K B O I I P O C Y OB O I I P E ~ E Y I E H H H T O q H O C T H H H T E P I I O S I H P O -

B A H H , q B F P A B H M E T P H q E C K H X K A P T A X

VINCENC VYSKOCIL

Feoo6u~uuectcu~ uncmumym tlCAH, Hpaea*)

1. BBE~EHHE

B ~arino~ CTaTbe paccMaTpaBaeTca MeTO~HKa BS~rlc3lemln 3Mrlrlpa~ecI~HX 3Ha-

aeaa~ xapa~TepacTaK TOqaOCTrt rtaTeplIOOlnposaHaa B rpaBttMeTprlaeCKaX ~apTax.

HCXO~HSlMrI 3~!ecb KaK a B [2, 4, 5, 7 - -12] pa3HOCTa

(1) el = Agi.t; i- dgl

3HaqeHH~ aHOMa.rla~ CH3IBI TaXeCTrt, IIOJIyqeHHblX HHTeEtIO.rlHEOBaHHeM, H BBIHHC-

~IeHHBIX IIO Ha6~II-O,~eHH~IM B ,~I4cKpeTHBIX ToqKax Pv B HBBeCTHbIX cIIOCO6ax HX

o6pa6OTKH He OT~IeJDLrOTCA C ]IOCTaTOqHOH CTpOFOCTBtO B2IHSIHH~ OLI_IH~OK aHO-

Ma.IIH~ B ,~IHCKpeTHBIX l'oqKax OT BJIHffFIHH OI.I.IH~OK CaMO~ HHTeplIOJIffI~HH, KOTOpBIe

B CBOrO o~Iepe~l, 3aBHC~IT OT FyCTOTB1 pa3MeIL~eHFt~ I'paB!,IMeTpHqeCKHX IIyHKTOB HOT

CTeIIeHH ~I.HOMa.rIbHOCTH IIO.JIa aHOMa.rlH~ CH~IM TgmeCTI, i B ,~aHH0~ o6~IaCTIa. B ,/IaH-

HO~ pa6oTe Ilpe~3IO)KeI~S~t~ CrlOCO6 rICXO~HT rI3 3T~IX r l o a o m e m ~ rt IIO3BOOlJteT

yTO~HHTb MeTO~rlKy o6pa6OTKa 3HaqeHrt~ e~. TaK ~aK nacToamaa CTaT~,a aB3IaeTca

HenocpeacTBeam, tM npo~o3IxeHaeM [3], TO neT Heo6xo~rlMOCTrt CaOBa O6~,aCHaTr,

rlptaHgr~Ie o6o3uaqem~a ri qbopMyslbI.

2. METO~HKA OBPABOTKH 3HAHEHHITI e

3HaqeHrta sf noolyqalOTCa H3m B pe3yJIbTaTe yBesta~emm rtOln me y M e n s m e m l a

.FyCTOTbI paBMemeHrlJt rpaBrtMeTp~IqeCKrlX IIyHKTOB ~aHr~Oft KapTr~I. B llOC3ie~irleM

cay~ae rmTeprlonapytoT Ilpa 66;IsmeM cpe~HeM paccToaamI S Mex<~y I'paB~t-

MeTpHqeCKgMH rlyHKTaMrI H ffO3TOMy 3HaqeHHa ef OT.riaqatOTC~I OT Tex, KOTOpbte

HMetOT MeCTO Ilprl IlepBOHaqaolbHOM cpe~rleM paccTO~tHrta S 0.

3aaqeHaa e i CtIHTaIOTCff O63Ias xapaKTepOM ~2e~CTBI4TedlbHBIX OIIIH6OK

MI, I 6y~IeM npe~rtoaaraTb, aTO OmI B o6meM Moryr co~epxaTs Ka~ c2Iyaa~ttyro

qaCTb c~ TaK H CHCTeMaTHqecKyrO qaCTA a~. ~I2Ia BblqHCOIeHH~ 3MlXHpHqecKHx 3aaqe-

Hn~ cpeznea KBaapa'rn~tecI~o~ cny'mfiHOfi oturt6Ka m(II), cr~cTeMaTw~ecI<o~ c(II) r~ o6me~ M(II), xapa~Tepa3yrou_tax TO,mocrt, CaMO~ aHTepnonumm, )Ieo6xo~nMo

HMeTB C TOHKt,I 3peHHJ~ TOqHOCTH O~HOpO,~HyIO COBOKMIIHOCTB 3HaHeHH~ e~. 9TO

3Haq~T, qTO Bce S i ~IOSlmHBI 6BITB rlon3rqeHBI c O,~HHaKOBBtM BecoM, H qTO HX cay'~a~-

HSm qaCTI~ 6~ ~OnXHbt 6StTS BnOnHe He3aBrtCI, IMSI ~pyr OT ~pyra. I/Ix artpaopHs~e

Beca M~,~ He 3HaeM, O~HaKO MOXHO rtpe~nonaraTs, ~TO OHa 6y~yr pa~m,~ B cay~ae

*) A.~pec: Bo6ni II, Praha 4 -- Spofilov.

