The is used to respect the misestimation of the measurement noise and theoretically, it can be derived from an instrument error model. However, in practice, an alternative method has to be proposed. If the expected MLE is estimated by the mean MLE, studies have shown that the mean MLE, as a function of cross-track index and wind speed(not statistically dependent on wind direction), can effectively respect measurement noise(Portabella and Stoffelen, 2001, 2002; Portabella, 2002)

Through reviewing the inversion procedure using the MSS method, the multiple solutions, together with the corresponding probabilities calculated by the EPF(Eq.2), retain all the direction solutions corresponding to the MLE cost function. However, for the case of the st and ard MLE procedure, only solutions at the minima of the MLE cost function— typically up to four minima—are used. This ignores neighboring wind solutions that have comparable probability of being the “true” wind. Apparently, the MSS method fully retains the information of the cost function and the observations.

The goal of the AR procedure is to select the best solution based on a set of ambiguous solutions and spatial consistency constraints and it is generally performed using many neighboring WVCs at once. Median fi ltering(Schultz, 1990) and 2DVAR are the two AR techniques used most commonly in scatterometry(Schultz, 1990; Stoffelen et al., 1997b; Stiles et al., 2002). In this paper, 2DVAR is used because it can explicitly use probability for the multiple solutions. The 2DVAR approach can be divided into two principal steps: fi rstly, an analysis wind is obtained using variational analysis, which combines background information(winds from NWP model)with the ambiguous solutions; and secondly, among all the ambiguous solutions, the one closest to the analysis wind in the wind direction is selected.

Using probability theory and Bayes’ theorem, the joint probability of the true state of the sea surface wind(x) and an ambiguous solution()can be expressed as(Lorenc, 1986):

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To increase computational effi ciency, an incremental formulation is used:

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Then, the cost function Eq.5 can be rewritten as:

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The observation term J o represent the misfi t between the ambiguous solutions and the control variable on a particular WVC. The contribution of the solutions is weighted by the solution probability. The observation term of the cost function can be expressed as(Stoffelen and Anderson, 1997a; Portabella and Stoffelen, 2004):

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The background term quantifi es the spatial context of the background fi eld errors and determines the spread of the observational information, which is expressed as(Vogelzang, 2007):

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The cost function of Eq.7 can be minimized using a conjugate gradient method, and the wind increments of the control variable vector are added to the background fi eld to obtain the wind analysis. The analyzed wind fi eld is then used for the AR procedure, as discussed earlier.

The MLE function for HY2-SCAT is defi ned as Eq.1, which refers to that of SeaWinds, and the NSCAT-3 GMF was used in this study. As wind speeds are overestimated above 15 m/s in NSCAT-2, a linear downscaling of such wind speeds was applied(Verhoef and Stoffelen, 2012).

The MLE cost function is computed as a function of wind direction from 0° to 360° with a step of 2.5°, and an array of solutions is produced(typically 144). Figure 2 shows an example of the MLE cost function for HY2-SCAT.

The probability of every solution being the “true” wind is given through the EPF. In order to provide the probability of each solution in the cost function, the parameters L and the expected MLE—the key parameters of the EPF—must be estimated using an empirical statistical method based on HY2-SCAT data.

To estimate the misestimation of the measurement noise, the mean MLE()was used as an alternative parameter. According to Portabella and Stoffelen(2001), the does not depend statistically on wind direction and therefore, the can be computed as a function of the crosstrack index and wind speed. In this study, the expected MLE was computed based on the winds retrieved by the st and ard MLE procedure using HY2-SCAT observations from 100 orbits. A rejection process was applied to fi lter noise, due mainly to geophysical effects such as rain. Finally, the expected MLE was smoothed using a 3×3 median fi lter to remove the noise(Portabella, 2002). The obtained values of used for HY2-SCAT are shown in Fig. 3.

Fig. 3 Expected MLE for HY2-SCAT as a function of crosstrack index and wind speed The wind speed binning is 1 m/s and the index binning is 1.

