Gravity gradient tensor, vertical pyramid model, right rectangular prism, parabolic density function. 2. Theory

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1 F gavit gaient tenso of a vetia pai oe of fat top & botto wit ept-wise paaboi ensit vaiation Anan P. Goka an abata G. Sast Depatent of Eat Sienes Inian Institte of Tenoog ookee Eai ID: gssast@gai.o / gssfes@iit.a.in. Kewos Gavit gaient tenso vetia pai oe igt etanga pis paaboi ensit fntion. Sa Gavit gaient tenso an tei iffeent tpes of obination ae ve sef to eineate te eges of te soe bo. It enanes te ig feqen gavit anoaies an otines te sape of te anoaos bo. F gavit gaient tenso an tei intepetation fo igt etanga pis an poea bo ae pesent in geopsia iteate. Hee we popose a vetia pai oe wit paaboi ensit fntion an tei gavit gaient tenso. Tese gavit gaient tensos epessions ae vaiate against tat of igt etanga pis oe. We ave ine two snteti eapes - A seti pai oe an igt etanga pis oe (ae b two pai oes) an tie to eineate te eges of te soe bo. Tese peiina ests ae ve enoaging an an be appie fo eineation of seienta basins in gete etai in fte.. Intotion Te se of gavit gaient ata is oe oon in epoation an is wie se to eineate te otine of te soe bo. Gavit gaient tensos ae pesent in geopsia iteate fo igt etanga pis wit onstant ensit as we as inea ensit vaiations. Copete gavit gaient tenso an aso eive fo te vetia oponent of te gavit sing Fast Foie tansfo teniqes (Miks an Hinojosa ). Diffeent tpes of ensit fntion ike eponentia (Cai an Hine 988) paaboi (ao et a. 99; Cakavati et a. ) an peboi ensit fntions (ao et a. 995) ave been se in seienta basin oeing. Hansen (999) Hostein () an Haan et a. (9) ave se poea oe in gavit fowa oeing wit a inea ensit vaiation ept-wise. Gavit gaient tensos ae sef to estiate te soe oation an ege etetion of soe bo (Bake 986; Zeng ; Bosetti 5; Beiki ; Beiki an Peesen Zo 5). Saa (6) eonstates te ope patten of te anoaies of sat oes stating fo te gavitationa potentia to te f gavit gaient tensos an tei obinations. Te etaie st of singaities enontee in gavit fowa oeing of iffeent oes as been aesse b Okabe (979) Poanka (988)Kwok (99) Petović (996) Tsois () Tsois an Petović () Hostein () an D so (). Staostenko(978) as popose an inoogeneos vetia pai oe of fat top an botto wit inea ensit vaiation ept-wise. Bt e was nabe to eive te opete anatia epessions fo its f gavit gaient tenso. We eive te opete anatia epessions of f gavit gaient tensos of te pai oe bt not ine e to page iitation. We ave potte a gavit gaient tensos an tei iffeent gavit gaient obinations to eineate te eges of te soe bo (snteti pai oe).. Teo Consie an isoate ega pai oe wit eptwise paaboi ensit ontast ABCDEFGH wit fat top ABCD an botto sfae EFGH (Fig. ). Te potentia effet of s a oe at an abita point ( ) in fee-spae is given b () t Biennia Intenationa Confeene & Eposition

2 F gavit gaient tenso of Pai oe wee paaboi ensit ontast ρ (ζ) an iit of vaiabes ξ an η ae given b. an () wee γ is te nivesa gavitationa onstant ρ is te ensit ontast obseve at te gon sfae in g/ α is a paaboi ensit fntion in g/ /k an ae te ept of te top an botto sfae of te pai an ζ efes to ept beow. Te vae of α an be obtaine b fitting paaboi ensit fntion (fist one of eqation ) to te known ensit ontastept ata of seienta oks (Cakavati et a. ). A(ξ η ) B(ξ η ) C(ξ η ) D(ξ η )E(ξ η ) F(ξ η ) G(ξ η ) an H(ξ η ) ae ones of te pai (Fig. ). B anging te vaiabes on te igt an sie (HS) in eqations () an () an we get () Wee. an () We an ewite te paaboi ensit fntion ρ ( ) an te iits an of eqation () as. an (5) wee an (6) A te gavit gaient oponents ae opte b taking a te possibe obe eivatives of te potentia (eqation ) in an -ietion. So tee ae tota nine gavit gaient tensos ot of wi five ae inepenent ( an ) an fo ae epenent ( an ) gavit gaient tensos. Epessions fo a inepenent gavit gaients in intega fo ae given beow: (7) (8) (9) () () Te epenent gavit tensos an be obtaine b te seti aateistis of te seon eivatives ( = = an = ) an sing te Lapae eqation ( = - - ). Fige () sows a -D pai oe an its geoet wit a ept-wise paaboi ensit vaiation. Fige () sows te opete gavit gaient tenso pots of vetia pai aong wit five inepenent ( an ) an fo epenent ( an ) gavit gaient tenso. an ae potte t Biennia Intenationa Confeene & Eposition

