A Comparison of the Radar Ray Path Equations and Approximations for Use in Radar Data Assimilation

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1 A Comparison of th Radar Ray Path Equations and Approximations for Us in Radar Data Assimilation Jidong Gao, Kith Brwstr and Ming Xu, Cntr for Analysis and Prdiction of Storms School of Mtorology, Univrsity of Oklahoma Submittd to Advanc in Atmosphric Scincs Rvisd: Jun 5 Abstract Th radar ray path quations ar usd to dtrmin th physical location of ach radar masurmnt. Ths quations ar ncssary for mapping radar data to computational grids for diagnosis, display and numrical wathr prdiction (NWP). Thy ar also usd to dtrmin th forward oprators for assimilation of radar data into forcast modls. In this papr, a stpwis ray tracing mthod is dvlopd. Th influnc of atmosphric rfractiv indx to ray path quations at diffrnt locations rlatd to an intns cold front is xamind against th ray path drivd from th nw tracing mthod. It is shown that th radar ray path is not vry snsitiv to sharp vrtical gradints of rfractiv indx causd by strong tmpratur invrsion and larg moistur gradint in this cas. In th papr, th rrors causd by using th simplifid straight ray path quations ar also xamind. It is found that thr will b significant rrors in th physical location of radar masurmnts if th arth s curvatur is not considrd, spcially at lowr lvation angls. A rducd form of th quation for bam hight calculation is drivd using Taylor sris xpansion. It is computationally mor fficint and also avoids th nd for using doubl prcision du to a small diffrnc btwn two larg trms in th original form. Th accuracy of this rducd form is found to b sufficint for modling us. Ky words: Dopplr radar, Ray path quations, Rfractivity Indx, data assimilation Corrsponding Author Addrss: Dr. Jidong Gao, Cntr for Analysis and Prdiction of Storms, Sarkys Enrgy Cntr, Suit, East Boyd, Norman, OK jdgao@ou.du

2 . Introduction Th oprational Dopplr NEXRAD (WSR-88D) radar ntwork is bcoming mor and mor important to improv th ral tim dtction and warning of hazardous wathr (Albrty t al. 99; Crum t al. 998; Srafin and Wilson ). It is viwd as an ssntial obsrving systm for initializing non-hydrostatic, storm-rsolving (i.., horizontal grid spacing of ordr km) numrical wathr prdiction (NWP) modls (.g., Lilly, 99; Drogmir, 99, 997). Attmpts to dmonstrat such capability bgan arly in th past dcad (.g., Sun t al., 99), and subsqunt fforts hav bn notably succssful (.g., Gao t al. 998; Wygandt t al. a,b; Crook and Sun ; Xu t al. ; Brwstr t al. ; Gao t al. ). To assimilat th radar rflctivity and radial vlocity data from wathr radar into an NWP modl, it is ncssary to us suitabl ray path quations to obtain th physical location of ach radar masurmnt and to hav accurat forward oprators to convrt modl winds to radial vlocity in data assimilation schms. Currntly, thr ar svral vrsions for ray path quations in th txtbooks (. g. Doviak and Zrnic 99). Most studis in th radar data assimilation us a vry simpl straight lin ray path quations to modl th forward oprator that projcts th D wind filds from NWP modl to th radial dirction (.g., Gao t al. 998; Shapiro t al. ; Wygandt t al. a,b). This papr aims at stimating rrors causd by using simplifid ray path quations. In nxt sction, w will rviw th radar ray path quations in diffrnt forms with and without considring arth curvatur undr th assumption that th standard atmosphr is considrd. In sction, w first driv a nw stpwis ray tracing mthod, and thn xamin th validity of ray path quations whn th vrtical gradints of rfractiv indx of air ar significantly diffrnt from that of standard atmosphr during a strong cold front outbrak. In sction and 5, w analyz th rror of th bam hight and horizontal distanc calculation whn arth curvatur is not considrd. Th rror for local slop angl θ of th ray path, which is important whn projct th thrdimnsional wind fild to radial dirction, is analyzd in sction. Finally a summary and furthr discussion is givn in sction 7.. Th Ray Path Equations and Som Othr Oprators Undr th assumption that tmpratur and humidity ar horizontally homognous so that th rfractivity is a function only of hight abov ground, Doviak and Zrnic (99) drivd th formulation that xprsss th ray path in trms of a path following a curv of a sphr of radius, a a = = ka, () + a ( dn dh ) whr a is th arth s radius and k is a multiplir which is dpndnt on th vrtical gradint of rfractiv indx of air, dn dh. Th rfractiv indx of air, n, is a function of its tmpratur, prssur and humidity and is usually takn, subjct to crtain assumptions, as (Bam and Dutton 98), 5 N = ( n ) = 77. P/ T +.7X T ()

