org/10.1007/s13351-017-6798-7
Article History
- Received December 22, 2016
- in final form August 2, 2017
2. Key Laboratory of Mesoscale Severe Weather of Ministry of Education, Nanjing University, Nanjing 210093;
3. Guangdong Institute of Tropical and Marine Meteorology of China Meteorological Administration, Guangzhou 510080;
4. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240
In recent decades, data assimilation systems have become highly advanced and complex (Lorenc et al., 2000; Rabier, 2005), and a large variety of space- and ground-based observations are assimilated through these systems. Effective evaluation of the performance of such complicated assimilation systems appears increasingly necessary. Traditionally, assimilation systems are evaluated by comparing analysis fields with reanalysis, radiosonde, and other independent observational data, or by the performance of the forecasts initialized by those analyses. Recently, attention has also been paid to the methods of self-evaluation, which evaluate the assimilation results using the features of the assimilation system itself rather than by comparison with third-party independent observations or by the performance of the forecasts. The sensitivity of analysis to observations is one such technique.
The sensitivity of analysis to observations is mainly used to quantify the observational impact on analysis fields, which show how different variables affect the assimilation system, how much influence is due to a particular observational type of data in the assimilation product, how much influence comes from observations and from the background, and so on. Evaluations of the observational impact on the analysis are important for understanding the data assimilation system itself. With the increasing number of observational data used in modern assimilation systems, evaluating the observational influence through sensitivity analysis is highly necessary for the optimal employment of these data.
The sensitivity of analysis to observations can be described by the derivative of the analysis with respect to the observations, and is obtained by the adjoint method (Daescu and Langland, 2013; Lorenc and Marriott, 2014; Boullot et al., 2016) or by the degrees of freedom for signal (DFS) in three- or four-dimensional variational data assimilation (3D/4D-Var) ( Lupu et al., 2012; Brousseau et al., 2014; Turner and Löhnert, 2014). The DFS is among the methods to describe the information content contained in observations for an inverse problem, and a detailed approach was presented by Rodgers (2000) to quantify the information content of observations using the DFS in inverse processing. For a complex assimilation system, since the DFS depends on the observational and background error statistics used in the assimilation, direct evaluation of the DFS is so costly for the size of the covariance matrices of the observational and background errors that some approximate calculation of the DFS is a necessity. Cardinali et al. (2004) used the DFS to evaluate the analysis sensitivity to observations in the 4D-Var system of the ECMWF, and developed an effective approximation method to calculate the DFS for a large-dimensional variational data assimilation system. The results showed that 15% of the global influence was due to the assimilated observations in any one analysis, and the complementary 85% was due to the influence of the background field. Chapnik et al. (2006) applied a perturbation method with the Action de Recherche Petite Echelle Grande Echelle 4D-Var system of Météo-France. By utilizing the algorithm of Bai et al. (1996), Fisher (2003) calculated the DFS in a version of the ECMWF 4D-Var system, assimilating many thousands of observations. Lupu el al. (2011) provided a practical method to estimate the DFS from a posteriori statistics, in which the consistency of the error statistics is not required in the analysis system. They estimated the observational impact on the analysis using observational departures from the analysis and forecast. Because it is a natural by-product of the assimilation process, the calculation by this method is conveniently simple. The Shannon information content method, a measure of the reduction of entropy, is another way to obtain information content. An example of the application of this method was presented in Du et al. (2010), in the context of High Resolution Infrared Radiation Sounder, Advanced Microwave Sounding, Atmospheric InfraRed Sounder, and Infrared Atmospheric Sounding Interferometer data, by evaluating the temperature and humidity information content. Some valuable applications in the design, evaluation, and comparison of the different instruments and the use of remote sensing data were introduced in their work.
