Errors Incurred in Sampling a Cyclostationary Field

Abstract

Low earth-orbiting satellites such as the Tropical Rainfall Measuring Mission (TRMM) estimate month-long averages of precipitation (or other fields). A difficulty is that such a satellite sensor returns to the same spot on the planet at discrete intervals, about 11 or 12 h apart. This discrete sampling leads to a sampling error that is the one of the largest components of the error budget. Previous studies have examined this type of error for stationary random fields, but this paper examines the possibility that the field has a diurnally varying standard deviation, a property likely to occur in precipitation fields. This is a special case of the more general cyclostationary field. In this paper the authors investigate the mean square error (mse) for the monthly averaging case derived from the satellites whose revisiting intervals are 12 h (sun synchronous) and off 12 h (11.75 h). In addition, the authors take the diurnal cycle of the standard deviation to be a constant plus a single sinusoid, either diurnal or semidiurnal. The authors have derived an mse formula consisting of three parts: the errors from the stationary background, the cyclostationary part, and a cross-term between them. The separate parts of the mse allow the authors to assess the contribution of the cyclostationary error to the total mse. The results indicate that the cyclostationary errors due to the diurnal variation appear small for both a 12-h and an off-12-h (11.75 h) revisiting satellite. In addition, the cyclostationary error amounts are similar to each other. For the semidiurnally varying field, the cyclostationary errors increase rapidly as the magnitude of the variance cycle increases for both the 12-h and off-12-h revisting satellites. However, the off-12-h sampling shows the cyclostationary error to be less than that of the exact 12-h sampling. Furthermore, the authors have evaluated the cyclostationary error as a function of the phase of the satellite visit as it is shifted from the phase of the diurnal cycles (the sun-synchronous case or the start of the month for the off-12-h case). It is found that the cyclostationary error observed from the off-12-h satellite is much less sensitive to the phase shift than the cyclostationary error from the exact 12-h satellite.