Estimating Advective Tendencies from Field Measurements

Abstract

The ability to estimate horizontal advective tendencies of environmental variables from measurements at a finite set of observation points has been evaluated. The observation points include a central point plus from three to six boundary points at a man distance of R0. Two methods of estimation were considered: either by a numerical approximation to the flux line integral, or by integrating a quadratic fit to the field function (the latter if there are five of more boundary points). Errors arise from random instrument errors (or turbulent fluctuations) and because of truncation errors. The latter results from a mismatch between the spatial distribution of the field being considered and the assumptions underlying the approximation algorithm. Both types of errors were considered. Random errors were considered using standard theory for the propagation of errors. Terms of the Fourier series were used as test functions to study truncation errors. The standard against which estimates were evaluated was multipoint numerical integrations. For truncation errors, the significant result is that the use of a triangle of boundary points yields estimates close to 10% of the exact value only for a wavelength greater than about 20R0; if cells have radius of 100 km, that would he a wavelength of 2000 km; for a square, the estimates are better than 10% at a wavelength of about 7R0 (700 km); and for a pentagon, the estimates are less than 10% for the smallest nonaliasing wavelength. For this application, the use of a quadratic fit added little to accuracy; for a symmetrical army of points, the quadratic term does not contribute to the advective tendency. When one considers joint random and truncation errors. The general result is that truncation errors are more important than random errors for small wavelengths, and the reverse for large wavelengths. The results indicate that there is a substantial gain in going from three to four or five boundary points. The improvement for each further increment is less dramatic. It is recommended that simple algorithms be augmented by the use of remotely sensed finescale observations of surrogate and by occasional periods of observations at higher spatial density.