On the indirect e ect in the Stokes±Helmert method of geoid determination

Transcription

1 Jounal of Geodesy (1999) 7: 87±9 On the indiet e et in the Stokes±Helmet method of geoid detemination L. E. SjoÈ beg, H. Nahavandhi oyal Institute of Tehnology, Depatment of Geodesy and Photogammety, S Stokholm, Sweden sjobeg@geomatis.kth.se; Tel.: ; Fax: eeived: 10 Mah 1998 / Aepted: 16 Novembe 1998 Abstat. The lassial integal fomula fo detemining the indiet e et in onnetion with the Stokes±Helmet method is elated to a plana appoximation of the sea level. A stit integal fomula, as well as some appoximations to it, ae deived. It is onluded that the apsize tunated integal fomulas will su e fom the omission of some long-wavelength ontibutions, of the ode of 50 m in high mountains fo the lassial fomula. This long-wavelength infomation an be epesented by a set of spheial hamoni oe ients of the topogaphy to, say, degee and ode 60. Hene, fo patial use, a ombination of the lassial fomula and a set of spheial hamonis is eommended. Key wods. Geoid Helmet ondensation Indiet e et emove±estoe that they ae limited to the seond o thid powe of elevation. A eent desiption of the Stokes±Helmet method fo geoid detemination was given by VanõÂ ek and Matine (1994). The spei poblem of detemining the indiet e et was teated by Matine and VanõÂ ek (1994), who pointed out that the lassial fomula may be seveely biased. In the pesent pape, we will stat to ompae the fomulas based on plana appoximation with those based on spheial hamonis. Futhe on, a igoous sufae integal fomula fo the indiet e et will be deived (assuming a spheial appoximation of the geoid and onstant topogaphi density). The stit fomula is nally expanded in a Taylo seies fo ompaison with the pevious appoximate solutions. The indiet e et to powe H 1 Intodution Geoid detemination by Stokes' well-known fomula equies that all topogaphi masses ae emoved o edued to a laye on o below the geoid. The e et of estoation of these masses is the indiet e et. The most ommon edution method is Helmet's seond ondensation method, whee the topogaphi masses ae edued to a sufae laye at sea level in a spheial appoximation of the geoid. This lassial method, using a plana appoximation, has been extensively disussed, by, fo example, Heiskanen and Moitz (1967), Wihienhaoen (198), VanõÂ ek and Kleusbeg (1987), Wang and app (1990) and SjoÈ beg (1994). SjoÈ beg (1994, 1995, 1997) and Nahavandhi and SjoÈ beg (1998) used a spheial hamoni appoah to deive the indiet e et. Common to all the above methods is Coespondene to: Las E. SjoÈ beg The lassial fomula (see e.g. Wihienhaoen 198) fo detemining the indiet e et on the geoid fo Helmet's seond ondensation method is N I P lassi ˆ plh P l H HP 6 `0 d 1 whee l ˆ Gq 0, G being the univesal gavitational onstant, and q 0 the density of topogaphy, assumed to be onstant; H; H P =othometi heights of the unning and omputation p points, espetively; ˆ 1 os w ˆ sin w ; = mean Eath sufae adius; = the unit sphee; w = geoenti angle between omputation point P and unning point on the sphee; = mean nomal gavity.

