Statistical methods in radar astronomy. Determination of surface roughness

R ecently some lunar theories b ased on the use of s tatistical theor y h ave been questi~ned [Siegal ?-nd Senior, 1962], even though mo~t o ~ th e .th eoretlCal work in the field of the determlllatIOn of planetary surface roughness h as b een statistically orien ted [Davies 1954 ; Cooper 1958; H ayre and M oore 1961 ; Hayr e 1961 and 1962; Winters 1962 ; E vans 1963; Millman 1963, etc. ], L ater on, E vans [1963] published so me fur th er da ta on lunar ech oes to show that lunar theories employing statistical theory offer feasible r esults. This is another simple illus tration of the direct use of the s ta tisti cal theory in surface roughness studies by r adar. A naturally occLUTing rough sur face may b e described eith er by infinitesimally closely spaced contour maps or b j; i ts statistical properties . . Wi thout much ado, i t is apparen t that the la t ter l~ probably the most logical and compact way 111 the absen ce of any other exact descrip tion unless some other form of microscopic scale descrip tion is n eeded. Almost any rough sUl'face can b e sh o\,:n to have a probability density function. of its he~ghts, and a height-distance autocorrelatIOn functIOn, It has been previously shown [Hayre, 1961] that a large number of naturally occurring rough ?ur~aces may be said to have their heights normally dlstnbuted above and below their mean v alue. It has also been shown that an exponential h eight-distance autocovariance function seems to describe m any such cases, or