3. ESTIMATION OF GRIDDED WIND FIELDS

The following items are presented in this rubric :

• 3.1 Objective method
• 3.2 Resultant mean fields of wind vector, wind stress vector, wind divergence and stress curl

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3.1 Objective method

Since wind estimated at a point can vary significantly over periods of a few hours, it is difficult to reconstruct the synoptic fields of surface winds at basin scales from discrete observations, without the use of an appropriate method. Thus we have developed a statistical technique for the objective analysis of remote sensor wind data. This statistical interpolation is a minimum variance method related to the kriging technique widely used in geophysical studies. The analysis scheme is based on determining the estimator of surface parameters derived from scatterometer measurements. Figure 6 shows an example of seven days of scatterometer coverage. The computational details in constructing a regular wind field from polar orbit satellite data are given by Bentamy et al (1996). Briefly, let V(X) be an observation at point X=(x,y,t), where x and y are the spatial locations and t indicates time. We suppose that V(X) is a realization of the variable <U>(X).

We assume that each measurement consists of the true value plus a random error :

V(X) = <U>(X)+ (X)

The analysis scheme is based on the determination of the estimator Û of <U>, at a grid point X₀, of the surface variables using N observations V at the point X_i :

Here X_i stands for spatial and temporal coordinates. The weights are determined as the minimum of the linear system named kriging system :

Where is the structure function, named variogram. It allows the spatial and temporal variability behavior of the variable to be estimated. It is defined as :

E() and C() indicate the statistical mean and covariance functions, respectively.

Furthermore, the kriging method provides an expression for variance error, named kriging variance, which indicates the accuracy of the estimated wind variable at each grid point. The solution of the kriging system is used to calculate the variance of the difference between the estimated value Û and the true value <U> of the surface parameter :

In order to resolve the kriging system it is necessary to acquire the best possible knowledge of the variogram . Several models exist to define the theoretical formulation of the variogram. In the scatterometer case, the exponential model appears suitable. Its expression in terms of space and time separation is given by the equation :

where a, named sill value, corresponds to the variogram value when there is no correlation between variables. b, named spatial variogram range, corresponds to the spatial lag beyond which there is no more structure or where variables are uncorrelated. c is used to indicate the time correlation between variables. Coefficient corresponds to the spatial noise on scatterometer wind vector estimates. The calculation of indicates that its value is close to zero.

For instance the estimated values of variogram parameters a, b and c for scatterometer wind speed, zonal component and meridional component in the tropical area are given by table 3.1.

These values indicate that for wind speed, for example, the spatial and temporal ranges are 1350 km and 3.18 days, respectively.

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3.2 Resultant mean fields of wind vector, wind stress vector, wind divergence and stress curl

The above method is used to construct weekly and monthly wind fields from ERS-1, ERS-2 and NSCAT scatterometer observations. the calculation of the scatterometer gridded wind fields, using the kriging method, is based on the following items :

• The scatterometer wind products, including backscatter measurements and retrieved wind vectors, are extracted from CERSAT data base. They have been generated as the off-line ERS-1/2 (Maroni, 1995) and NSCAT (JPL, 1998) scatterometer wind products. Only validated data, according to standard quality controls, are used.

• At each ERS-1/2 scatterometer cell (50km), a new wind speed is estimated from the three backscatter coefficients and using the new C-band model function.

• The flags defined in WNF and NSCAT level 2.0 products are used to choose the "  best " wind vector among the solutions of the inverse problem. However, for low wind conditions, a comparison between each scatterometer wind direction solution and ECMWF wind direction, interpolated in space and time on scatterometer cell, is performed. The closest scatterometer wind direction from ECMWF is selected. The zonal and meridional wind components are estimated from scatterometer wind speed and direction.

• Wind stress and components are estimated over each ERS and ADEOS scatterometer cell.

• The land and ice flags included in scatterometer products are also used.

• The world oceans is divided into boxes with constant grid spacing in latitude and longitude. In this study, the spatial resolution is 1° ´ 1° in latitude and longitude.

• For each scatterometer swath, the data (wind speed, zonal component, merdional component, wind strees, zonal wind stress, and meridional wind stress) are averaged in each 1° ´ 1° grid point in order to reduce spatial dependency between the variables. The standard deviation and the number of observations in each box are recorded. The sampling distributions of these ERS 1° ´ 1° scatterometer "  observations " are summarized in Figure 7. They are evaluated for eigth ocean areas in dicated by table 3.2. On average, three or four 1° ´ 1° "  observations " are found in each grid point during one week. The distribution of the observation number is different in North Atlantic (Figure 7). The mean value is the lowest and then events are under-sampled. This is due to Synthetic Aperature Radar (SAR) which operates routinely in this region. In tropical areas, the scatterometer sampling scheme is appropriate to calculate averaged wind fields (Legler, 1991), (Halpern, 1987).

• At each grid point, the determination of neighborhood containing the scatterometer data used to estimate the wind vector (wind speed, zonal and meridional components) is performed. It is a quite sensitive step. Indeed, due to highly irregular spatial and temporal arrangement and the density of the scatterometer wind observations, the determination of a local neighborhood is not straightforward. In the operational method (CERSAT, 1998), the neighborhood is determined as a successive circles centered on grid point. The radius of these circles correspond to the variogram parameters. the maximum number of observations in grid point neighborhood is 20, which is a compromise between an adequate spatial and temporal sampling number and time computing duration. For monthly and especially NSCAT wind fields, this criteria is not acceptable. Indeed, NSCAT has a better spatial coverage than ERS-1/2. Therefore, the new procedure takes into account all samples located within the neighborhood. Their number reaches 1200. This data set is then sorted by time and for each hour the closest scatterometer observations from the grid point are used for wind vector estimation.

An example of gridded wind fields (wind speed and direction), derived from ERS-2 and NSCAT wind observations, are shown in Figure 8.

Figure 8

The raw weekly and monthly wind vector and wind stress fields are then analyzed objectively. The result is the determination of twelve regular fields : wind speed, W, zonal component, u, meridional component, v, magnitude of stress , | |, eastward component, _x, and northward component _y. and the errors corresponding to each previous parameter.

The wind divergence, Div(V), and the stress curl, curl( ), at each 1° x 1° grid point are then evaluated from the resultant wind fields. Finite difference schemes are used to estimate the two parameters.

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