The effect of tropospheric refraction on the Doppler shift of a satellite signal

Abstract

An expression for the contribution made by tropospheric refraction to the Dop pler shift of a satellite signal is derived. It is assumed that the troposphere is horizontal ly stratified and does not vary during the time of a pass; a two-parameter quadratic expression is used as an approximation to the refractivity profile. The qualitative effect of the troposphere during any satellite pass is to steepen the slope of the Doppler shift "ersus time curve, making the tra. eking station appear slightly closer to the orbit than it a. ctually is. When Doppler data from severfll passes are used for orbit computation, the resultant effect of the troposphere in general is not zero; thus it may bias the orbit slightly, the amount depending on the geometry of the selected passes. Conversely, if the orbit is assumed to be known, the station position as detem1ined by a single satellite pa. ss may be shifted towa. rd the orbit by 50 meters or more. Such errors are not negligible if precise geodetic work is to be dOlle with satellites. The residual tropospheric error in the Doppler shift cun probably be reduced by an order of m agnitude by using a computed tropospheric correction of the type described here. Some Doppler data from observed satellite pas ses are presented in corroboration of the theory.

Dynamic Pattern Generation in Behavioral and Neural Systems

“…Our approach also shares some common features with recent work on artificial neural networks that stresses the computational capabilities of collectives of "neurons" (70). Unlike the present operational attitude, however, the linkage between theory and cxperimental observables in neural network modeling is neither a prerequisite nor a necessary source of constraint.…”

“…Most neural network modeling starts with model-neurons and circuits and then attempts to produce various learning, memory, and pattem recognition schemes. For instance, prescribed sets of states can be made stable states of such networks that correspond to patterns or content-addressable memories (70). It is not clear, however, how to define such prescriptions a priori.…”

Review of the Precise Orbit Determination for Chinese Lunar Exploration Projects

“…Relativistic acceleration correction Moyer (2005) Earth tropospheric correction Hopfield model (Hopfield, 1963) spacecraft, the quality of the lunar gravity field model will dominate the accuracy of lunar spacecraft orbit determination. Moreover, we can evaluate the accuracy of different lunar gravity models through the POD.…”

A formula for computing refraction effects on range measurements

“…Based on a constant temperature lapse rate, a flat earth, and the perfect gas law, Hopfield [1963Hopfield [ , 1969Hopfield [ , 1971 where T is the surface temperature (Kelvin). It should be noted that the linear nature of (1)enables us to treat the 'dry' and 'wet' parts of the atmosphere separately [Hopfield, 1969].…”

“…Writing It can be seen from the definition of t that t = 1 when E 0 = 0 ø, and then decreases toward f/(1 -f) at E 0 = 90 ø. The rapidity of the decrease can best be seen in Table 1. It was observed by Hopfield [1963] that (1) does not take bending into account, but although the resulting error reaches 10% at the horizon, it is negligibly small above 3 ø or 4 ø . For this reason (and since t is very small above 3.5ø), the series expansion for /• may be terminated after only a few terms.…”