Gravity　anomalies　and　the　corresponding　subterranean　mass　distribution,with　special　reference　to　the　Nobi　plain　and　its　vicinity,Japan

ABSTRACT

　A　theoretical　formula　for　calculating　subterranean　mass　distribution　from　the　gravity　values　at　the　earth’s　surface　is　presented.A　formula　is　approximately　expressed　by　a　linear　combination　of　gravity　value　and　its　first　and　second　vertical　derivatives　at　the　earth’s　surface.The　coefficients　of　this　linear　equation　were　determined　by　the　best　response　of　the　Fourier　components.The　numerical　coefficients　are　also　presented　under　the　consideration　of　the　grid　spacing　or　the　wave　length　and　the　depth　for　calculation.The　evaluation　of　the　vertical　derivatives　involved　was　examined　by　a　characteristic　equation　connected　with　wave　length,with　special　reference　to　the　formulas　given　by　Elkins,Henderson　and　Zietz,and　Kato.The　subterranean　mass　distribution　is　determined　by　the　present　formula　within　about　ten　percent　error.
　The　present　calculation　method　was　applied　to　the　gravity　values　which　were　measured　in　the　Ndbi　plain　and　its　vicinity　in　south　central　Honshu,Japan.The　gravity　values　were　determined　at　539points　in　this　district　by　means　of　a　North　American　gravimeter.The　Bouguer　anomalies　were　determined　and　the　subterranean　mass　distribution　for　average　depth　of　1km　was　obtained.The　Bouguer　anomaly　is　negative　in　the　greater　part　of　this　district,and　its　negative　values　increase　as　one　moves　westward.The　greatest　negative　anomaly　zone　is　located　at　the　east　side　of　the　Yoro　mountains,suggesting　a　geosynclinal　underground　structure　where　thick　young　formations　have　developed.
　There　is　a　positive　gravity　anomaly　in　the　southern　area　of　this　district,suggesting　that　the　young　formations　lying　in　this　area　thin　out　southwards　in　connection　with　the　geologic　tectonic　structure.Two　main　basins,Nobi　and　Mikawa　and　some　fault-like　structures　were　interpreted　in　this　district.

INTRODUCTION

　The　methods　for　determining　a　subterranean　mass　distribution　from　gravity　values　at　the　earth’s　surface　have　been　studied　by　a　number　of　investigators　[2,5,11,12].The　Fourier　series　method　[12]　for　interpreting　gravity　anomaly,particularly,has　opened　a　possibility　to　calculate　directly　the　underground　mass　distribution　from　observed　gravity　anomalies.We　therefore　used　this　method　very　much　in　the　past.The　computation　of　the　Fourier　coefficients　entails,however,much　hard　work　and　is　troublesome:In　order　to　reduce　this　complexity,we　tried　theoretically　to　find　a　computing　formula　[7]　which　approximately　represents　the　true　subterranean　mass　distribution　responsible　for　the　gravity　anomaly.This　was　determined　from　a　linear　combination　of　gravity　value　and　its　first　and　second　vertical　gradients　at　the　earth’s　surface.Since,in　this　case,it　is　a　question　of　how　reasonable　values　for　the　first　and　the　second　vertical　derivatives　are　given,a　characteristic　equation　determining　their　evaluations　was　established,and　the　degree　of　approximation　in　the　computation　formula　was　also　examined.
　Thus,one　part　of　the　present　paper　deals　with　the　establishment　of　the　computation　formula　for　subterranean　mass　distribution　responsible　for　gravity　values.The　other　part　deals　with　the　gravity　measurements　in　the　Nobi（濃尾）plain　and　its　vicinity　and　also　with　the　application　of　the　present　method　to　its　gravity　data.Using　a　North　American　gravimeter　we　commenced　gravity　measurements　in　Nagoya　city　and　the　Chita（知多）peninsula　in　January,1955.In　addition,during　April,1956　and　in　March　and　April,1957　the　gravity　survey　was　carried　out　over　the　whole　area　of　the　Nobi　plain　adjacent　to　Nagoya　city.