228 Studia geoph, et geod. 7 (1963)

K aonpocy 06 onpec)e.teuuu moqHocmu unmepno,tupoaauua a epaauu.,,tempu~tecKttx Kapmax

TaKHX To'teK P~, B KOTOpbIx nO.rlyqatoTC~ 14HTepHo~HpOBaHHble 3HaqeHHR dg~nt; ~

C O~HaroBof i cpenHe~ KBa~paTHqecKofi OmH6KO~ m(Ag~,,,). Tpe6oBaHne B3aHMHOH He3aBHCHM0CTH 6yaeT Bt~II'IOJIHJtTbCR .rlHl.I/lb B cay~ae,

Korea HHTepno~npOBaHnbte 3HaqeHHfl Agent; f 6y~yr noLtyqeHt,I no aHOMaYlHRM Ag

B pa3HbIX, npn BblqHC~eHH~X 3HaqeH/41~ e~ B O~HO~ a TO~ a<e COBOKynH0CTH He HO-

BTOp~ItOLI2HXC~I FpaBHMeTpHqeCKHX HyHxTaX.

I-Io3TOMy Heo6xo~HMO, qTO6bI nyHKTbI P~ Haxoj/HJIHCb .rlHl.IJb B Taxnx qburypax

HHTepHoYtaIIHH, KOTOpbte He HMe~OT O6LtIHX CTblKOB. HaH6onee tIenecoo6pa3HI, iM

c npaKTHqeCKo~ CTOpOHbI ~IBJIJleTC~I BbI6op Tpex rpaBHMeTpHqeCKHX IIyHKTOB JIe-

a<amHx Ha o~no~ npaMo~, npaqeM P~ naxo~an~CZ B cepe~n~e Me~Ay O6OnMr~ ~pa~-

HHMI4 HyH~TaM~, cpeAHee paCCTOaHHe rOTOpl,~X paBHO S.

170 3HaqennzM e~ B TO~ ~ e O~HOpO~nO~ c o ~ o r y n n o c r n o6pa3yeTca apHqbMeTn-

qec~oe cpe~r~ee n

(2) = _1 2 11 i=1

H cpe~Hee 3HaqeHI, le i.tx /(Ba~IpaTOB

1 " (3) E(e~i } =-ni=E~ e~ "

3na,~enHe E{e~} 4= 0, roTopoe ne ,,~oa~ez 6~,m, caencT~r~eu n a r o n a e m m cayqa~m,~x

OUIH6OK ~l IlpI~ HX I(OHeqHOM qHC~e //, MO~KHO CqHTRTb cpe)lHe~ CHCTeMaTHq~CKOH

omn6~o~ c(s). KpaTepun ~_~ Bs~aeaenaa cncreMarnqec~ax omn6o~ npaBeaeas[

B [1]. 3Haaenne ropHa ~/E{e~} npencTaBnaeT co6o~ o6 t t ty~ cpeznroro ~BaapaTn-

~ec~y~o omn6Ky M(e) pa3nocTe~ e~.

FlpHaexTeHnble pa3HOCYn

(4) 6, = e , - E{,,}

MO~KHO CtIHTaTb O63~a~IarotttHMn xapaKTepOM cny~a~H~,tx otm~6oK, O~HaXO neo6-

xo~m'~o ncc.ae~oBaT~,, ecna nx pacnpe~enemte a~naeTen nopMam, n~,IM [6, 13-1. Cpe~-

H ~ ~Ba~paTHqec~a~ c3~y~a~Ha~ omH6~a re(e) nony~aeTca n3 COOTHOmeHHa

, , t

tl ~ i=

3Mnnpnaec~oe 3Ha~enne m(Ag~,,) MOXnO noay~aT~, no qbopMyae

(6) mZ(Agi,,) = m2(e) - m2(Ag)

n cpennaolo ~na:Ipa'raaecKyro C.rlyqa~nyro oma6~y CaMO~ ~nTepno.r~ttm~ m(lI), ec:m re(I) nOCTOaHHO ~;tZ Bcex TO'~e~ Pi CO~o~ynHocrH, no qbopMyne

(7) m~(tl) = m Z ( e ) - m Z ( A g ) - m2(1).