2 88 SjoÈ beg (1994, 1995, 1997) developed the indiet e et in tems of (sufae) spheial hamonis to the powe H, and Nahavandhi and SjoÈ beg (1998) extended this appoah to the powe H. The esults an be summaized as N I P ˆ pl X 1 n 1 n 1 H n P pl X 1 n n 1 n 1 H n P whee Hn m 1 P ˆn H m P n os w d; m ˆ ; 4p and P n os w is Legende's polynomial of ode n. Equation () an be efomulated as an integal simila to Eq. (1). To ahieve this, we st ewite Eq. () as follows: " N I P ˆpl HP H P X1 n 1 H n P X1 n 1 # 1 H n P 4 n 1 whee we have used the notation H m P ˆ X1 Hn m P ; m ˆ ; Inseting Eq. (), and onsideing that X 1 P n os w ˆ and (Heiskanen and Moitz 1967, p. 9) 1 X 1 nhn H P ˆ HP d p `0 we aive at N I P ˆpl HP H d 4p H P HP H H 6p `0 4 d In view of the fat that d ˆ 1 4p we nally obtain N I P ˆpl H P l H HP d 4 l H H P 1 `0 6 `0 4 d Compaing the lassial fomula [Eq. (1)] with the new one [Eq. (9)], we obtain the di eene dn I P ˆN lassi I l 4 NI new o, in view of Eq. (8) dn I P ˆ l 4 o, in spetal fom dn I P ˆ pl whee H m P ˆ X1 ˆ pl H P H H P H H P pl 1 n 1 H m n P d l 8 d l 8 H P H P H H P H H P d d Fomula (1) shows that thee ae some long-wavelength di eenes of powe H between the lassial and new fomulas. (The tems of powe H ae less than a few entimetes, and they an theefoe be negleted in most ases.) The most likely explanation fo this di eene is that the lassial method su es fom the plana appoximation. This an be seen fom the following example. Fo a smooth topogaphy, the st tem on the ight-hand side of Eq. (11) an be appoximated by l 4 H d ˆ: plh P s 0 ˆ: 0:07s 0 H P mm whee s 0 is the maximum pola adius (in km) and H P is also given in km. Fo example, with s 0 ˆ 555 km (oesponding to a geoenti adius of about 5 ) and H P =1 km, this tem beomes 1:5 m, but fo H P ˆ 6 km it anges to 50 m! (Cf. Matine and VanõÂek 1994). Hene dn I in Eq. (1) an be egaded as a oetion to the lassial method, whih leads to the fomula N new I ˆ N lassi I dn I 14 A stit fomula fo the indiet e et At the point P on the sphee of adius, the topogaphi potential at point P an be witten (fo onstant density) Z H V t d P ˆl d 15 ˆ ` p whee ` ˆ t, is the geoenti adius of topogaphi point and t ˆ os w. The oesponding

3 89 potential of a ondensed laye aoding to Helmet's seond ondensation method beomes V P ˆl H d 16 oesponding to a sufae laye of density q 0 H on the sphee of adius. By di eening and dividing by (Buns' fomula), we obtain the indiet geoid e et N I P ˆ1 V t P V P ˆ l f H; t H d 17 whee we have intodued the kenel funtion f H; t ˆ Z H d ` 18 This fomula an be dietly integated to (f. Matine and VanõÂ ek 1994) f H; t ˆ H t ` t H t ` P t ln t 19 o 8 0 if H ˆ 0 >< t f H; t ˆ ` H H ` 0 H` >: P t ln H t ` if H > 0 t 19a Theefoe, Eq. (17), with the funtion f H; t povided by Eq. (19) o Eq. (19a), is a igoous integal fomula fo the indiet e et. 4 Taylo expansion of the stit fomula Fo ompleteness, we now deive the powe seies of N I fom the stit fomula of Eq. (17) above. We stat fom a Taylo expansion of f H; t aound f 0; t : f H; t ˆf 0; t X1 kˆ1 f k 0; t H k k! 0 whee f k is the k-th adial deivative (i.e. with espet to H) off H; t. It follows dietly fom Eq. (18) that f 0; t ˆ0 and f 1 0; t ˆ = Thus we an ewite Eq. (17) as X N I P ˆl 1 f k 0; t H k d kˆ k! Fom (with the notation ˆ H) 1 f 1 H; t ˆ` ˆ X1 we obtain n P n t f H; t ˆ ` ` ˆ X1 n 1 n P n t and f H; t ˆ 4` 5 4 ` ˆ 1 X 1 n n 1 n 1 P n t In the limit H! 0, o equivalently!, we obtain f 0; t ˆ d 1 t ˆ X1 n 1 P n t 4 and fom the last sum of Eq. () X f 0; t ˆ1 1 n n 1 P n t ˆ 1 X 1 n 1 n n 1 P n t ˆ 1 D t d 1 t whee we have intodued the notations d 1 t ˆX1 n 1 P n t and D t ˆX1 n 1 np n t 5 Fom Eq. (6), we obtain HP H d ˆ X1 nhn p `0 P ˆ 1 X 1 n 1 np n t H d 4p ˆ 1 D t H d 6 4p and we also have 1 d 1 t F Q d Q ˆ F P 4p fo any ontinuous funtion F and t ˆ os w PQ, whih de nes the popeties of D t and d 1 t. By inseting Eqs. (4) and (5) into Eq. (1), we nally aive at