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In order to derive Eq.2 empirically, the ambiguous solution from the st and ard MLE procedure closest to the ECMWF wind was considered as the “selected” wind and initially, only those cases that had just two solutions were considered. The relative probability of selecting the 1 st and 2 nd, as a function of Rn 1 and Rn 2, can be given by:

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We processed 100 obits of HY2-SCAT data and obtained the exact two-solutions case. The relative probability can be calculated by using Eq.11 and the statistical results are shown in Fig. 4a. The best-fi t function was obtained by adjusting the L parameter and it can be represented by the following function:

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Fig. 4 Relative probability of selecting the 1 st and 2 nd as a function of Rn 1 – Rn 2 (a), EPF based on HY2-SCAT data (b)

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Equation 12 can be used to determine the probability of the solutions for HY2-SCAT and it can express the probability of any number of solutions of MLE. Finally, the 2DVAR technique was applied, together with the background fi eld, to each WVC in the AR procedure. Generally, the background fi eld is obtained from the ECMWF model by interpolation.

In this section, the wind data retrieved from HY2- SCAT using the st and ard MLE inversion and median fi lter AR(called MLE-winds), and those derived using the multiple solution scheme with 2DVAR AR(called MSS-winds)are used to compare the performance of the two different algorithms for wind retrieval. We processed fi ve months’(Jan. 2012, Feb. 2012, Nov. 2012, Dec. 2012, and July 2013)HY2-SCAT data and collected the operationally released HY2- SCAT wind products from the National Satellite Ocean Application Service with wind components from the ECMWF.

Figure 5a shows the ambiguous solutions for HY2- SCAT using the st and ard MLE inversion output, and the 7×7 median fi lter AR results are shown in Fig. 5b. The two solid lines identify the nadir region of the HY2- SCAT swath. Several WVCs within the nadir region exhibit wind directions that are clearly spatially inconsistent(Fig. 5b). The reason for this problem is most likely attributable to the ambiguous solution distribution for which no relative accurate solution is available, as shown in Fig. 5a. Therefore, there is no mean by which to select a consistent wind fi eld from such a solution pattern. In order to avoid the plotted lines are too dense, WVCs in both Figs.5 and 6 are sampled with one third along longitude and latitude separately.

Fig. 5 HY2-SCAT wind fi eld using standard MLE inversion output (a) and the 7 × 7 median fi lter AR (b) Area between the two solid lines is nadir region of HY2-SCAT swath.

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Figure 6a shows the multiple ambiguous solution distribution for the same case as in Fig. 4, and the variational analysis AR results are shown in Fig. 6b. In Fig. 6a, only solutions with normalized possibilities above 0.01 are shown. In comparison with Fig. 5a, the MSS inversion provides much more information to the AR. As discussed earlier, the 2DVAR is able to use the information in a more appropriate manner to provide a spatially consistent wind fi eld(Fig. 6b).

Fig. 6 Same as Fig. 5, but for the HY2-SCAT ambiguous wind fi eld using the MSS inversion (a) and variational analysis (b) Only solutions with normalized probabilities above 0.01 are shown.

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To validate the accuracy of the wind speed and direction at different ocean states, the bias and st and ard deviations of the wind speeds with respect to the ECMWF winds were calculated as a function of the ECMWF wind speeds in bins of 1 m/s, as shown in Fig. 7a, and the mean absolute differences of wind direction are shown in Fig. 7b.

Fig. 7 Wind speed (a) and direction (b) bias referred to ECMWF winds as a function of wind speed in bins of ECMWF wind speed of 1 m/s

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Tab. 1 Comparison of MLE-winds (left column) / MSSwinds (right column) with buoy data

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Figure 7a and Figure 7b shows that the -deviations of MSSwinds -vary with increasing wind speed. In Fig. 7a, the bias decreases from 2.0 to 0.2 m/s as the wind speed increases from 0 to 5 m/s, while the RMS error remains about 2 m/s. The wind speed RMS error becomes changeable at speeds above 17 m/s, although the absolute bias remains within 1 m/s up to a wind speed of 24 m/s. In Fig. 7b, the bias of wind direction also decreases as the wind speed increases from 0 to 9 m/s and then it falls to 20° at a wind speed of 5 m/s. The bias shows little variation for wind speeds above 9 m/s.

A comprehensive analysis of Fig. 7a and Figure 7b reveals that wind speed is noisy at both low wind speeds(below 4 m/s) and high wind speeds(above 17 m/s), whereas the wind direction is reproduced well at wind speeds between 5 m/s and 24 m/s or higher.