The assessment of the contribution of each type of observation to the analysis for a specific data assimilation system is helpful toward understanding the data assimilation system itself, and provides guidance for further development of the system. Besides, studying the relationship between the observational influence and the characteristics of the observational and background errors is useful in comprehending the nature of the observational influence. The main goal of the present study is to evaluate the observational influence on the analysis using the DFS calculated by thea priori and a posteriori methods for a 2D-Var sea surface wind blending scheme, and the relationship between the observational influence and the observational and background errors. Section 2 describes the concept of the sensitivity derivative and DFS, and the methodology for computing the DFS. Section 3 firstly introduces the 2D-Var sea surface wind blending scheme, and then describes the calculation of the DFS and quantification of the impacts of the NASA Quick Scatterometer (QuikSCAT) data on the analysis. Conclusions and discussion are presented in Section 4.
2 Self-sensitivity and the DFSConsider a measurement equation in an assimilation system:
$y=H\left(x\right)+{\epsilon}_{o},$ | (1) |
where y is the observational vector, x the state vector, H the observational operator, and ε_{o} the vector of observational errors. The optimal analysis x_{a} can then be written as:
${x}_{\text{a}}={x}_{\text{b}}+K\left[y-H\left({x}_{\text{b}}\right)\right],$ | (2) |
where
$K=B{H}^{\text{T}}{\left(R+HB{H}^{\text{T}}\right)}^{-1}={\left({B}^{-1}+{H}^{\text{T}}{R}^{-1}H\right)}^{-1}{H}^{\text{T}}{R}^{-1}\text{\hspace{1em}}\text{\hspace{0.17em}}$ | (3) |
is the optimal gain matrix expressed by the background error covariance matrix B, the observational error covariance matrix R, and H the tangent linear model of H, linearized in the vicinity of x_{b}.
For the linear case, projecting x_{a} onto the observational space yields the corresponding observational vector:
$\widehat{y}\equiv H{x}_{\text{a}}=HKy+\left(I-HK\right)H{x}_{\text{b}}.$ | (4) |
It can be seen that the analysis in the observational space (Hx_{a}) is a weighted sum of the observational vector y and background state in the observational space (Hx_{b}).
From Eq. (4), the derivative of the analysis with respect to the observations is:
$S\equiv \frac{\partial \widehat{y}}{\partial y}={K}^{\text{T}}{H}^{T},$ | (5) |
where S is called the influence matrix (Fisher, 2003; Cardinali et al., 2004). The elements of S are
The DFS is defined, according to inverse problem theory, as the trace of S:
$\text{DFS}=\text{tr}\left(S\right)=\text{tr}\left\{\frac{\partial \left(H{x}_{\text{a}}\right)}{\partial y}\right\}=\text{tr}\left({K}^{\text{T}}{H}^{\text{T}}\right)=\text{tr}\left(HK\right),$ | (6) |
where tr{·} denotes trace of {·}. The DFS is a measure of the total sensitivity of the analysis to observations, and also represents the information content extracted from the observations in an inversion or data assimilation application. Therefore, the DFS can be regarded as an important diagnostic index. Similarly, the derivative of the analysis to the background can be derived as:
$\frac{\partial \widehat{y}}{\partial \left(H{x}_{\text{b}}\right)}=I-{K}^{\text{T}}{H}^{\text{T}}=I-S,$ | (7) |
which is called the “degrees of freedom for background” (Cardinali and Healy, 2014).
Because the size of the matrices in Eq. (6) is generally large and the gain matrix in a large variational assimilation system is not available, a straightforward evaluation of the DFS in Eq. (6) is hard to achieve. A variety of ways have therefore been proposed to compute the DFS.