3 whr P is air prssur in hpa (including watr vapor prssur), is watr vapor prssur in hpa, and T is air tmpratur in dgr K. It is convnint to us th quantity N calld radio rfractivity instad of n. N rprsnts th dpartur of n from unity in parts pr million. N has a valu of about (at th surfac) and its variations can b considrd mor convnintly. In th abov quation, th first trm on th right hand sid is known as th dry trm, th scond trm is th moist trm. Th valu of N can b computd from masurmnt of P, T,. Whn th Standard Atmosphr is considrd, it is found that k is qual to / (Doviak and Zrnic 99). This is oftn rfrrd to as th four-thirds arth radius modl. If N dcrass mor (lss) rapidly with hight than th Standard Atmosphr, th bam may b rfractd mor (lss), and in such cass, th hight of a targt may b ovrstimatd (undrrstimatd) by th four-thirds arth radius modl. In an xtrm condition (sharp rfractivity gradint N/km blow m hight), a ray snt at a positiv lvation angl may actually dcras in hight with rang and vntually strik th arth Doviak and Zrnic 99). Th following two quations rlat h and th surfac rang (distanc along th arth s surfac), s, to radar-masurabl paramtrs, th slant path, r and radar lvation angl, θ (Doviak and Zrnic 99), s = k r cosθ a sin, () ka + h [ r + ( k a) + rk asin ] k a h = θ. () To driv an xprssion rlating th radial vlocity, masurd by th radar, to th wind at th masurmnt point ( r, θ, φ ), whr φ is azimuth angl, a combind sphrical and Cartsian coordinat systm is usd with x and y as arc distancs from th radar along two orthogonal grat circl paths. W choos y to b along a longitud, with north th positiv dirction. z masurs th hight abov th antnna hight of th bam. Th horizontal componnts u and v of th vctor wind v ar tangnt to th grat circl arcs at x and y and ar dirctd astward, rspctivly. Th vrtical componnt w of v is along z, with z= at th hight of th radar fdhorn. Som symbols usd in Eqs. ()-() ar listd in Tabl I. In thir txtbook, Doviak and Zrnic (99) also show that if r k a, th coordinats x, y and z ar rlatd to th radar coordinats ( r, θ, φ ) by, x rcosθ sinφ, (5a) y rcosθ cosφ, (5b) / z= h= ( ka + r + rka sin θ) ka, (5c) whr θ, th angl btwn th radar bam and th tangnt plan blow th data point, is th sum of two trms xprssd as th following (Brwstr ), θ = θ + tan [( rcos θ/( ka+ rsin θ)]. () From (5a) and (5b), on can asily driv th distanc along th arth s surfac as, s rcosθ. (7) Th radial vlocity v r is th projction of v onto r, th vctor from th radar to th point ( r, θ, φ ). Again, if r k a,