3 F gavit gaient tenso of Pai oe b seti aateistis of te gavit gaient tenso ( = = an = ). A seon vetia eivative is eive fo Lapae eqation (= - -). Diffeent obinations of gavit gaient oponents ae se in gavit gaient intepetation to sipif tei ope patten (Saa 6). Diffeentia vate o oionta ietive tenen (HDT) of te pai oe wit paaboi ensit fntion is opte b te obination of an. Diffeentia vate agnite (DCM) = / () Hoionta an tota gaient agnite of of pai oe is te obination of an wi ae given beow: Hoionta gaient agnite of (HGM) = / () Tota gaient agnite of (TGM) = / () A tese obinations geat enane te effet of te saowe soes.. ests an isssion Fige sows o -D vetia pai oe wit ept-wise paaboi ensit vaiation. We ave sown pots of a gavit gaient tensos (Fig. ) an tei obine effets (Fig. ) fo a pai oe wit paaboi ensit fntion wi seves as fist snteti eape. Wite ase an te bak soi ines iniate te top an botto sfae of te pai oe espetive (Fig. ). Gavit gaient tensos (Fig. a) fo igt etanga pis (ae b two pai oe soi bak ine an ase wite ine sow te fist an seon pai oe) seve as a seon snteti eape fo te pai oe wit onstant ensit wi is aso se fo vaiation ppose.. Vaiation To vaiate o gavit gaient tensos fo a pai oe we ave onsiee gavit gaient tensos of a singe vetia pis (Fig. a) of onstant ensit (Nag et a. ) in te absene of optabe gavit fo vetia pis oe wit paaboi ensit fntion. We ave onvete paaboi ensit fntion fo te pai oe into onstant ensit b onsieing α =. We ave onstitte a singe vetia pis eqivaent to Nag et a. () b te obination of te two pai oe wit onstant ensit (Fig. b). Aoing Fig. a oespons to te gavit gaient tensos of a vetia pis oe of onstant ensit (Nag et a. ) wie Fige b oespons to tat of o onstitte vetia pis oe (b te obination of two pai oe wit onstant ensit sown b soi bak an ase wite ine espetive in Fig. b). O oe esponse ates we (MS eo of 7.* - ga) wit tat of a vetia pis oe.. Disssion Te opete gavit gaient tenso an tei anoa esponse pots (Fig. an ) aong wit iffeent obinations of gavit gaients (Fig. ) ae se to eineate te eges of a -D pai oe wit paaboi ensit fntion an igt etanga pis (ae b o pai oes) wit onstant ensit. Te wite ase ine sows te ppe etanga sfae an bak soi ine sow te botto etanga sfae of te vetia pai oe (Fig. ). pot (Fig. ) sows te positive-negative-positive tipet patten wit stong negative at te ente. Te eo ontos efine te west-east eges of te ppe etanga sfae an te positive peaks on bot sies of te negative peak efine te west-east eges of te owe etanga sfae of te pai oe aong te -ais. Siia te pot (Fig. ) efines te sot-not eges of te ppe an te owe etanga sfae of te vetia pai oe aong te -ais. Bot te pots an (Fig. ) eibit te negativepositive-negative-positive qapet patten wi eat efine te ppe fo ones of te vetia pai oe bt not efine te botto ones. Pots of an (Fig. ) ae sae e to te seti aateistis of te gavit gaient tenso an siia fo an (Fig. a). an (Fig. ) efine te west-east an sot-not eges of te ppe etanga sfae of te pai aong te -ais an -ais espetive bt o not give te infoation abot te owe eges (Fig. ). Te ppe etanga sfae infee b an ae siia to tat of an (Fig. ). is opte via an b sing te Lapae eqation in soe-fee egion. an be se to efine te ente of te pai oe. Fige aso sows te anoa pot of iffeentia vate agnite wi oet eineates te eges of te ppe etanga sfae bt not eat t Biennia Intenationa Confeene & Eposition