Rodgers (2000), for instance, suggested a method based on the singular value decomposition (SVD) to evaluate the DFS. The measurement equation, Eq. (1), is transformed into the following form:
$\tilde{y}\text{}=\text{}\tilde{H}\tilde{x}+\tilde{\epsilon},$ | (8) |
where
$\begin{array}{l}\tilde{y}={R}^{-1/2}\left(y-H{x}_{\text{b}}\right),\\ \tilde{x}={B}^{-1/2}\left(x-{x}_{\text{b}}\right),\\ \tilde{H}={R}^{-1/2}H{B}^{1/2}.\end{array}$ | (9) |
The DFS is available in terms of the singular values of
$\text{DFS}=\underset{i=1}{\stackrel{m}{{{{\displaystyle \sum}}^{\text{}}}^{\text{}}}}\frac{{\lambda}_{i}{}^{2}}{1+{\lambda}_{i}{}^{2}},$ | (10) |
where λ_{i} is the ith singular value of
Another approach (Lupu et al., 2011) for calculating the DFS is based on the statistics of the a posteriori information. Desroziers and Ivanov (2001) proposed a set of diagnostics according to the following three differences: between the observation and background [
$\text{E}\left[{d}_{\text{a}}^{\text{o}}{\left({d}_{\text{b}}^{\text{o}}\right)}^{\text{T}}\right]=\tilde{R}=R{D}^{-1}\tilde{D},\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ | (11a) |
$\text{E}\left[{d}_{\text{b}}^{\text{a}}{\left({d}_{\text{b}}^{\text{o}}\right)}^{\text{T}}\right]=H\tilde{B}{H}^{\text{T}}=HB{H}^{\text{T}}{D}^{-1}\tilde{D},$ | (11b) |
$\text{E}\left[{d}_{\text{b}}^{\text{a}}{\left({d}_{\text{a}}^{\text{o}}\right)}^{\text{T}}\right]=H\tilde{A}{H}^{\text{T}}=HK\tilde{D}{D}^{-1}R,\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ | (11c) |
$\text{E}\left[{d}_{\text{b}}^{\text{o}}{\left({d}_{\text{b}}^{\text{o}}\right)}^{\text{T}}\right]=\tilde{D}=H\tilde{B}{H}^{\text{T}}+\tilde{R},\text{\hspace{1em}}\text{\hspace{1em}}$ | (11d) |
where E[·] is the statistical expectation operator,
Based on Eqs. (11a)–(11d), the a posteriori gain matrix can be defined as:
$\begin{array}{l}\tilde{K}=\tilde{B}{H}^{\text{T}}{\left(\tilde{R}+\text{}H\tilde{B}{H}^{\text{T}}\right)}^{-1}\\ \text{\hspace{1em}}={\left({H}^{\text{T}}{\tilde{R}}^{-1}H+{\tilde{B}}^{-1}\right)}^{-1}{H}^{\text{T}}{\tilde{R}}^{-1}=\tilde{A}{H}^{\text{T}}{\tilde{R}}^{-1}.\end{array}$ | (12) |
The a posteriori DFS is then
(13) |
From Eqs. (11a) and (11c), we have
$\begin{array}{l}{\tilde{R}}^{-1}{\left(H\tilde{A}{H}^{\text{T}}\right)}^{\text{T}}={\tilde{R}}^{-1}{(}^{H}\\ \text{\hspace{1em}}={\tilde{R}}^{-1}R\tilde{D}{D}^{-1}{K}^{\text{T}}{H}^{\text{T}}={K}^{\text{T}}{H}^{\text{T}}\uff0c\end{array}$ | (14) |
and consequently
$D\tilde{F}S=\text{tr}({\tilde{K}}^{\text{T}}{H}^{\text{T}})=\text{tr}\left({K}^{\text{T}}{H}^{\text{T}}\right)=\text{DFS}\text{.}$ | (15) |
This shows that the DFS can be obtained either from the a priori information, or from the a posteriori statistics. Substituting Eq. (11c) into Eq. (13), the a posteriori can be written as
(16) |
which indicates that the a posteriori DFS can be estimated by the a posteriori differences and observational covariance matrix
In order to generate a regional high-resolution sea surface wind product, Xiang et al. (2015) applied a 2D-Var scheme to blend the Level-3 sea surface vector winds from QuikSCAT (with a horizontal resolution of 0.25° × 0.25°) and sea surface vector winds from a high-resolution mesoscale forecast model. The high-resolution mesoscale model was the Guangzhou Mesoscale Model (GZMM) and covered a limited area with a horizontal resolution of 0.125° × 0.125°. In the GZMM, the data assimilation module is from the Global/Regional Assimilation and Prediction System’s 3D-Var system; and conventional observations, Advanced TIROS (Television and Infrared Observation Satellite) Operational Vertical Sounder satellite radiances, and radar radial velocity data, amongst others, are directly assimilated. However, scatterometer sea surface vector wind products, such as QuikSCAT data, are not assimilated.
