DETERMINATION OF THE MELTING LAYER FROM METEOROLOGICAL RADAR DATA
Statistical methods in radar astronomy. Determination of ... · Statistical Methods in Radar...
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JOURNAL OF RESEARCH of the Nati onal Bureau of Sta ndards-D. Radio Propagation Vol. 67D, No.6, November-December 1963 Statistical Methods in Radar Astronomy. Determination of Surface Roughness H. S. Hayre Contribution from Department of Electrical Engineering, Kansas State University, Manhattan, Kans . (Recei ved Jun e 13, 1963) An ex pe rimentally esta blished stat ist ical mod el of a rough s urface is u sed to show t hat sufficient info rmat i on about the roughness of such a s urface In the form of Its st a !l da rd devIa- t ion, mean horizo n ta l si ze of lump s, a nd ave ra ge slope can be ob ta In ed f rom. exp cl'l ITI cn tal data when uscd in conju nct ion wi t h a theory based on s tat I st Ica l ana l ,v sls. 1. Introduction R ecently some lun ar th eori es based on th e use of s ta tistical th eory h ave been [Si egal ?- nd Senior , 1962], even th ough the .th eor etlCal work in the fi e ld of the d etermlll atIOn of pl an etary surf ace roughn ess h as b ee n sta tistically ori en ted [D avies 19 54; Cooper 1958; H ayre a nd M oor e 1961 ; Hayr e 1961 a nd 196 2; Win ters 1962; E va ns 1963; Millman 19 63, etc. ], Later on, E vans [1963] pub- lished some fur ther data on lun ar echoes to show tha t lun ar theories empl oying statisti cal theory off er feasible res ult s. This is an ot her simpl e illus tr at ion of th e dir ect use of th e statis ti cal th eory in s urfa ce roughn ess st udi es by radar. A n at urally occ LUTing rough sur face m ay be describ ed either by infinites imally closely spaced contour maps or bj; i ts stati sti cal pr operti es . . Wi t h- out mu ch ado, it is a pp arent th at th e latter prob- ably th e m ost logical a nd comp ac t way 111 th e absence of any o th er exa ct descrip tion unl ess some oth er form of mi crosco pi c scale descrip tion is needed. Almo st any rough s Ul'f ace can be sho\,:n to hav e a probability density fun c ti on. of i ts a nd a height-di st anc e auto c orr el at IOn functIOn, It has be en pr eviously shown [Hayr e, 1961] that a larg e numb er of naturall y occurring rough may be said to have their heights normall y dlstnbuted abov e and below th eir mean v alue. It has also been shown t hat an exponential h eight-di s tance auto covarianc e function seems to describ e many such cases, or (7) __ 1_ -( h -m) 2/2u2 p /, - /-- e , '\ 27T'0' 2 wher e p(h) = prob ability densit y fun c tion of height s h= height . m= mean heIght O' = standard devia tion of height s oCr) = height-distan ce au tocovari ance 696- 013- 63--13 (1) (2) 13 = horizon tal di stan ce at whi ch au tocova l'i ance dr ops to l /eth of its value at zer o 1'. It is now necessary to establish the physical meanil 1D' of th ese t hr ee stat i st i cal p arameters m, 0' a nd 13. b Th e usefulness of mC!1n height, m, a nd it s rel at ionship to a physical roughn ess is undoub t,e d,l y seH-explana tor y. Now l et us see on e m(m: a bou t t he m ea n slope of the sur/ace, mean size of lumps along the surface, aJ!d, io general, rou ghn ess in te rms of ve rti cal an d h Ol'lzon tal roughn ess para m- eLm's after Lh ese consLa nLs have been determin ed from exp erim ent al d ata, Th e classical the ories [Taylor, 1935] very clearly that Cf( /13 ) is a measur e o( th e ave rage slop e of t be surface, because 0' is the sta nd ard d e vi at ion of ve rti cal rou o'hness while B is a meas ur e of hori zon tal si ze of . b - , surface pert urb at ions" , . Th e roughness parameters 13 a! ld 0' can speCIf y many types of Fo!' lI1 st ance, a large value of 13 would lIlchcate reJ at llrely smoo th va na- tl 'on of heio'hts wh ereas a lar o'e va lu e of 0' would b , "'. S II represent th e rugge dn ess o( the te rram .. m,a valu es of 13 a nd 0' indi ca te a very rough terr am "nth small scale (v ertic al) roug hn ess, wh ereas a large 13 !1nd l arge 0' repl'6sent a rela ti vely smoo th s urfa ce with large scale perturb ations, On the ot l,t er .ha nd , a small 13 and a large 0' would seem to an extremely rough surf ace with large size undul at IOn s, while a laro'e v alue of 13 a nd a small v alu e of ()' are chara cteristi c of