4 r cosθsinφ cosθcos φ ( t) sinθ v = u + v + w w (8) For som applications, whr r<< k a, prviously publishd NWP rsarch has typically ignord th influnc of arth curvatur and furthr rducd th abov quations, trating th radar ray path as a straight lin ovr a flat arth. In such a cas, Eqs. (5), (7), and (8) ar simplifid as, x rcosθ sinφ, (9a) y rcosθ cosφ, (9b) z rsinθ, (9c) v = ucosθ sinφ+ vcosθ cos φ+ ( w w ) sinθ, () r t s rcosθ. () Equations (7) and () ar two diffrnt forms of approximation of ray path quation (). Equation (5c) uss xactly th four-thirds arth bam hight quation (), and (9c) is an approximation of (). In th following sctions, w will first xamin th influnc of rfractiv indx to th ray path quations basd on a stpwis ray trac mthod, thn invstigat whthr th simplifid ray path quations ar appropriat.. Influnc of Rfractiv Indx Typically, th four-thirds arth radius modl has bn usd for radar ray paths, assuming th rfractivity indx is linarly dpndnt on hight in th first kilomtr of th atmosphr. Howvr, th gradint of th rfractiv indx is not always a constant, and dparturs from linarity may xist whn thr ar strong tmpratur invrsions or larg moistur gradints with hight. In th following study, w will xamin th influnc of svral diffrnt nvironmntal thrmodynamic profils to th radar ray path. To accuratly stimat th radar ray path, w dvlop a stpwis ray tracing mthod as follows: a) Starting from th first gat nar th radar location, for ach radar masurmnt, calculat th rfractivity N i- for th prvious gat according to Eq.() using diffrnt thrmodynamic profils and gradints of rfractiv indx according to th diffrntial of Eq. () with rspct to bam hight, dn dn =. () dh i dh i Hr i is th indx of th gat. b) Calculat ai, = ki, a according to Eq. () using th gradint of rfractiv indx from stp (a) at ach gat; c) Calculat th angl btwn th radar bam and th tangnt plan blow th data point, θi using Eq. () for ach radar bam gat;,

5 d) Finally, th radar bam hight h and th surfac rang s can b calculatd using th following formulation, hi = hi + r sin θ, i, () s = s + r cos θ, i i, i whr, r is gat spacing, with 5m for NEXRAD radial winds. Variabls h i and s i ar th bam hight and surfac distanc for ach gat, rspctivly. Stps (a) through (d) ar rpatd from th first (i=) until th last gat of th radar masurmnt. As an xampl, w apply th abov procdur to an intns cold front outbrak in th Southrn Plains of th Unitd Stats in th wintr of 99 documntd in th txtbook of Blustin (99). Figur shows four diffrnt tmpratur and dw point profils within th Southrn Plains at UTC, Dcmbr, 99. At this tim a surfac analysis indicats an intns cold front along th boundary of Oklahoma and Arkansas, and th astrn part of Txas. Two of th profils (Figs. a, b), Lak Charls, Louisiana (LCH) and Longviw, Txas (GGG) wr on th warm sid of th front; th othr two stations, Norman, Oklahoma (OUN) and North Platt, Nbraska (LBF) ar on th cold sid of th front (Figs. c, d). Th air in th frontal zon is humid on both sids of th front; and th air abov th frontal zon is rlativly warm and dry. At th Norman and North Platt sits, thr is a pronouncd frontal invrsion btwn 9 and 85 hpa. Th profils of vrtical radio rfractivity gradint ar shown in Fig.. It might b xpctd that thr xist larg rfractivity gradints btwn 9 and 85 hpa bcaus th strong tmpratur invrsion and sharp moistur gradints at Norman and Lak Charls sits (Figs. a and c). Howvr, th gradints of rfractivity ar clos to normal for th North Platt sit, which only has a strong tmpratur invrsion, whil th othr soundings xhibit larg dviations from th standard atmosphr at lvls with strong moistur gradints. So it is sn that th gradints of radio rfractivity ar fairly snsitiv to vrtical variations in humidity. To quantitativly stimat such snsitivity is vry difficult bcaus th rfractivity gradint is a complicatd function of humidity. Figur shows th variations of radar bam hight with th rang gat for a.5º lvation angl. Th solid lin is th ray path calculatd from Eqs. () and () with th standard atmosphr condition and th dashd lin is th ray path calculatd using th stpwis mthod with th rfractivity gradints drivd from th obsrvd thrmodynamic profils. It is shown that for all four soundings, th calculatd bam hights ar gnrally clos to th ray paths that ar drivd from th standard atmosphric condition, with som small diffrnt variations, dspit strong dpartur from th standard atmosphr in som layrs. On can s that for th Lak Charls and Norman sits (Figs. a and c), th radar bam rfractd downward toward arth surfac du to th sharp rfractivity gradint at th -km lvl. Th largst diffrnc in th bam hight is about mtrs and occurs with th Lak Charls sounding at about.5 km hight. Th rlativ rror with rspct to bam width is about 7%. For Longviw and North Platt, th calculatd ray path is vry clos to that for th standard atmosphric condition. For th highr lvation angls, th radar ray paths ar vn lss snsitiv to th rfractivity gradints (not shown). Suppos that th rror for bam hight rlativ to bam width should b no mor than 5%, thn w can s that for this cas which had profils 5