Because the present paper focuses on the evaluation of the observational influence on analyses in the blending of sea surface wind data, the 2D-Var scheme is briefly reviewed below. The daily QuikSCAT data are obtained by averaging the ascending and descending passes and daily model data by averaging 0–24-h GZMM forecast data. The research area is 10.12°–17.92°N, 110.12°– 117.92°E (Fig. 1), a domain located in the South China Sea. The blending of the two kinds of sea surface winds via the 2D-Var method can be achieved by minimizing the cost function, as defined by:
${J}_{\text{total}}={J}_{\text{q}}+{J}_{\text{g}},$ | (17) |
where
$\begin{array}{l}{J}_{\text{q}}=\frac{1}{2}{\left({H}_{\text{q}}u-{u}_{\text{q}}\right)}^{\text{T}}{Q}_{u}^{-1}\left({H}_{\text{q}}u-{u}_{\text{q}}\right)\\ \text{\hspace{1em}}+\frac{1}{2}{\left({H}_{\text{q}}v-{v}_{\text{q}}\right)}^{\text{T}}{Q}_{v}^{-1}\left({H}_{\text{q}}v-{v}_{\text{q}}\right),\end{array}$ | (18) |
$\begin{array}{l}{J}_{\text{g}}=\frac{1}{2}{\left(u-{u}_{\text{g}}\right)}^{\text{T}}{M}_{u}^{-1}\left(u-{u}_{\text{g}}\right)\text{\hspace{1em}}\text{\hspace{1em}}\\ \text{\hspace{1em}}+\frac{1}{2}{\left(v-{v}_{\text{g}}\right)}^{\text{T}}{M}_{v}^{-1}\left(v-{v}_{\text{g}}\right),\end{array}$ | (19) |
where u and v are the zonal and meridional components of the wind vector (on the GZMM grid); the subscripts q and g represent QuikSCAT and GZMM, respectively;
In any variational data assimilation system, among the key issues are the selection of observational operators and the specification of the error covariance matrices of the backgrounds and observations. In the present case, both the observations and analysis are the sea surface wind; H_{q} is the interpolation operator via bilinear interpolation. For simplicity, we assume that observational error variances are uniformly distributed. Therefore, the observational error variances can be obtained by comparing QuikSCAT measurement data and the automatic weather station in situ observations. The National Meteorological Center (NMC) method (Parrish and Derber, 1992) is used to estimate the background error covariance matrix. For the NMC method, the model-prediction errors are approximated as the difference between two forecasts of different length (e.g., 24 and 48 h, or 12 and 24 h) valid at the same time, and then the prediction error covariance matrix is determined statistically. For the present case, the background error covariance can be calculated from the following statistical expectation:
$M=\text{E}\left[{x}^{\prime}{{x}^{\prime}}^{\text{T}}\right],$ | (20) |
where
In this section, the DFS concept is applied to the 2D-Var sea surface wind blending scheme described above, and the a priori method based on the SVD of the transformed observational operator [Eq. (10)] and the a posteriori method based on statistical diagnostics [Eq. (16)] are used to compute the DFS, which are referred to as DFS_{PRIO} and DFS_{POST}, respectively. Blending of sea surface wind data is performed from 1 August to 31 December 2008 on a daily basis, for a total of 147 days.