a surf ace which is very s moo th a nd has small size disturban ces in it s contours, A s umma ry of t h e physical of th e sta tisti cal rou ghn ess parameters IS gIven 111 t he following t abl e. . ' This int erpr etation, wh en used WI th Pe tt engIll's [1960] experiment al data on lun ar echoes Hayre a nd Moore's [19 6 1] lun ar echo th eory, an avera ge slope of 1 in ] 0 [Hayr e, 1963, to be It is in th e o' eneral rano 'e for th e me an slope of 1 III 7.4 and 1 in 11 ob ta ineci"'b y Dani els [1963] and E,r ans a nd Pe tt engill [1963], respec tively , This Seel?S, to indi cat e that th e lunar th eories bas ed on statIstI cal analysis alone caD yield reasonable and experiment ally 763
Transcript of Statistical methods in radar astronomy. Determination of ... · Statistical Methods in Radar...
JOURNAL OF RESEARCH of the National Bureau of Standards-D. Radio Propagation Vol. 67D, No.6, November-December 1963
Statistical Methods in Radar Astronomy. Determination of Surface Roughness
H. S. Hayre
Contribution from Department of Electrical Engineering, Kansas State University, Manhattan, Kans.
(R eceived Jun e 13, 1963)
An experimentally established statist ical mod el of a rough surface is u sed to show t hat sufficient information a bou t t he roughn ess of such a surface In t he form of Its sta !l da rd dev Iat ion, mean horizo nta l size of lumps, a nd average slope can be obta In ed f rom. expcl'l ITI cn tal data when uscd in conju nction wi t h a t heo ry based on statIstIca l anal,vsls.
1. Introduction
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
(7) _ _ 1_ -(h -m) 2/2u2 p /, - /-- e , '\ 27T'0' 2
where p(h) = probability density function of heights
h= height . m = mean heIght
O' = standard deviat ion of heights oCr) = height-distance autocovariance
696- 01 3- 63--13
(1)
(2)
13= horizon tal distance at which au tocoval'i ance drops to l /eth of its value at zer o 1'.
It is now n ecessary to establish t he physical meanil1D' of these t hree statistical parameters m, 0'
and 13. b The usefulness of mC!1n heigh t , m, a nd its relationship to a physical ro ughness is undoub t,ed,ly seH-explan a tory. Now let us see ~vha L one ca l~ m(m: a bout t he mean slope of t he sur/ace, m ean size of lumps along t he surface, aJ!d , io gen eral, roughness in term s of vertical and h Ol'lzontal roughness parameLm's a fter Lhese consLanLs have b een determined from experimental data, Th e classical th eories [Tay lor, 1935] ver y clearly establish~d t ha t Cf(/13) is a m easure o( the average slope of t be surface, b ecause 0' is the standard de viation of vertical rouo'hness while B is a m easure of horizontal size of . b - ,
surface per t urbatio ns" , . The roughness p ara meters 13 a!ld 0' can speCIf y
m any types of ro.ug l~ness, Fo!' lI1stan ce, a large value of 13 would lIlchcate reJatllrely smooth vana-tl'on of heio'hts whereas a la ro'e value of 0' would
b , "'. S II represent the ruggedness o( t he terram . . m,a values of 13 and 0' indicate a very rough terram "nth small scale (vertical) roughness, whereas a large 13 !1nd large 0' repl'6sent a rela tively smooth surface with large scale perturba tion s, On the ot l,ter .hand, a small 13 and a large 0' would seem to slgl1lf~T an extremely rough surface with large size undulatIOns, while a laro'e value of 13 and a small value of ()' are characteristic of a surface which is very smoot h and has small size disturbances in its contours,
A summary of t he physical inte~'Pre t,a tion, of the statistical roughness param eters IS gIven 111 t he following table. . '
This interpretation , when used WI th P ettengIll 's [1960] experimental da ta on lunar echoes an~ Hayre and Moore's [1961] lunar echo theory, Ylel~s an average slope of 1 in ] 0 [Hayre, 1963, to b e pubhshe~]. It is in the o'eneral rano'e for the mean slope of 1 III 7.4 and 1 in 11 obtaineci"'by Daniels [1963] and E,rans and Pettengill [1963], respectively , This Seel?S, to indicate that the lun ar theories b ased on statIstIcal analysis alone caD yield r easonable and experimentally
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verifiable results . It may not be out of place to add that the use of statistical theory in radar return from a rough surface is a step in the right direction when other deterministic infoTmation is not available.