6 typical of a strong cold front, with strong tmpratur invrsion and vrtical moistur gradint, that th us of th four-thirds arth radius modl prdicts bam hight with sufficint accuracy for numrical modling using wathr radar data. In Doviak and Zrnic (99), it stats that for wathr radar applications, th four-thirds arth radius modl can b usd for all ray paths, if bam hight, h, is rstrictd to th first - km and if rfractiv indx, n, has a gradint of about /a in th first kilomtr of th atmosphr. It is still tru for this xtrm cas that rfractiv indx has gradints that significantly lss than /a in som lvls. Crtainly thr xist cass whr ducting and strong dparturs from th fourthirds arth radius modl can and do occur. W ar using th abov procdur ()-() to vrify th bam path quation () and () undr a rang of locations across th Unitd Stats for svral dcads to quantify th rlativ occurrnc of significant dparturs. This climatological study will b publishd sparatly.. Error Analysis of th Bam Hight Calculation Bcaus th total numbr of radar data to b usd in a data assimilation application can b quit voluminous, it would b practical to us th simplifid quations for th bam path to improv computational fficincy. Simplifid quations ar also usd in th litratur at tims to improv tractability. In this sction, w discuss th rrors of bam hight calculation using simplifid Eq. (9c) as compard to Eq. (). To xamin th rrors for th lowr lvation angl, w first choos lvation θ =.5, and rangs from to km with a rang rsolution of 5 m. Figur a shows th comparison of radar bam hights calculatd by Eqs. () and (9c), whil Fig. b shows th absolut diffrncs in bam hights calculatd using ths two formulas. On can s that two curvs stay rlativly clos to ach othr only for rang lss than km. At km rang, th rror is about 5 m; at km, th rror rachs 5 m. At km, th rror has grown to m. This dmonstrats that it may not appropriat to us Eq. (9c) to calculat th bam hight for low lvation angls. Whn th lvation angl θ is incrasd to, it is shown that th two bam hight curvs rmain within 5 m of ach othr until th rang of km (Fig 5a). Although th rlativ rror is small, th absolut diffrnc btwn th two stimats is narly sam as that of low lvation angl cas. Fig shows th rlativ rrors to radar bam width, calculatd using Eq. (9c) instad of th four-thirds arth ray path quation Eq. (). It is shown that for both lvation angls of.5 and, th rlativ rrors ar quit larg for most of gats. Though at rang km, th rlativ rror is only about %; as th bam rachs km, th rlativ rrors xcd 7%. So for most applications, it is ncssary to us th curvd ray path quation () to calculat th hight of radar masurmnt locations, instad of th simplifid Eq. (9c). Howvr, numrically valuating th bam hight of ach radar masurmnt using th original formula, Eq. (), is complicatd and doubl prcision is usually rquird bcaus th right hand sid of Eq. () is a small diffrnc btwn two larg trms. A possibl rduction of Eq. () for computational fficincy is xamind in th following. Rarranging Eq. () yilds,