Table 1 gives the estimated DFS for the u and v components by the three different evaluation methods: the analytic method (DFS_{ANALYTIC}), the a priori method based on SVD (DFS_{PRIO}), and the a posteriori method based on statistical diagnostics (DFS_{POST}). Clearly, the differences between the DFS results of the three methods are so small that they can be considered the same. In addition, for each evaluation method, the DFSs for the u and v components are nearly equal, which indicates that although the variances of the observational and background errors for the u and v components are different, so almost the same DFS can be obtained. This shows that the u and v components provide nearly identical degrees of freedom, i.e., from the whole area, the same impact on the analysis.
u | v | Total | |
DFS_{ANALYTIC} | 414.7619 | 414.5477 | 829.3096 |
DFS_{PRIO} | 414.7621 | 414.5479 | 829.31 |
DFS_{POST} | 414.7639 | 414.5429 | 829.3068 |
For comparing the time cost of the different DFS calculation methods, we calculate the DFS using Eq. (6), which is denoted as DFS_{ANALYTIC}. As shown in Table 2, because of the heavy computational burden for calculating the gain matrix, the time cost in the direct calculation of the DFS using Eq. (6) is the highest, at up to 4905 s; whereas, for the a priori and a posteriori methods, the time costs are 1795 and 322 s, respectively. Clearly, the a posteriori method takes the least time, at 18% of that of the a priori method and only 6% of that of the analytical one.
Equation (6) indicates that the DFS is associated with the estimation of the error covariance matrices of the backgrounds and observations. Thus, we conduct four additional experiments to the above (marked Exp1): experiment 2 (Exp2), with overestimated observational error variance and accurate background error variance [
The self-sensitivities S_{ii} are used to measure the rate of change for the projected observational estimate
As shown in Fig. 4, the self-sensitivities for QuikSCAT observations range from 0.1 and 0.6. In the northeast of the domain, they are large, which mean that an individual observation has large influence, while data with low influence are located in southern and western regions. It is interesting to compare the error variances of the background/observations and the self-sensitivities, since the sensitivity is closely related to the distribution of the error variances of the background and observations. Because the observational error variances are assumed to be uniform (constant) in the blending area, we only consider the relationship between the background error variances and sensitivity. It is obvious from Figs. 2 and 4 that the large- and small-value areas are roughly consistent; namely, large (small) background error variances often correspond to large (small) self-sensitivities.
In order to further verify the above conclusion, we conduct two other experiments: experiment 6 (Exp6), with double the background error variances in the left part of the region in Fig. 2; and experiment 7 (Exp7), which reduces the background error variances to half in the right-hand part. In both experiments, the observational error variance is kept unchanged. The results are shown in Figs. 5, 6, respectively. It can be seen that an increase in the background error variance means an increase in self-sensitivity; and conversely, a decrease in the background error variance means a decrease in self-sensitivity. Self-sensitivities in the left in Figs. 5c, d are larger than in their counterpart in Fig. 4 because of the doubled background error variances. In Figs. 6c, d, due to decreased background error variances, the self-sensitivities decrease as well.
The general relationship mentioned above between the self-sensitivities and background error variances can be simply interpreted as follows: Eq. (4) indicates that the analysis in the observational space (Hx_{a}) is a weighted sum of the observational data y and background state in the observational space (Hx_{b}). The derivative of the analysis to the observation, i.e., the matrix of influence, S can be written as follows:
$S={K}^{\text{T}}{H}^{\text{T}}={\left(R+HB{H}^{\text{T}}\right)}^{-1}HB{H}^{\text{T}},$ | (21) |
and the derivative of the analysis to the background in the observational space is
$\frac{\partial \widehat{y}}{\partial \left(H{x}_{\text{b}}\right)}=I-{K}^{\text{T}}{H}^{\text{T}}=I-S={\left(R+HB{H}^{\text{T}}\right)}^{-1}R.$ | (22) |
It can be seen from Eqs. (21) and (22) that, for fixed, R large background error variances lead to a small derivative of the analysis to the background
From Eqs. (3) and (5), we have
$S={R}^{-1}H{\left({B}^{-1}+{H}^{\text{T}}{R}^{-1}H\right)}^{-1}{H}^{\text{T}},$ | (23) |
which also shows that, for fixed R, B is positively correlated with S, and the change of the ith background error variance may affect the self-sensitivities of the surrounding observations.