HCC;Ie21OBaHneM c~c'reMaT~a,~ec~nx OtUH60~ C(e) MbI 3al~mvta'r~,ca 3,aecr~ ne 6yaeM.

OTMeTHM JIHnlb, qTO OHH B HeKOTOpO~ Mepe 3aBHC~tT OI; qi~cJIa 3HaqeHH~ e~ B s

Studia geoph, et geod. 7 [1963] 229

V, Vysko(il

�9 COBOKylIHOCTH HOT pa3MettteHn~ FIyHKTOB ~i" TaK KaI( MO>KHO r lpeanoJIaraTb, qTO

cny~ae 6nn3Krtx rpaBHMeTprlaeCKHX ny~KTOS CnCTeMaTHqeCKHe OInHOXH c(ae) aHoManafi npH6JIn3riTenbBO paBHbl, TO OHH B pa3HOCTaX g~ npaKTHqeCKH IHCqe3HyT,

rt 3~a~enHe c(e) rlpa~T~,tecxr~ paBno cpe~Hefi CHCTeMaTrt~eCKOfi Omn6Ke CaMO-~

HHzepnonara~ c(II). Hoxa rleT ~aHnblx, n o 3 s o n m o m a x cy~;HT~, O TOM, S KaKOfi

Mepe 3Ha,teHwa c(ll) 3asHcm OT cTerleHH at~oia.rtm~OCTVt non.~ aHoMa0ia~ cmrtst

T~x<ecTH H OT cpeaHero paccroaHHS~ FpaBHMeTpHqeCKHX nyHI~OB.

EcnH H3SeCTHO aMnHpH.ec~oe 3Ha.eH.e c(ll), TO MO~dm no [3, qbopM. (1.5)]

BI~ItlHCnHTb 3MnHpHqecKoe ~HaqeHHe o6mefi cpe~ne~ KBa/IpaTHNeCKO~ OILIH6KH

CaMO~ . n T e p n o n m m ~ M(H).

Q 3. CPABHEHHE C CYIIIECTBYtOLUHM METO~OM

B [2, 4, 5, 8] Bce pa3HOCTH e i ,onpe~eneHH~,/c npri H3BeC'THO~ cTeneHrt yMeHbme-

HH~I ryC'l'OTbI pa3MettleHH~I, BXO,/LqT B o,~ny COBOKyffHOCTb, ,/1.fir rOTOpO~ BblqRCnffeT-

Ca cpe•uee rBaapaTa,tecKoe aHaqeHrte x/E{s2}. HocnezHee rmCaT n a 3 a a m t e , ,non-

Hag OttIH6Ka nrlTepno,rt~IIHrI". 3Ha,leflrle E, BbtqHCneHHoe no qbopMyne

(8) E2 _-- E { d } - ,,,2(Ag)

rm3Ba~o ,,,~Hc'ro~ Om~6KO.~ IaHTepnonaL~mt". B [2, 8] ~ccne~oBa~acb 3aBrtCrtMOCTb

3naueHH~ E OT cpe,auHx paccroaHrl~ s coce~H~tX rpaBHMe'rpH,~ecKrix nymcroB n p a

paBHblX crenen~x yMeHbtUCHH~ ryCTOTbl pa3MCKlen~Lq. Ha 3THX HCcJIe~IOBaHHfi

BblTeKaeT, qTO qbyHKIIHIO E(S) MOXKHO FIpe~aCTaBHTb COOTHOLLIeHHeM

(9) IE(s)l = Ks k ,

no KOTOpOMy rIonyqHTbC~I 3HaqeHHe ,,KHCTO~ OIIlH6KH HHIepnoJI.qlLHH" ~JI~ nepBo-

HaqaJIsHOrO cpe~Hero paccTosHrta rpaBnMeTpHqeCKHX nyHKTOB. IlapaMeTpl,~ K, k

xapalcreprtaylOT cTeneHs aHoManbHoc'rrl noah auoMa_rtrl~ crl.rroi T~IXeCTrI B ~aHrIo~

o6naerH. ~ n a s < 25 ~c.vt MOXnO no [2, 8] npaKTH,~ecKn npnna-rr~ k = 1.