4 F gavit gaient tenso of Pai oe eineate te botto etanga sfae of te vetia pai oe. A of te fo ones of te ppe etanga sfae pass tog te peak points of te anoa. Hoionta an tota gavit gaients agnite (Fig. ) ae opte b te obination of te an an ain se as an ege eteto of te anoaos soe. Bot oionta an tota gavit gaients eineate te otine of te ppe etanga sfae of te vetia pai oe (eges ae passing toog te peak vaes) bt fai to eineate botto sfae of te pai oe. Siia poee is appie to etet te eges of igt etanga pis oe. Te eges of te botto etanga sfae of te igt etanga pis ae te sae as te eges of te ppe etanga sfae so tee ae no iffities to etet te botto eges of te igt etanga pis oe. Consions Hee we ave netaken a vetia pai oe wit paaboi ensit fntion an eive ose fo epession of a te gavit gaient tenso qantities bt e to page iitation we ave sown epessions on in intega fo. A te gavit gaient pots (Fig. ) of vetia pai sow te ope patten (singe obet tipet an qapet) eept (onopoa). We ave eineate te eges of te ppe etanga sfae of te vetia pai oe wit te ep of iffeent gavit gaient pots bease of te enaneent of te saow sfae feates. Sigt positive peaks of te an (Fig. ) give te infoation abot eges of te etanga botto sfae of te vetia pai. Diffeent obinations of te gavit gaients ike iffeentia vate agnite oionta gaient agnite an tota gaient agnite of (Fig. ) epasie te effet of te saow soes an an be se as an ege eteto of te anoaos soe ike o pai oe. O gavit gaients pots fo igt etanga pis (ae b two pai oes) at we wit tat of igt etanga pis oe (Nag et a. ) at vaiation stage. So it is eas tootine te ppe etanga sfae of te vetia pai oe an tat of igt etanga pis oe bt it is ve iffit to eineate te botto sfae. Aknowegeents Te fist ato gatef aknoweges te finania ep eeive fo MHD Govenent of Inia Dei. Fige.A seati -D pai oe an its geoet wit ept-wise paaboi ensit vaiation. Fige. Te f gavit gaient tenso an tei iffeent obinations of pai oe wit paaboi ensit fntion. Te paaetes of te pai oe fo a ases ae as foows: ξ = ξ = ξ =8 ξ =6 η = η = η =8 η =6 Δρ = -.56 g/ α =.g/ /k =.5 = 5 an =. A engt paaetes an station istanes ae epesse in k. t Biennia Intenationa Confeene & Eposition

5 F gavit gaient tenso of Pai oe Banes G. an J. Le () Poessing gavit gaient ata Geopsis 76 I I7 oi:.9/.5858 Beiki M. () Anati signas of gavit gaient tenso an tei appiation to estiate soe oation Geopsis 75 I59 I7 oi:.9/.969. Beiki M. an L. B. Peesen () Eigenveto anasis of gavit gaient tenso to oate geoogi boies Geopsis 75 I7 I9 oi:.9/.898 Bake. J. an. W. Sipson (986) Appoiating eges of soe boies fo agneti o gavit anoaies Geopsis Ekat E. A. (9) A bief isto of te gavit eto of pospeting fo oi Geopsis 5. Esag M. an L. E. Sjobeg (9) Atospei effets on sateite gavit gaioet ata Jona of Geonais 9 9 oi:.6/j.jog MiksK. L. an J. H. Hinojosa () Te opete gavit gaient tenso eive fo te vetia oponent of gavit: a Foie tansfo teniqe Jona of Appie Geopsis Fige. Vaiation of o gavit gaient tensos fo pai oe wit tat of igt etanga pis oe. a) Gavit gaient tensos fo igt etanga pis (Nag et a. ). Te paaetes of te igt etanga pis oe fo a ases ae as foows: = =5 = =5 ρ = -.56 g/ =.5 an = 5. A engt paaetes an station istanes ae epesse in k. b) Gavit gaient tensos fo igt etanga pis ae b te obination of te two pai oes. Te paaetes of te pai oe fo a ases ae as foows: ξ = ξ =5 ξ =8 ξ =8 η = η =5 η =8 η =8 Δρ = -.56 g/ α = g/ /k =.5 = 5 an =. A engt paaetes an station istanes ae epesse in k. MikaiovV. G. Pajot M.Diaent an A. Pie (7) Tenso eonvotion: A eto to oate eqivaent soes fo f tenso gavit ata Geopsis 7 I6 I69 oi:.9/.797. Peesen L. B. an T. M. asssen (99) Te gaient tenso of potentia fie anoaies:soe ipiations on ata oetion an ata poessing of aps Geopsis Saa A.H. (6) nestaning gavit gaients a ttoia SaaGeoConsting ion Teas SA Te Leaing Ege Yan Y. an Q. Y () Ege Detetion in Potentia-Fie Gaient Tenso Data b se of Ipove Hoionta Anatia Signa Metos Pe App. Geopsis oi:.7/s--88- Seete efeenes t Biennia Intenationa Confeene & Eposition

6 F gavit gaient tenso of Pai oe t Biennia Intenationa Confeene & Eposition