An MLE objective function (Freilich, 2000) is used for the RFSCAT wind retrieval because of its advantages mentioned above. It can be written as follows:

where z and M represent the measured backscatter coefficient and the model value, respectively; V_r is the measurement variance; and w, Φ, φ, θ, and p denote the wind speed, the wind direction, the radar azimuth, the incidence angle, and the polarization, respectively. The purpose of the wind retrieval is to find the appropriate wind vectors that maximize Eq. (1).

A wind direction extension is an important part of the algorithm design. The change rate of the objective function along the wind direction is adopted as the criterion for the wind direction extension and is defined as follows:

where J_r and J_l are the local maximum objective function values at neighboring right and left wind directions, respectively; and △d is the wind direction interval.

The wind direction extension algorithm consists of two procedures, namely, left-extension and right-extension. The left-extension algorithm contains the following steps.

(1) Take the wind direction of the ambiguity solution as the right wind direction d_r and calculate the current wind direction d_c using the following equation:

(2) Determine the maximum value of the objective function J_c at the current wind direction using Eq. (1).

(3) Similarly, calculate the left wind direction d_l using the following equation:

(4) Determine the maximum value of the objective function J_l at the left wind direction.

(5) Calculate the change rate of the objective function with the wind direction using Eq. (2).

(6) Update the right and current direction with the current and left direction, and repeat Steps (2) to (6) until the change rate k is larger than the given threshold k₀.

Thus, the last current wind direction is the left-extension boundary when the iterative process stops. The right-extension has similar processing steps.

It is difficult to determine a rate threshold that is appropriate for all regions of the RFSCAT swath. Experiments show that the best method is to set a different rate threshold for each wind vector cell position across the swath. In this paper, we determined the rate threshold value according to the distribution histogram of the bias between the retrieved wind direction and the “real” wind direction. Only the first and second ambiguity wind directions were considered in the histogram analysis.

Here, the wind direction bias refers to the difference between the retrieved wind direction and the reference, or “real”, wind direction. The retrieved wind directions were obtained by using the traditional MLE wind vector retrieval algorithm. There is a tight relationship between the change rate of the objective function and the width of the bias distribution.

The wind direction extension values are the wind direction boundary values after the wind direction range extension, which is an important part of the modified algorithm. Therefore, the extent of similarity between these two histograms can reflect if the change rate threshold values are suitably set or not. Simulation experiments indicated that the rate threshold in the middle region should be larger than that in the nadir and outer regions of the swath in order to achieve good retrieval performance.

The proportion of the wind vector cells with a certain wind direction bias can be defined as follows:

where N_b is the number of wind vector cells that have the wind direction bias b; and N is the total wind vector cells in the statistical analysis. The wind direction bias is calculated by the following formula:

where d_m and d₀ represent the retrieved wind direction and the true wind direction, respectively. Note that the wind direction bias b should be transformed into the range of [–180°, 180°] as follows:

Similarly, the proportion of the wind vector cells with a certain wind direction extension boundary value d can be calculated by

where N_d is the number of the wind vector cells associated with the wind direction (d) extension value, which is the last current wind direction d_c after extension.

As discussed above, the proportion of the wind vector cells depends on and is sensitive to the rate threshold k₀ value. The larger rate threshold is, the larger wind direction extension is, and the wider the distribution histogram of the wind direction extension value becomes. In theory, the similarity between the two histograms of the wind direction bias and the wind direction extension value should be high in order to obtain high wind retrieval performance. Therefore, it is a key step to adjust the given threshold to make the two histograms more similar in the advanced algorithm design.

The threshold values of the change rate of the objective function for each wind vector cell column used in this paper are shown in Fig. 1. From Fig. 1, it can be seen that the curve of the threshold value shows an M shape, which is opposite to the shape of the wind retrieval error curve. This indicates that larger change rates are required to obtain a wide enough range of the wind direction in the middle regions of the swath.

Figures 2 and 3 give the distribution histograms of the wind direction bias and the corresponding wind direction extension values respectively. In the simulation experiments, the NCEP wind direction from the SeaWinds L2B data file is taken as the reference wind direction, which is one of the data input into the simulation software.

In addition, modifications should also be made to a circle median filter algorithm in order to let it select more possible wind directions in the process of the ambiguity removal.

According to the distribution characteristics of the histograms, the swath can be divided into three regions in the ambiguity removal (Xie et al., 2010). The swaths of the RFSCAT are partitioned as follows:

(1) In the nadir region: Columns 16 to 31;

(2) In the middle region: Columns 7 to 15 and 32 to 40;

(3) In the outer region: Columns 1 to 6 and 41 to 46.

The concept of a circle median filter was first extended and applied to wind vector data by Shaffer et al. (1991). According to Shaffer et al.(1991), the definition of a circle median can be written as follows:

where (i, j) is the center of the filter window with size N (N is an odd number); h=(N–1)/2; $A_{ij}^k$ denotes the kth wind vector ambiguity solution of the window center; ${A_{mn}}$ is the current ambiguity solution located at position (m, n) in the filter window; and ${W_{mn}}$ is the corresponding weight coefficient at the same position.

The modified circle median filter algorithm for an ambiguity removal is composed of three steps.

(1) Initialize the wind field using the first ambiguities and apply the circle median filter along the column direction in the middle regions.

(2) Take the filtered wind field from Step (1) as a reference, and apply the extending filter from the boundaries of the middle regions to the nadir and outer regions. The moving direction of the filter window is also along the column.