7 [ ] h = ( ka) + x, () whr, r + rkasinθ x =. (5) ( ka ) Obviously, x, using Taylor sris xpansion and kp th first ordr, h can b xprssd as, r h rsinθ +. () ka This quation is a much bttr approximation than Eq. (5c). Figur 7 shows th diffrnc btwn th bam hights givn by () and (), as th function of bam rang for lvation angl.5. Th maximum diffrnc at rang km is only about.5 m. Equation (), a much simplifid form, is thrfor a good approximation to Eq. () and suitabl for us whn fficincy is important and many calculations must b don. 5. Th Surfac Rang (Distanc along th Earth s Surfac) In last sction, w prformd rror analyss for bam hight. In this sction, w will stimat if th calculation for th horizontal location of ach radar masurmnt using th rducd Eqs. (9a) and (9b) giv good approximations to Eqs. (5a) and (5b). To do that, w only nd to dtrmin if Eqs. (7) and () ar good approximations to ray path Eq. (). Figur 8a shows absolut rrors of Eqs. (7) and (), that is, th variations of s s and * s s as a function of rang at low lvation angl.5. On can s that Eq. (7) givs an xcllnt stimat of surfac rang as compard to Eq. (). But Eq. () givs a good stimat only for radar masurmnts lss than km. Whn radar lvation angl is, on can s that th rrors for surfac rang calculatd using both Eqs. (7) and () ar largr, but Eq. (7) still givs a vry good stimat of th surfac rang (Fig 8b) as vn at distant rangs th diffrnc is lss than a singl rang gat. Equation () introducs som rror, but can still b usd bcaus for this lvation angl, th radar ray is at a hight of abov km at 5 km rang (not that at 5 km rang, th horizontal position rror is about m, about on quartr of th rang rsolution of th NEXRAD). So w can conclud that for dtrmining th horizontal location of radar masurmnt, Eq. (7) is a vry good approximation, and Eq. () may also b accptabl for rangs lss than 5 km.. Local Slop Angl θ of th Ray Path Equation () dscribs th calculation of th local slop angl of radar bam, θ, th angl btwn th radar bam and th arths tangnt plan blow th data point. This is an important variabl bcaus w us it not only to calculat th location of radar masurmnt, but also to calculat th radial vlocity in th forward oprator within radar data assimilation, as in Eq. (8). Figur 9 shows th variation of local slop angl, θ with th rang. For low lvation.5, θ incrass with rang, bcoming.8 at km, almost tims th original lvation angl at th radar antnna; whil for highr lvation 7

8 , θ rachs. at rang km (not shown). So th rlativ changs of local slops ar mor significant for lowr lvation angls than that for high lvation angls. Suppos at on radar gat, w hav a horizontal wind u=v= ms -, a vrtical vlocity w = 5 ms - and a trminal fall vlocity w t = 5 ms -, with th azimuthal angl bing φ = 5. Substitut ths valus into Eq. (), w gt V r =.5 ms - for lvation angl.5 and V r =.7 ms - for lvation angl.8. So th diffrnc in radial vlocity is only. ms -, much lss than th xpctd rror of th masurmnt itslf, vn though th local lvation angls vary by a factor of (s Fig. 9). Sinc th NEXRAD radar usually oprats at rlatly low lvation angls (gnrally blow ), and sinθ is small compard to othr trms in lft hand sid of forward oprators (8) and (), th contribution of vrtical vlocitis and trminal fall vlocitis to radial vlocitis rmains rlativly small. Th variation of cosθ is also not vry snsitiv to θ with th incras of rang gat for a fixd lvation angl; this lads to Eq. () bing an accptabl approximation to Eq. (8). 7. Summary and Discussion Th radar ray path quations ar ndd to dtrmin th forward oprators for th assimilation of radar data into forcast modls. In this papr, w hav rviwd th ray path quations in svral forms and dvlopd a nw stpwis ray tracing mthod. Th influnc of atmosphric rfractiv indx to ray path quations at diffrnt locations rlatd to an intns cold front is xamind. It is shown that th radar ray path is not vry snsitiv to th rlativly larg vrtical gradints of rfractiv indx causd by larg tmpratur invrsion and moistur gradints in this intns cold front nvironmnt. In som publishd work, radar ray paths hav bn approximatd as straight lins. This simplifis th quations usd to dtrmin th physical location of ach radar masurmnt, but introducs rrors that ar significant for rangs byond km. It is found that th calculation of th physical location of ach radar masurmnt may hav significant rror if th four-thirds arth ray path quations ar not usd, spcially whn th radar is opratd at low lvation angls. A rducd form of th quation for bam hight is drivd using Taylor sris in this papr, which is computationally mor fficint and also avoids th nd for using doubl prcision du to a small diffrnc btwn two larg trms in th original form. This form is found to b rathr accurat. It is dmonstratd that for horizontal location of radar masurmnt, Eq. (7) is a vry good stimat, and Eq. () can also b accptabl for radar masurmnts within th 5 km in rang. W also find that th radial vlocity forward oprator that projcts th thr componnts of wind to th radial dirction undr th assumption of straight-lin ray paths givs a rasonabl approximation undr typical oprating conditions. Th rsults of this papr provid usful guidancs to radar data analysis and assimilation applications in which both fficincy and accuracy ar important. Acknowldgmnts. This work was supportd by US NSF ATM-989, ATM-75, ATM-59 and EEC-77, and DOT-FAA grant NA7RJ7-. Graphics plots wr gnratd by th GNUPLOT graphics packag. 8