3.3.2 Average observational influencesIn order to quantitatively describe the influence of the observations (QuikSCAT) on the analyses (the blended sea surface winds), we calculate the average observational influence (AOI), introduced by Cardinali et al. (2004) and given by:
$\text{AOI=}\frac{\text{DFS}}{p}\times \text{100\%,}$ | (24) |
where p is the total number of observations. From Eq. (7), the sum of the AOI and average background influence is 1. Table 3 presents the results of the AOI from DFS_{POST} for both the u and v wind components. The impact of the observations on the analysis is 40.5%; thus, the impact of the background on the analysis is about 60%. This result seems to be different from that of Cardinali et al. (2004), i.e., a global observational influence of 15% and background influence of 85%, for the 4D-Var system of the ECMWF. The difference can be similarly explained according to Eq. (23). Since the ECMWF numerical weather prediction system is the most advanced in the world, its model output (forecast) is certainly more accurate than that of GZMM, and therefore, the background error variances from the ECMWF model should be smaller than those from GZMM; thus, the AOI reported by Cardinali et al. (2004) is smaller. In addition, the ECMWF data assimilation system is more complex, in which the number and types of observations used are much more than those used in the 2D-Var sea surface wind blending system of Xiang et al. (2015). This is also a possible reason for the different AOIs.
The analysis of sensitivity with respect to observations quantifies the influence of observations on the analysis in data assimilation systems, and is therefore an important research topic. There are various methods for the analysis of sensitivity with respect to observations, such as observing system sensitivity experiments, the adjoint sensitivity method, and the DFS-based method. The DFS-based method is a self-evaluation procedure, in which the matrix of influence S is utilized to indicate the self-sensitivity and cross-sensitivity of observations. In the present paper, the DFS-based method is applied to evaluating the influence of QuikSCAT sea surface wind field observations on the analysis.
For the 2D-Var sea surface wind blending system in Xiang et al. (2015), the DFS is calculated to assess the analysis sensitivity to QuikSCAT sea surface wind observations, and both the a priori and a posteriori methods are used to estimate the DFS of the u and v components of the sea surface wind field. The a posteriori method obtains almost the same results as the a priori method. Due to the use of by-products in the 2D-Var system for calculating the DFS in the a posteriori method, its computation time is the least, at only 18% and 6% of that for the a priori and analytical methods, respectively. The magnitude of the DFS is critically related to the observa-tional and background error statistics. To illustrate this, we carried out a set of experiments using different evaluation methods of the background error covariance matrix, and changing the observational and background error variances. Overestimated observational error variances or underestimated background error variances correspond to a larger number of DFS, and vice versa.
The self-sensitivities calculated show the observationalinfluence at each observational location for the QuikSCAT sea surface wind data: the observations, in the northeast of the domain, where there are larger background error variances, have larger influence; while in the west of the domain, where there are smaller background error variances, they have smaller influence. In addition, the total observational influence of the u and v components with respect to the analysis is about 40%, meaning that the influence of the background with respect to the analysis is about 60%. Thus, on average, the observational influence is lower compared with that of the background.
The above results show that the self-sensitivity is attributable mainly to the contribution of the background error variances; namely, for fixed observational error variances, large background error variances usually yield large self-sensitivities of observations, and small background error variances yield small self-sensitivities of observations. When the observational error variances are assumed to be constant, large background error variances mean large errors in the background, and therefore more weight is placed on the observations in the analysis, which yields large observational self-sensitivities.
The DFS is a useful tool for design and evaluation in systems of multi-source data blending and assimilation, comparisons of observational types in data assimilation, and so on. In the future, we will calculate the DFS for general cases, where error covariance matrices for backgrounds and observations are non-diagonal and there are more different types of observations, such as those from scatterometers, radiometer, and altimeters. Besides, the DFS can only be used to evaluate the observational impact on the data assimilation system itself. Further work should be done to access the impact of observations on the forecast.
Acknowledgments. We thank the anonymous reviewers, whose comments were helpful and greatly appreciated.
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