4 90 N I P ˆpl H P l H HP d 4 l H HP 1 `0 6 `0 4 d O H 4 i.e. the thid-ode Taylo expansion of the stit fomula of Eq. (17) equals ou pevious fomula Eq. (9). 5 Numeial investigations In ode to numeially investigate the indiet e et, we have applied the lassial and new integals, Eqs. (1) and (7) espetively. We have also applied the lassial fomula oeted with the hamoni expansion of Eq. (14) as well as the stit integal of Eq. (17) with f H; t given by Eq. (19a). The esults of the spheial hamoni appoah ae limited to the thid powe of elevation H. The hamoni oe ients of heights Hn and H n ae detemined fom Eq. (). Fo this, a Digital Teain Model (DTM) is geneated using the Geophysial Exploation Tehnology (GETECH) DTM (GETECH 1995a). This DTM was aveaged using aea weighting. Sine the inteest is in ontinental elevation oe ients, the heights below sea level ae all set to zeo. The spheial hamoni oe ients ae omputed to degee and ode 60. The paamete l ˆ Gq is omputed using G ˆ 6: m kg 1 s and q ˆ 670 kgm. =671 km and = 981 Gal ae also used in the omputations. Stitly, Buns' fomula equies to be omputed fo geodeti latitude /. Howeve, sine N I eahes at most m, is set to a onstant. This appoximation is of the ode of the attening of the ellipsoid ( 0:00), whih is at most 6 mm. Two test aeas of size ae hosen. Both of these test aeas ae in Sweden. The aea A is limited by latitudes 57 and 59 N and longitudes 1 and 15 E, loated in the south of Sweden. The topogaphy in this aea vaies fom 40 to 40 m. The aea B is limited by latitudes 6 and 65 N and longitudes 1 and 15 E, loated in noth-west Sweden. The topogaphy in this aea is moe ugged than in test egion A and vaies fom 76 to 1051 m. Fist, the stit fomula is omputed with a global integation. The integation aea is extended outside the test aeas and limited to ap size (w) of0.a:5 :5 0 DTM (GETECH 1995b) is used inside the ap size of 0. Futhe out, we have applied a DTM. This esult is then used as the `tue' solution to ompae with all othe methods. Figue 1 shows the indiet e et omputed by the stit fomula [Eq. (17)] fo aea B. It anges fom 1:5 to 7.4 m. Fo the lassial and new integal appoahes, a :5 :5 0 DTM (GETECH 1995b) is also used. The integation aea is again extended outside the test aeas, but limited to ap sizes of and 1. Figues ±4 show the indiet e et omputed by the lassial [Eq. (1)] and new integal [Eq. (7)], as well as the lassial fomula oeted with the hamoni expansion [Eq. (14)] methods with 1 integation aea in test egion B. Figues and ae simila in shape with mino diffeenes. The plots ange fom 4:0 to 1.0 m (Fig. ) and :99 to 1.79 m (Fig. ). Both of them inlude the loal ontibutions with pue integal fomulas, but they Fig. 1. The indiet e et omputed by stit method [Eq. (17)] with a global integation in test egion B. Contou inteval is 0. m

5 91 Fig.. The indiet e et omputed by lassial method [Eq. (1)] with 1 integation aea in test egion B. Contou inteval is 0. m Fig.. The indiet e et omputed by new integal method [Eq. (7)] with 1 integation aea in test egion B. Contou inteval is 0. m ae laking the fa-zone ontibution outside the ap of adius 1. Figues 1 (`tue' solution) and 4 (oeted lassial solution) ae simila in shape and only slightly di eent in magnitude. In both these fomulas, the loal infomation and long-wavelength ontibutions ae pesent. Finally, Fig. 5 depits the smooth long-wavelength spheial hamoni epesentation of the indiet e et [Eq. ()], anging fom 4. to 6.96 m. This suggests that the lassial fomula oeted with the hamoni expansion [Eq. (14)] is the best altenative to the stit fomula in this study.

6 9 Fig. 4. The indiet e et omputed by lassial fomula oeted with the hamoni expansion [Eq. (14)] with 1 ap size in test egion B. Contou inteval is 0. m Fig. 5. The indiet e et omputed by the spheial hamoni method [Eq. ()] in test egion B. Contou inteval is 0. m In ode to obtain futhe insight into how the methods di e, the lassial, new integal and hamoni expansion appoahes and the lassial fomula, oeted with the hamoni expansion, ae subtated fom the stit method. These di eenes ae pesented fo the two test aeas using integation aeas of and 1 in Tables 1 and, espetively. These tables show eos of the lassial fomula, Eq. (1), as well as the new integal fomula Eq. (7). We note also that the eos even inease with ap size (see Tables 1 and ), evealing that an integation aea of 1 is too small to aount fo the long-wavelength signal. Also, the pue hamoni ex-