To validate the MSS-winds and MLE-winds, we also compared these winds with in situ observations from 29 offshore buoy stations. The buoy data were collected from the National Data Buoy Center for the same fi ve months(as mentioned earlier)as the MSSwinds; and the MLE-winds use the same time period. Only wind speeds are above 2 m/s and below 30 m/s are considered in this validation. The matched winds are limited to less than 10 min and 0.25° for the temporal difference and spatial separation, respectively. Buoy data are converted to 10-m height using the logarithmic profi le method(Peixdto and Oort, 1992). The results of the comparison are shown in Table 1.

A total of 4 936 points were matched and the data are presented in Fig. 8a for wind speed and Fig. 8b for wind direction. In the comparison of the MLE-winds with buoy winds, the RMS errors of wind speed and wind direction are 1.3 m/s and 24.0° for all swath data, respectively. In the comparison of the MSS-winds with buoy winds, 4 812 points were matched and the data are shown in Fig. 9a for wind speed and Fig. 9b for wind direction. The RMS errors of wind speed and wind direction are 1.3 m/s and 17.4° for all swath data, respectively. Comparisons for sweet swath(nodes 9–28 and 49–68)winds and nadir swath(nodes 29–48)winds are also calculated. The results listed in Table 1 shows that MSS-winds are more improved in wind direction domain. Besides, the MSS-winds narrow the differences of accuracy between sweet swath data and nadir swath data. All matched points within triple st and ard deviations were considered in this study.

Fig. 8 Comparison of MLE-winds with buoy winds from the National Data Buoy Center a. Wind speed; b. wind direction.

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Fig. 9 Comparison of MSS-winds with buoy winds from the National Data Buoy Center a. Wind speed; b. wind direction.

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Finally, we compared two wind fi eld data sets of MLL-winds and MSS-winds for the case study of typhoon Soulik. Soulik was a powerful typhoon that caused widespread damage in Taiwan Isl and and East China in July 2013. The storm originated to the northeast of Guam on July 6, and became a tropical depression early on July 7. The depression underwent a period of rapid intensifi cation starting on July 8 that culminated in Soulik attaining its peak strength early on July 10. Figure 10 shows a HY2-SCAT-retrieved wind fi eld using the st and ard MLE inversion and median fi lter AR. In the region of the red rectangle, it is clear that the wind vectors do not conform to the vortex structure. Figure 11 shows the improved winds using the MSS inversion in combination with 2DVAR for the same case as shown in Fig. 10. In order to avoid the plotted lines are too dense, WVCs in both Fig. 10 and Fig. 11 are sampled with odd number along longitude and latitude separately.

Fig. 10 Wind fi eld retrieved using standard MLE inversion and median fi lter AR for the case of typhoon Soulik MLE on July 10, 2013

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Fig. 11 Wind fi eld retrieved using MSS inversion in combination with 2DVAR for the same case as shown

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In this study, the MSS algorithm is used in combination with the 2DVAR technique to improve HY2-SCAT wind retrievals. The current operational system using st and ard MLE wind retrieval procedures, which only considers the local MLE cost-function minima as ambiguous solutions, produces inaccurate winds in the HY2-SCAT nadir region.

The MSS algorithm retains all the direction solutions corresponding with the MLE cost function. The probability of each ambiguous solution being the “true” wind is calculated using the EPF, the parameters of which are derived empirically based on HY2-SCAT data. In the AR procedure, a variational analysis AR(i.e., 2DVAR)is used instead of a median fi lter AR, because it is capable of explicitly using the probabilities for multiple solutions and ensuring the spatial consistency and meteorological balance of the retrieved winds.

The study shows that the MSS wind direction is substantially better than the MLE wind in the nadir region. The comparison between the MSS winds and ECMWF winds shows that wind speed is noisy at low wind speeds(below 4 m/s) and at high wind speeds(above 17 m/s), while the wind direction shows lower bias and stability, even at high wind speeds above 24 m/s. Furthermore, the wind fi eld of a vortex structure is reproduced well by the MSS winds, showing spatial consistency. However, the quality of derived wind speeds above 20 m/s is still poor and further study is required. This should include a greater number of observations of high wind speeds(e.g., from buoys, dropsondes, oil rigs, and WindSat)to refi ne the geophysical model function of the scatterometer.

We greatly appreciate the European Organization for the Exploitation of Meteorological Satellites(EUMETSAT), National Data Buoy Center(NDBC), and National Satellite Ocean Application Service(NSOAS)for their freely available data sets. We thank the Royal Netherl and s Meteorological Institute(KNMI)scatterometer team for answering our detailed questions.