9 Rfrncs Albrty, R. L., T. D. Crum, and F. Topfr, 99: Th NEXRAD program: Past, prsnt, and futur--a 99 prspctiv. Prprints, 5 th Intl. Conf. on Radar Mtorology, Paris, Amr. Mtor. Soc., -8. Ban B. R. and E. J. Dutton, 98: Radio Mtorology, Dovr Publication, 5 pp. Brwstr, K.A., : Phas-corrcting data assimilation and application to storm scal numrical wathr prdiction. Part II: Application to a svr storm outbrak. Mon. Wa. Rv., Blustin, H. B. 99: Synoptic-Dynamic Mtorology in Midlatituds. Volum II, Oxford Univrsity Prss, Inc. Crook, N. A., J. Sun, : Assimilating Radar, Surfac, and Profilr Data for th Sydny Forcast Dmonstration Projct. J. of Atmos. and Ocanic Tchnol., 9, Crum, T. D., R. E. Saffl, and J. W. Wilson, 998: An Updat on th NEXRAD Program and Futur WSR-88D Support to Oprations. Wa. and Forcasting,, 5. Doviak, R.J. and D.S. Zrnić, 99: Dopplr Radar and Wathr Obsrvations, Acadmic Prss, nd Edn., 5 pp. Drogmir, K. K., 99: Toward a scinc of storm-scal prdiction. Prprint, th conf. on Svr Local Storms, Kananaskis Park, Albrta, Canada, Amr. Mtor. Soc., 5-., 997: Th numrical prdiction of thundrstorms: Challngs, potntial bnfits, and rsults from ral tim oprational tsts. WMO Bulltin,, -. Gao, J., M. Xu, Z. Wang and K. K. Drogmir, 998: Th initial condition and xplicit prdiction of convction using ARPS forward assimilation and adjoint mthods with WSR-88D data. Prprints, th Confrnc on Numrical Wathr Prdiction, Phonix, AZ, Amrican Mtorol. Socity, Gao, J., M. Xu, K. Brwstr, and K. K. Drogmir : A thr-dimnsional variational data assimilation mthod with rcursiv filtr for singl-dopplr radar, J. Atmos. Ocanic. Tchnol.,, Lilly, D. K., 99: Numrical prdiction of thundrstorms - Has its tim com? Quart. J. Roy. Mtor. Soc.,, Srafin, R. J., and J. W. Wilson, : Oprational Wathr Radar in th Unitd Stats: progrss and opportunity. Bull. Amr. Mtor. Soc. 8, Shapiro, A., P. Robinson, J. Wurman, and J. Gao, : Singl-Dopplr Vlocity Rtrival with rapid-scan radar data, J. Atmos. Ocanic. Tchnol., Sun, J., D. W. Flickr, and D.K. Lilly, 99: Rcovry of thr-dimnsional wind and tmpratur filds from simulatd singl-dopplr radar data. J. Atmos. Sci., 8, Sun, J. and N. A. Crook, : Ral-tim low-lvl wind and tmpratur analysis using singl WSR-88D data. Wa. and Forcasting.,, 7-. Wygandt, S.S., A. Shapiro and K.K. Drogmir, a: Rtrival of initial forcast filds from singl-dopplr obsrvations of a suprcll thundrstorm. Part I: Singl- Dopplr vlocity rtrival. Mon. Wa. Rv,, -5. 9

10 Wygandt, S.S., A. Shapiro and K.K. Drogmir, b: Rtrival of initial forcast filds from singl-dopplr obsrvations of a suprcll thundrstorm. Part II: Thrmodynamic rtrival and numrical prdiction. Mon. Wa. Rv,, 5-7. Xu, M., D.-H. Wang, J. Gao, K. Brwstr, and K. K. Drogmir, : Th Advancd Rgional Prdiction Systm (ARPS), storm-scal numrical wathr prdiction and data assimilation. Mtor. and Atmos. Physics. 8, 9-7.