(3) On the basis of the filtered results from Steps (1) and (2), apply the circle median filter to the entire swath along the row direction.

It is noted that the filter algorithm should also be improved to make it consider more possible wind directions when determining the circle median in Steps (1) and (2). The traditional circle median filter algorithm is adopted in Step (3) to prevent it from not converging.

The procedures of the improved filter algorithm are as follows.

(1) Construct the extended wind direction array using the boundary values after extension and the wind direction interval to save the possible wind direction set. Assume there are four ambiguities in the current wind vector cell, with the wind directions represented by d₁, d₂, d₃, and d₄. Then, the extended wind direction ranges of the first and second ambiguities can be denoted as below:

where d_1l and d_1r are the left boundary and right boundary values of the first ambiguity after extension; and d_2l and d_2r are the corresponding values of the second ambiguity. The whole set of possible wind directions to be used in the ambiguity removal can be expressed as

(2) Determine the circle median of the window center using Eq. (9), and record its array index.

(3) Determine which ambiguity the circle belongs to based on its array index.

(4) If the circle median wind direction lies in the extended wind direction of the first ambiguity or the extended wind direction of the second ambiguity, then search for the wind speed that maximizes Eq. (1) for the current wind direction. The calculated wind speed and the circle median wind direction compose the circle median ambiguity. If the circle median wind direction is equal to the third ambiguity or the fourth ambiguity, then the third or fourth ambiguity is the circle median ambiguity.

(5) Replace the current ambiguity at the window center with the circle median ambiguity.

To avoid the effects of the random factors from some specific orbit data on the retrieval results, the new algorithm was validated by using 100 orbits of simulated RFSCAT L2A data and the collocated NCEP wind field data contained in the SeaWinds L2B file. Here, the NCEP wind data are referred to as the numerical weather prediction (NWP) wind products, which are routinely generated by a global forecast system (GFS) developed and maintained by the National Center for Environment Prediction (NCEP) (http://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs). These wind data are also called GFS winds. The NCEP wind product has a spatial resolution of 0.5°×0.5° and needs to be temporally and spatially interpolated into the 25 km×25 km wind vector cell grids for collocation. Because the SeaWinds L2B files contain the quality collocated NCEP winds with a swath-based spatial framework, it is convenient for us to use these NCEP winds as the input wind fields for the RFSCAT data simulation. The SeaWinds L2B data can be obtained via the PO.DAAC FTP site: ftp://podaac. jpl.nasa.gov,in the pub/ocean_wind/seawinds/L2/directory (Lungu, 2001; Dunbar, 2006; JPL, 2003). Compared with the SeaWinds L2B winds, the wind directions of the NCEP have a better spatial consistency. However, the derived SeaWinds L2B winds have more errors and noise in their wind directions, which will influence the simulation effect. Therefore, the NCEP data were used as the input for the “real” wind field in this paper. The orbit numbers of the L2B data files used for the simulation are from 10 163 to 10 262. The maximum iteration times and window size for the circle median filter are 100 and 7×7, respectively.

The simulation software uses the input “reference” wind field data and instrument parameters of the RFSCAT to produce the simulated noisy backscatter measurement data. Then, the modified algorithm is run to perform the wind vector retrieval with the simulated backscatter coefficients.

The wind vectors (speed and direction), mean absolute errors (MAEs), and root mean square errors (RMSEs) of the two algorithms for three different regions of the swath are calculated and given in Table 2. Table 2 shows that the modified algorithm can effectively improve the wind direction retrieval accuracy in the nadir and outer regions of the swath.

Figures 5 and 6 illustrate the variation curves of the wind speed and wind direction MAEs versus the wind vector cell location across the swath. Figure 4 clearly shows that the wind direction errors of the modified algorithm are lower than those of the traditional algorithm in the nadir and outer regions. Approximately 1.6° and 9° improvements in the wind direction retrieval accuracy are achieved for the wind vector cell nearest to (Column 23 or 24) and farthest from (Column 1 or 46) the nadir track, respectively. Figure 5 indicates that the difference in the wind speed retrieval errors between these two algorithms is not very significant, in spite of a slight improvement being made by the modified algorithm.

All the statistical results in Table 2, Figs 5 and 6 were calculated by comparing the retrieved wind vectors with the input reference wind fields. In Table 2, we divided the whole data set into three groups, according to the three regions of the swath, and computed the statistical parameters for each region. Thus, the wind vector errors in Table 2 are actually the average levels for the corresponding swath region, whereas, in Figs 5 and 6, the errors are computed on the basis of each wind vector cell column, representing the average level of the wind vector cells belonging to that column. Hence, Figs 5 and 6 reflect the variations of the wind retrieval errors with the wind vector cell position across the swath.

Figures 7 and 8, respectively, present the portions of the wind fields for orbit 10165 retrieved by the traditional MLE algorithm and the advanced algorithm. For comparison, the corresponding portion of the input wind field is given in Fig. 9. The rows of the wind field shown in the figures are from 1 080 to 1 140. Figure 7 shows that some of the wind directions located in the red rectangular box (left area of the wind field) clearly exhibit inconsistencies. In contrast, a more continuous and coherent wind field is obtained by the new integrated algorithm in the same region, as displayed in Fig. 8. In comparison with Fig. 9, the wind field retrieved by the integrated algorithm is much closer to the input wind field.