S pec1:jication oj surface rou ghness in terms of characteri stics constants Band rr
ilorizontal size Vert ical size of roughness of roughness
(a verage length of areas at one Small M ed ium Large
elevation)
Large Very small slope M edium slopes Smallu MediumCT Large u Large B L arge B L arge B
Medium Small slopes M edium slopes Small u Mediumq Large (J'
M ediumB M edium B Medium B
Small Large slopes Very large slopes Small u Medium u Large (f"
Small B Small B Small B
2. References
Cooper , J . A. ("May, 1958), Comparison of observed and calculated near-vertical radar ground return intensities and fading spectra, Univers ity of New Mexico Engr. Exper. Station Tech . Report RE. 10.
Daniels, F. B. (Jan . 15, 1963), Radar determination of t he Rms slope of the lunar surface, J . Geophys. Res. 68, No. 2, 449- 453.
Davies, H . (Aug. 1954), The reflection of electromagne tin
waves from a rough surface, J. Inst . Elcc. E ngrs. (London), pt. 4, 101, 209- 215.
E vans, J. V. (J an .- F eb. 1963), A lunar t heory reasserted- a rebu ttal, J . Res. NB S 67D (Radio Prop.), No. 1, 1- 4.
E van s, J . V., and G. H. P ettengill (Jan . 15, 1963), The scattering behavior of the moon at wavelengths of 3.6, 68, a nd 784 cent imeters, J. Geophys. R es. 68, No. 2, 423- 447.
Hayre, H. S. (1961 ), Radar scattering cross section applied to moon return, Proc. IRE 49, 1433.
Hayre, H. S. (Feb. 1962), Roughness and radar return, Proc . IRE 50, No. 2, 223- 224.
Hayre, H. S. (1963), S tatistical estimate of th e lunar surface (to be published).
I-I a yre, H. S., and R. K . Moore (Sep t.- Oct. 1961) , Theoretical scat t ering coefficien t for near-ver t ical incidence from contour maps, J. R es . NBS 65D (Radio Prop.), No.5, 427-432.
NOTE: On page 430, 2d col, 11th line, (3.8) should read (3 .9) ; in line Hi. (3.9) should read (3.8)
Millman, G. H ., and F. L. Rose (Mar.- Apr. , 1963) , Radar re flect ions from t he moon at 425 Mc/s, J . Res. NBS 67D (Radio Prop.), No . 2, 107- 117.
Pettengill, G. H . (Ma y , 1960), Measurement of lunar reflect ivity using t he millstone radar, Proc . IRE 48, No. 5, 933.
Siegle, K. M ., and T. B. A. Senior (May- June, 1962), A lunar theory reasser ted, J. Res. NBS 66D (Radio Prop.), No.3, 227- 229 .
Taylor, G. J . (1935), Statistical t heory of t urbulence, pt. 1- 4, Proc. Roy. Soc. A151, 421- 478.
Winter, D. F. (May- June, 1962), A theory of radar reflect ions from a rough moon, J. Res. NBS 66D (Radio Prop.), No.3, 215- 226.
(Pap er 67D6-305)
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