**Abstract**

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 areal coverage of the 2-sided NSCAT is 2.4 times greater than that of the 1-sided ERS scatterometer.
- The areal coverage of the single, wide-swath SeaWinds scatterometer is 40% greater than that of NSCAT and 3.2 times greater than that of ERS.
- QSCAT sampling is the same as that of SeaWinds.

Figure 1: (!Click on the thumbnail to see a bigger
version of the Figure!) |

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.

Figure 2: (!Click on the thumbnail to see a bigger
version of the Figure!) |

**2. Methodology**

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).

Figure 3: (!Click on the thumbnail to see a bigger
version of the Figure!) |

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).

Figure 4: (!Click on the thumbnail to see a bigger
version of the Figure!) |

**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*:

- The errors decrease toward higher latitudes because of the improved sampling shown in Figure 2.
- The inhomogeneities of the mapping errors drift longitudinally over the 41-day repeat period of the NSCAT orbit in concert with the space-time structure of the satellite ground track pattern.

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).

Figure 5: (!Click on the thumbnail to see a bigger
version of the Figure!) |

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.

Figure 6: (!Click on the thumbnail to see a bigger
version of the Figure!) |

**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:

- decrease as either
*dt*or*ds*is increased. - are dramatically larger for the 1-sided ERS scatterometer than for NSCAT or SeaWinds.
- are several times smaller for the wide-swath SeaWinds than for NSCAT.

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.

Figure 7: (!Click on the thumbnail to see a bigger
version of the Figure!) |

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.

Figure 8: (!Click on the thumbnail to see a bigger
version of the Figure!) |

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.

Figure 9: (!Click on the thumbnail to see a bigger
version of the Figure!) |

**Conclusions**

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:

- seasonal for the ERS scatterometer.
- 2--3 weeks for NSCAT.
- about one week for SeaWinds or QSCAT.
- a few days for a tandem QSCAT/SeaWinds mission.
- about 2 days for a triplet QSCAT/SeaWinds/ERS mission.

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).

**References**

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.

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