Winds vary over a continuum of space and time scales. Accurate determination of the short space and time scales requires frequent and closely spaced observations. Moreover, undersampling alsocorrupts estimates of the long space and time scale signals through aliasing. Cognizance of sampling-imposed limitations is an essential prerequisite to the interpretation and application of wind fields constructed from scatterometer observations.
A formalism for determining the effects of sampling errors on maps constructed on a specified space and time interpolation grid from irregularly sampled observations has been developed by Schlax and Chelton (1992). The method has been applied by Chelton and Schlax (1994) and Greenslade et al (1997) to investigate the relative merits of single and multiple satellite altimeter datasets. It is applied here to investigate the sampling errors of fields of a wind velocity component constructed from single and multiple scatterometer datasets.
The five scatterometer mission scenarios considered here are:
1) the ERS mission.
2) the NSCAT mission.
3) a SeaWinds mission (effectively equivalent to a QSCAT mission).
4) a tandem QSCAT and SeaWinds mission.
5) a triplet QSCAT, SeaWinds and ERS mission.
1. Scatterometer Sampling Patterns
The sampling swaths of the ERS, NSCAT and SeaWinds scatterometers are shown in Figure 1:
The average sample interval between successive observations at any given location (see Figure 2) is directly proportional to the areal coverage of the sample swath. The decrease in sample interval with increasing latitude results from ground track convergence and the overlap of neighboring sample swaths at high latitudes.
The technique for estimating sampling errors is applicable to any linear interpolation and smoothing algorithm (e.g., block averages, weighted averages, successive correction, optimal estimation, etc.) The details of the formalism are of secondary importance. Any specific formulation can be characterized by the half-power filter cutoff frequency and wavenumbers that are effectively prescribed by the parameters of the linear estimate.
The quadratic loess smoother used here is a locally weighted least-squares fit to a quadratic. Its filtering properties are defined by "smoothing parameters" dt and ds, which correspond to the temporal and spatial half spans of the data incorporated in the estimates. A convenient property of the quadratic loess smoother is that the filter cutoff frequency and wavenumber are approximately the reciprocal of dt and ds, respectively.
In addition to the prescribed smoothing parameters and the spatial and temporal distribution of observations within the span of the smoother, mapping errors depend on the statistical characteristics of the wind variable of interest (i.e., the standard deviation and the autocorrelation function of the wind variable).
The standard deviation of a wind velocity component considered here is estimated from NSCAT data to be about 7 m/s at mid latitudes (Figure 3).
For the present analysis, the space-time autocorrelation function has been somewhat arbitrarily prescribed to be exponential with an isotropic decorrelation spatial scale of 600 km and a decorrelation time scale of 2.5 days (Figure 4).
3. Errors in Maps of a Wind Component
The geographical characteristics of sampling errors are illustrated in Figure 5. The three panels show maps of the expected error of a smoothed wind component at a particular time constructed from the NSCAT sampling pattern with three choices of smoothing parameters dt and ds.
Mapping errors vary dramatically both geographically and temporally for small dt and ds:
Spatial and temporal inhomogeneity of the mapping errors decreases as the smoothing parameters are increased. A latitudinal variation of the mapping errors still exists with large smoothing parameters (right panel of Figure 5).
The reduction of mapping errors for a tandem QSCAT/SeaWinds scenario is shown in Figure 6. Although the errors are much smaller in maps constructed from the tandem scenario, spatial inhomogeneities are still evident, especially when dt and ds are small.
4. Error Dependence on Spatial and Temporal Smoothing
The dependence of mapping errors on temporal smoothing dt and isotropic spatial smoothing ds is summarized in the left column of Figure 7 for smoothed estimates of a wind component at 30 deg latitude constructed from the three single scatterometer missions. The dots in the NSCAT and SeaWinds panels correspond to the smoothing parameters dt and ds used to construct the example error maps in Figures 5 and 6.
The mapping errors ensemble averaged over space and time:
The spatial and temporal inhomogeneity of mapping errors can be characterized by the standard deviation of the mean errors, ensemble averaged over space and time (right column of Figure 7). The far superior sampling for SeaWinds is readily apparent from the much smaller spatial and temporal inhomogeneities of the mapping errors.
The improved mapping accuracy that can be achieved for smoothed estimates of a wind component at 30 deg latitude constructed from simultaneous scatterometer missions is shown in Figure 8. The mean mapping errors, as well as the inhomogeneities of the mapping errors, are much smaller for a tandem QSCAT/SeaWinds mission or a triplet QSCAT/SeaWinds/ERS mission than for single scatterometer missions.
The contrasts between the mapping resolution capabilities of the various scatterometer mission scenarios considered here are summarized in Figure 9. The two lines correspond to the temporal smoothing required to obtain a spatial resolution of 2 deg with a mapping accuracy of 0.72 m/s (heavy line) or an error inhomogeneity of 0.12 m/s (thin line) for smoothed estimates of a wind component at 30 deg latitude.
1. The effects of sampling errors must be understood so that artifacts of inadequate smoothing of scatterometer data are not misinterpreted as real features in the wind field.
2. The time resolution for high spatial resolution maps of a wind component constructed from scatterometer data can be summarized as follows:
The results presented here are preliminary, as they are based on an ad hoc specification of the space-time autocorrelation function for a wind component. We expect qualitatively similar conclusions when a more precise estimate of the autocorrelation function is used.
Future extensions of this analysis will consider the sampling errors for wind variables other than the wind component fields considered here. Of particular interest are the wind stress components and derivative fields (e.g., wind divergence, relative vorticity, and wind stress curl).
Chelton, D. B., and M. G. Schlax, 1994: The resolution capability of an irregularly sampled dataset: with application to Geosat altimeter data. J. Atmos. Oceanic Technol.,11, 534-550.
Greenslade, D. J. M., D. B. Chelton and M. G. Schlax, 1997: The midlatitude resolution capability of sea level fields constructed from single and multiple satellite altimeter datasets. J. Atmos. Ocean. Tech., 14, 849-870.
Schlax, M. G., and D. B. Chelton, 1992: Frequency domain diagnostics for linear smoothers. J. Amer. Stat. Assoc., 87, 1070-1081.