3HaqeHne E, Bbt,tr~cneHnoe no qbopMyne (8), cooTne-rc'rByeT npn6n r ixe H no cpez-

nea rBa~apaTaaecro~ om,6Ke m(Agi.,) a (6). H o n , o e co~na,~em~e 3neck, 6y, aeT

HMeTb Mec'I'O B cnyqae, KOr,/la BCe paaHoc-ra e~ HMelOT paBn~Ic Beca npr~ ortpeaesteH-

nO~ creneHn rycro 'n , t pa3r~ememm ,a ~or,aa E{e~} = 0. T a r ~aK ,,,~Hcraa OLUa6xa

nHTepnoaaunu" , onpe~eaeHna~t COOTnOtuearteM (8), co,aepxnT cpe~mo~o xBa~pa-

Th',mcrym OtUri6~y re(I), TO uenecoo6pa3Hee a~ecTrf a (9) qbyH~ttHrO m(II; s) BMeCTO

E(s); ~ynruna In(H; s)[ Bl,~paxaeT 3aaHCH~OCT~, aMnrtpHqecrrix 3Ha,~enrt~ m(ll), ebv~c~enm, tx no qbopM. (7) an~ cpeannx TO'~e~ P~ Ha paccToa'Hrtax s. ECnH rtaaecTno

~MnHpa.ec~oe 3Haqerllle K B n .He~aof i qbyH~a. , lm(ll; s)[, TO XOXHO onpezeaHT~

pa3aoxeHHe 3 .a .enHf i m(ll) B . . repnonatmOHHOM none rlo COOTHOmeH.SM (3.3)

n (3.7) H3 [3 3, B o6nleM MO~HO CKa3aTb, tlTO 3JIecb OngCaHH~,k~ MeTOB omnHqae'rca OT H3BeCr-

HOrO TeM, '~TO HMeeTCN B B~/Iy pa3HbI~ xapa~-rep cpe/IHHX KBaBpa'rHt~eCKHX OtI.~6OK

230 Studia geoph, et geod. 7 [1963)

K aOn pOCy O6 ort3~e~e,ietlttu moqttocmu u~tmepno/tupooalttt.q ~ epaau.~emputtec~ux Kapmax

HblX TOqKaX OT TOtlHOCTH CaMO~ HIITepIIOfl,qLIHH, TO MO)KHO 6o.rIee ~eTa.rlbHO IlCCJIe-

~OBaTb TOHHOCTb rpaBHMeTprlqeCKHX KapT.

4. 3AKJIIO HEHHE

B CTaThe n p e ~ t n o ~ e r m M e T O ~ g a BbltltlC.iIeHH~ 3Mnr~paaec rax 3HaqeHH~t xapaK're-

pnCTHg TOq8oc'rr~ nHTepnoa r~poBaarm B rpaBHMeTpHaec~ax ~apTax. )~aany~o KapTy

HeO6XO~HMO pa3~eJIaTb Ha O~J'IaCTH, 110 BO3MO}KHOCTH O~HOpOLIHble xax c TOqXa

3peH~n x a p a g T e p a nO~L'~ a H o M a n r ~ cnsu,~ TaXeCTrt (O ,~eM MOb, nO cy~HTr~ n o ryCTOTe

a qbopMe aBoartoMaJ~), TaK H C TO'~Kr~ 3peHna TOaHOCTH aaoMa~Tafi A# B ~Ir~cKpeTrab~X

"roqxax. C r l o c o 6 o M onrlcaHHblM B ,~acra 2 Br~l,~ncnarorca n n a r a ~ o f i O6JIaCTH 3Mna-

p r i a e c r n e 3 n a q e n n n m(l I ) npH p a 3 n ~ x p a c c T o ~ n n a x s r p a B n M e T p n q e c x a x nyHrTOa

H onpe~eznTr~Ca qbym<t~a~t Im(lI; s)l. C T p o r o OTaaqaroTca B. r tasaaa omr~6oK ar~o-

Ma.ai4fi B gHcKpeTHbXX TOqXax OT BJ~HaHH~ OtUrt6OX CaMO~ HnTep~oJLqI~nrI. 13oJ~ee

~eTa.a~,h'yro p a 3 p a 6 o T ~ y MO~KHO 6y~eT O~HaKO naT,, naLW, Ha OCHOnanr~n pe3yn~,-

