Ray Height Computation for a Continuous Nonlinear Atmospheric Refractive‐Index Profile

Abstract

An algorithm is described for machine computation of the height of a ray originating at or near the earth's surface, when the initial elevation angle and the ray-path length (as measured, for ex· ample, by a radar-antenna elevation angle and the echo signal delay time) are given, and the refractiveindex profile is specified as a continuous but nonlinear function of the height. The numerical precision of the result is limited only by the number of digits carried in the computation, although faster computation is achieved with reduced precision. The computing time is less than ls for six significant figures using the CDC-3800 computer. The equations used (which include those for the ground range, true geometric range, and true elevation angle) have been manipulated to avoid formulations that destroy computational precision. The method requires that the refractive index be spherically symmetric with respect to the earth's center and that the height profile be representable as a piecewise continuous function. In the work described, a negative-exponential profile was assumed. Two algorithms for higher speed computation with reduced precision, suitable for radar-target height cal· culation, are also described.

Computer modeling of multipath propagation: Review of ray‐tracing techniques

“…The concept of an effective earth radius was introduced by Schelleng et al, [1933] and is also found in the works of Elmore [1975], Inoue [1975], Comit• Consultatif International des Radiocommunications (CCIR) [1978a], and Sasaki and Akiyarna [1979]. The use of Snell's law for rays refracted in a spherical atmosphere is applied by Schulkin [1952], Bean and Thayer [1959], Bean [1964], Armand and Kolosov [1965], Thayer [1967], Blake [1968], Bean and Dutton [1968], Plotnikov [1975], Gallop and Telford [1975], and Vickers and Lopez 1-1975].…”

Increase of error in range correction with elapsed time, evaluated by ray tracing through radiosonde‐generated atmospheric models

“…The computer program for ray tracing described by Blake [1968] was chosen for use in this investigation because it is designed for automatic operation over a wide range of elevation angles, yet retains computational precision. The principal features that make it generally applicable to tracing through varied atmospheres are: (a) no approximations ar'e made in the integrand, (b) the step size employed in the integration is varied automatically by an adaptive integration subroutine that uses Simpsoh's rule, and (c) special computations are carried out for the grazing ray at low altitudes., and N.0 is the refractivity times 10 -ø, where ce is the exponential constant.…”

“…The measurements were made in early April in Boston during a period of extensive overcast, drizzle, and occasional rain. The computations were performed using a computer program derived from the one described by Blake [1968]. The exponential model of refrac-Copyright ¸ 1973 by the American Geophysical Union.…”

A Guide to Basic Pulse-Radar Maximum-Range Calculation. Part 1. Equations, Definitions, and Aids to Calculation

“…In other cases the noise factor F, may be given. The relationship between the noise factor and the effective input noise temperature of the receiver, or of any transducer, is given (16) by T = To(F-, -1) , (41) where To is by convention 290 0 K. In this formula F,, is a power ratio (often given in decibels).…”

“…Unfortunately, this model does not result in simple formulas for the range-height-angle relationship; a numerical integration is required to trace a ray path (41,42). However, machine-computed results can be used to plot a range-height-angle chart for this model, on which the raypath lines are straight; this result is accomplished by suitably distorting the contours of constant height (42).…”