7 9 Table 1. Eo of appoximate fomulas fo indiet e et in test egions A and B with ap size Test egion A Test egion B Eq. (1) Eq. (7) Eq. (14) Eq. () Eq. (1) Eq. (7) Eq. (14) Eq. () Max ) )0.5.1 ).69.5 ) Min ).5 ).8 )1.01 ).1 )4.61 ).85 )1.81 )4.0 Ave ).5 )1. ) ).60 ).14 ) SD Units in m Eq. (1) = lassial fomula; Eq. (7) = new fomula; Eq. (14) = oeted lassial fomula; Eq. () = hamoni expansion Table. Eo of appoximate fomulas fo indiet e et in test egions A and B with 1 ap size Test egion A Test egion B Eq. (1) Eq. (7) Eq. (14) Eq. () Eq. (1) Eq. (7) Eq. (14) Eq. () Max ) ) ) ) Min ).1 ).94 )0.71 ).1 )8.4 )9.75 )1.55 )4.0 Ave ).74 )1.5 ) )6.8 )5.04 ) SD Units in m Eq. (1) = lassial fomula; Eq. (7) = new fomula; Eq. (14) = oeted lassial fomula; Eq. () = hamoni expansion pansion of the indiet e et [Eq. ()] is not a good altenative, beause it laks loal details. In both test egions (A and B), the lassial fomula oeted with the hamoni expansion [Eq. (14)] agees bette with the stit fomula than the othe two methods, espeially when the ap size is ineased fom to 1. Tables 1 and show that the mean of di eenes between two methods deeases fom 1.01 to 0.84 m. This justi es ou belief that the best altenative fo the omputation of the indiet e et is the use of Eqs. (14) and (1). 6 Conlusions The indiet geoid e et is omposed of both loal e ets and long-wavelength ontibutions. This implies that most fomulas studied in this pape may have some numeial poblems in epesenting these di eent signals. Ou stit fomula, Eq. (17), as well as its appoximations, Eqs. (7) and (9), equie that the integation aea oves most of the globe to inlude the long wavelengths. Howeve, a pue set of spheial hamonis, Eq. (), tunated to, say, degee 60 will not ontain the loal details. Fo the test aeas A and B in Sweden, the long-wavelength ontibution fom the hamoni expansion is of about the same signi ane as the shot-wavelength signal of the loal integals. In the lassial fomula, Eq. (1), suh long-wavelength infomation, anging to 50 m, is missing even fo a global integation aea. We onlude that ou Eqs. (14) and (1) may be a suitable ompomise between the loal ontibution [epesented by the lassial fomula of Eq. (1)] and a set of spheial hamonis, Eq. (1), that is subtated to aount fo the long-wavelength signal. Finally, all the sufae integals above ould be slightly genealized fom a onstant to a lateally vaiable topogaphi density (l), simply by putting l inside the integals. Aknowledgement. The authos wish to thank D Huaan Fan, who assisted in omputing the hamoni oe ients H n and H n. efeenes Geophysial Exploation Tehnology (1995a) Global DTM5. Geophysial Exploation Tehnology (GETECH), Univesity of Leeds, Leeds, UK Geophysial Exploation Tehnology (1995b) DTM.5 of Euope. Geophysial Exploation Tehnology (GETECH), Univesity of Leeds, Leeds, UK Heiskanen WA, Moitz H (1967) Physial geodesy. Feeman, San Faniso Matine Z, VanõÂek P (1994) The indiet e et of topogaphy in the Stokes±Helmet tehnique fo a spheial appoximation of the geoid. Manus Geod 19: 1±19 Nahavandhi H, SjoÈ beg LE (1998) Teain oetion to powe H in gavimeti geoid detemination. J Geod 7: 14±15 SjoÈ beg LE (1994) On the teain e ets in geoid and quasigeoid deteminations using Helmet's seond ondensation method. ep 6, Depatment of Geodesy, oyal Institute of Tehnology, Stokholm SjoÈ beg LE (1995) On the quasigeoid to geoid sepaation. Manus Geod 0: 18±19 SjoÈ beg LE (1997) On the downwad ontinuation eo at the Eath's sufae and the geoid of satellite deived geopotential models. To appea in Bull Geol Si A VanõÂek P, Kleusbeg A (1987) The Canadian geoid ± Stokesian appoah. Manus Geod 1: 86±98 VanõÂek P, Matine Z (1994) The Stokes±Helmet sheme fo the evaluation of a peise geoid. Manus Geod 19: 119±18 Wang YM, app H (1990) Teain e ets in geoid undulation omputations Manus Geod 15: ±9 Wihienhaoen C (198) The indiet e ets on the omputation of geoid undulations. ep 6, Depatment of Geodesti Siene and Suveying, The Ohio State Univesity, Columbus