11 TABLE. LIST OF SYMBOLS FOR EQUATIONS ()-() Symbol Dscription s h x, yz, θ θ Surfac rang of ray path using standard ray path quation () Bam hight Cartsian Coordinats for curvd ray path whn th arth curvatur is considrd Radar lvation angl Angl btwn th radar bam and th tangnt plan blow th data point s Surfac rang of ray path using approximat (7) v r Radial vlocity for th curvd ray path x, y, z Cartsian Coordinats for straight ray path whn th arth curvatur is not considrd v r s θ Radial vlocity for th straight lin ray path Surfac rang of ray path for th straight lin ray path Radar lvation angl

12 Prssur (hpa) (a) (b) Prssur (hpa) (c) (d) o o Tmpratur ( C) Tmpratur ( C) Figur. Th tmpratur (solid) and dw point (dashd) profils for UTC, Dcmbr, 99, at (a) Lak Charls, Louisiana (LCH), (b) Longviw, Txas (GGG), (c) Norman, Oklahoma (OUN), and (d) North Platt, Nbraska (LBF), abscissa is tmpratur ( o C); ordinat is prssur (hpa).

13 5 (a) 5 (b) Hight (km) Hight (km) 5 (c) 5 (d) Th gradint of rfractivity ( km - - ) Th gradint of rfractivity ( km ) Figur. Th rfractivity gradint profils (km - ) for UTC, Dcmbr, 99, at (a) Lak Charls, Louisiana (LCH), (b) Longviw, Txas (GGG), (c) Norman, Oklahoma (OUN), and (d) North Platt, Nbraska (LBF). Th solid lin is th rfractivity gradint for th standard atmosphr; th dashd lins rprsnt obsrvd conditions.

14 5 (a) 5 (b) Hight (km) (c) 5 (d) Hight (km) (b) 5 5 Bam rang (km) 5 5 Bam rang (km) Figur. Th radar ray paths calculatd for.5 lvation angl using th rfractivity gradints drivd from (a) Lak Charls, Louisiana (LCH), (b) Longviw, Txas (GGG), (c) Norman, Oklahoma (OUN), and (d) North Platt, Nbraska (LBF) using th stpwis mthod (dashd lins), as compard to thos drivd from standard atmosphr rfractivity gradint (solid lins).

15 8 (a) Hight (km) (b) Hight diffrnc (km) Bam rang (km) Figur. (a) Th variations of radar bam hight with th rang gat for lvation angl.5. Th solid lin is calculatd by th /rds arth ray path quation (c), and th dashd lin is calculatd by straight lin ray path approximation Eq (7c); (b) Th absolut diffrnc in bam hight calculatd from th two formulas givn by Eq. (c) and Eq. (7c). 5

16 8 (a) Hight (km) (b) Hight diffnnc (km) Bam rang (km) Figur 5. As in Figur, but for th cas whn lvation angl is. Bam hights gratr than km ar not shown bcaus typically thr is no wathr signal abov km lvl.

17 8 Hight (m) 5 5 Bam rang (km) Figur. Th rlativ rrors of bam hight to radar bam width calculatd by straight lin path approximation (7c). Th solid lin is for lvation angl.5 whil th dashd lin is for. 8 Hight (m) 5 5 Bam rang (km) Figur 7. Th diffrnc btwn th bam hights givn by approximat formula () and tru bam path quation (). 7

18 5 5 (a) Errors of surfac rang (m) (b) Errors of surfac rang (m) 5 5 Bam rang (km) Figur 8. (a) Th diffrncs in surfac rangs givn by th tru ray path quation and its two forms of approximation at lvation angl.5. Th dashd lin is for curvd ray path formula (7) and th solid lin is for straight lin ray path formula (). (b) Th sam as (a), but for lvation angl. Not that th dashd lin in (a) suprpos horizontal axis. 8

19 .8 Elvation angl (dgr) Bam rang (km) Figur 9. Th variation for local slop angl as a function of bam rang. 9