TaTOB npaK-raqecx~x n c c a e ~ o n a m ~ .

l-loc-tym~o 13.2. 1963 Petlen3e,m: M. Burga

J7umepam ypa

[1] J. B S h m : Kriteria systematick~ch chyb. Geod. a kartogr, obzor, 7/49 (1961), 61. t2] M. B u r g a : Odvozeni chyby z interpolace Bouguerov#ch anom~tlii na ~zemi (~ech. Travaux

Inst. G6ophys. Acad. Tch~osl . Sci. No 148, Geofysik~ilni sbornik 1961, N(~SAV, Praha 1962. (3] V. V y s k o ~ i l : K BoIIpocy 06 OLteHKe TOqItOCTH rpaBHMe'rprtnec~mx KapT. Studia geoph, et

geod., 7 (1963), 134. [41 A. H. Bai~6opo~HH: I~ sonpocy O BblBo~e OUlrI6OK r~peacraBMTenJ, cTBa ~ HHTepnorlJIIIHM.

Tp. 12HHFA~K, Bbn~. 86 (1951), 52. I5] B. K. S a n a n a ~ 3 e : FpanHTaU~OHHOe zone n C"t'poeaHe 3eMHOI~ xopbt n 1-'py3H~. H3s AH

FCCP, T6HnHcn 1957. [6] B. ]I. Bonr~maxon: O5 otleHKe cveneHa npn6nH~eHHa nel~CTBHTe~noro paanpe~enenHa

OtUrI6OK X HopMam~HoMy. Tp. MHHFAHIK, B~m. 27 (1957), 53. [7] H. H. FpylIIttHCIr 0 6 yCTaHOBneHHH paL~HOHa~qZbHO~ ryCTOTb[ pa3MemeHH.q nyHKTOB

H HeOSXO~HMO~[ TOqHOCTH onpe~eneHtlJt aHOMa./IH~ CHill3[ T~DKeCTR llpH rpaBllMeTpOBbIX

cT, eMXaX. Pa3B. ri npoM. reod~a3axa, Brim. 15 (1956), 40. [8] H. H. Fpyma~cKn~t : O6 omrtSKax rmTepnon~an n npeacTaB,Te~crBa aeTan~H~,tx rpa~rt-

MeTpaaecKax e~,eMox, TO~OCT~ KapT a t~enecoo6pa3noM pa3Memerma myacroB. AeTpOHOM. ~zypHaa, XXX (1957), 267.

[9] A. K. ManOBH,~XO: OcaonaoR ~ypc rpaBHpa3~e~x~, 1. aaeT~,. 1-IepM~, 1960, 78--84. [10] M. C. MOaOZ(eHCK~R: OcHO~Hr, te BOnpOC~t reoae3a,~ecxoit rpa~nMeTpm~. Tp. I2HHFAr~K

B~ra. 12 (1945). [11] M. C. Mo.rlo~er~cKail, B. ~ . EpeMeeB, M. I4. IOpxaHa: Meron~,z rl3y,~enna eHemHero

rpa~trattaoHHoro non.~ a d~,rypbt 3e.~c~m. Tp. I2HHHFAaK, sstr~. 131 (1960). [12] H. B. C a x a a a : l'IpaMep paaaoaa.ra, Horo m, xSopa ceaeaaa rl3oai-iOMZ,'i ~ rpaBnMerpa-

aecxnx gapT. Pa3s. n rrpoM, reo~Haa~a, B~,m. 26 (1958), 40. [13] A. C. q e 6 o T a p e s : Cnoco6 nal~MeabmHX gsar~paToB c O~CHO~tr, m Teopr~a Bepoaxnoc're~.

Feo~e~a3~aT, MOCKea 1958.

Stud[a geoph, et geod. 7 [1963] 231

1I. Vysko~il

S u m m a r y

O N T H E D E T E R M I N A T I O N O F T H E A C C U R A C Y

O F I N T E R P O L A T I O N I N G R A V I T Y M A P S

VINCENC VYSKO(~IL

Geophysical Institute, CzechosL Acad. Sci., Prague

A method for the practical calculation of mean square errors, characterizing the accuracy of interpolation in gravity maps, is discussed in this paper. The differences e be tween the inter- polated values of anomalies and those calculated directly from gravity measurements are con- sidered as starting quantities. The influence of the errors of anomalies in the observa t ion points is separated from the error of interpolation itself.

Received 13.2. 1963

232 Studia geoph, et geod. 7 {1963}