THEORY　FOR　CALCULATION　OF　SUBTERRANEAN　MASS　DISTRIBUTION　FROM　GRAVITY　ANOMALIES

　We　take　x-　and　y-axis　at　the　earth’s　surface　and　z-axis　vertically　downward　as　a　coordinate.Assuming　that　the　gravity　anomaly　△g　at　the　earth’s　surface　is　expansible　into　a　double　Fourier　series　according　to　the　Tsuboi’s　method　[12],we　can　write

　【数式】（1）

If　we　assume　that　the　subterranean　density　distribution　△p　that　is　responsible　for　this　gravity　anomaly　△g　is　condensed　at　a　depth　d　from　the　earth’s　surface,we　can　take　the　form

　【数式】（2）

in　which　r^2　is　the　universal　constant　of　gravitation.The　i-th　vertical　derivative　of　gravity　at　the　earth’s　surface　is　generally　obtained　from　a　given　distribution　of　△g　such　as

　【数式】（3）

On　the　other　hand,exp√（m^2　+　n^2d）is　expansible　into　the　following　series　for　its　finite　values:

　【数式】（4）

Particularly,the　convergency　of　this　series　is　good　for　the　condition　of　√（m^2　+　n^2d）＜1.From（2）,（3）,and（4）,we　get

　【数式】（5）^3

Now,let　the　terms　on　the　right-hand　side　of（4）equal　approximately　to　an　equation　of　second　degree　in　d.That　is,we　can　write　in　general　such　that

　【数式】（6）

in　which　a,b,c　are　constants.（5）is　therefore　expressed　by

　【数式】（7）

Since（6）does　not　hold　for　all　positive　values　of　t,it　must　be　determined　by　the　way　that　the　approximation　of（6）would　be　the　best　for　t　in　the　ranges　0＜t＜k.Using　the　method　of　least　squares,a,b,and　c　are　obtained　as　the　roots　of　the　following　equations:

　^3（5）can　be　obtained　also　by　another　method.If　the　gravity　anomaly　at　a　depth　d　from　the　earth’s　surface　is　assumed　to　be　△g_d（x,y）,the　corresponding　surface　density　is
given　by　

　【数式・1】

Since　△g_d（x,y）is　an　analytical　function,it　is　possible　to　compute　continuously　the　vertical　derivatives.Assuming　that　△g（x,y）is　the　gravity　anomaly　at　the　earth’s　surface,△g_d（x,y）can　be　directly　calculated　by　means　of　the　Naclaurin　expansion.Therefore,

　【数式・2】

　【数式】（8）

The　values　of　a,b,and　c　for　each　of　several　values　of　k　thus　obtained　are　given　in　Table1.These　results　are　shown　in　Fig.1.Putting　D＝a+bt+ct^2−e^t,the　values　of　D　for　various　values　of　a,b,and　c　are　shown　in　Fig.2　from　which　it　can　be　found　that　the　good　approximation　of（6）could　be　obtained　when　k　would　be　nearly　equivalent　to　unity,namely　the　range　of　t　would　be　0＜t＜1.
　To　determine　the　underground　density　distribution　responsible　for　components　larger　than　the　wave　length　λ　from　the　gravity　values　at　the　earth’s　surface　which　are　read　at　the　grid　spacing　s,it　is　desirable　to　get　an　approxi-mate　value　with　respect　to　m　and　n　that　satisfy　the　condition　such　that

　　　√（m^2+n^2）＜2π/λ,　（9）

because　of　the　wave　length　of【数式・3】　being　equal　to　2π/√（m^2+n^2）.Therefore,k　that　gives　a　range　of　t（＝　√（m^2+n^2d）can　be　given　by　the　form　of

　　　k＝2πd/λ,　（10）

（7）gives　some　values　for　the　components　which　are　smaller　than　the　wave　length　λ,but　it　is　out　of　question　owing　to　the　following　condition:

　　　[e^t]_t>k　＞　[a+bt+ct^2]_t.　（11）

Consequently,if　the　ratio　of　d　to　A　is　given,the　values　of　a,b,and　c,which　are　to　be　adopted,are　inevitably　determined.Since　the　density　distribution　can　not　be　obtained　for　the　components　which　are　smaller　than　two　times　a　grid　spacing,the　grid　spacing　can　　not　be　enlarged　too　much　from　the　relation　λ＞2s.As　seen　in　Fig.2,when　k　equals　0.5.the　error　in　the　approximation　in　question　can　be　controled　so　as　to　be　less　than　10　per　cent.Taking　the　unit　grid　spacing　as　a　unit　and　using　the　values　of　a,b,and　c　in　Table1　in　the　case　of　k＝0.5,from（7）we　can　now　get　approximately　the　equation　such　that

　△ρ（x,y）＝1/2πr^2{1.001△g（x,y）+0.9702△g_z（x,y）+0.643△g_zz（x,y）}.（12）

　Now　the　first　and　second　vertical　gradients　of　gravity　are　generally　expressed　by

　【数式】（13）

where　g（s）,g　（√2s）,etc.are　the　averages　of　the　gravity　on　rings　of　radii　s,√2s,etc.respectively,and　g（0）the　gravity　value　at　the　observation　point　where　the　derivative　is　to　be　sought.
　Then（7）becomes

　【数式】（14）

The　density　distribution　△ρ（x,y）can,therefore,be　replaced　by　the　relief　△h（x,y）of　a　boundary　surface　on　which　the　subterranean　mass　is　distributed,that　is

　【数式】（15）

where　ρ_1,ρ_2　are　the　densities　in　the　upper　and　lower　media　of　the　boundary　surface　respectively.
　For　instance,taking　ρ_2　−ρ_1＝0.4,d/s＝1,k＝0.5（λ＝4πkm）and　using　Kato’s　equation　[8]　such　as

　【数式】（16）

（15）then　becomes

　【数式】（17）

where　g（0）is　the　observed　gravity　value　at　the　center　of　a　ring　on　which　the　average　of　observed　gravity　values　is　taken,【数式・4】the　sum　of　four　gravity　values　each　on　the　rings　of　radii　s　and【数式・5】the　sum　of　the　eight　gravity　values　on　the　ring　of　radius　√5　s.

COMPUTATION　OF　FIRST　AND　SECOND　VERTICAL　DERIVATIVES　OF　GRAVITY

　The　various　methods　of　treating　the　first　and　second　vertical　derivatives　of　gravity　have　been　presented　[1,3,6,8,10,13].The　accuracy　of　the　approxi-mation　depends　not　only　on　the　number　of　rings,but　also　on　the　number　of　points　read　in　each　ring.The　error　induced　here　may　be　reduced　by　increasing　the　number　of　rings,so　that　the　computing　equation　with　a　grid　of　three　rings　of　radii—s,√2s,√5s—around　the　origin　for　observation　point　is　generally　used.Whatever　computing　equations　may　be　taken,the　derivatives　for　all　the　wave　lengths　can　not　be　obtained　exactly　as　they　are　but　they　have　filtering　characteristics　that　can　be　compared　to　that　of　an　electric　filter.As　a　consequence,4g_z,or　△g_zz　might　not　be　given　just　as　it　would　be,but　more　distortions　might　be　induced　in　it.These　kinds　of　influences　on　the　derivatives　have　been　discussed　by　Elkins　[4]　and　Nettleton　[9]　in　detail.Their　discussions　centered　on　the　problem　of　how　a　small　structure,such　as　a　sphere　or　a　cylinder,could　be　detected,but　not　on　the　exact　truth　of　derivatives.Therefore,we　tried　to　understand　the　characteristics　of　vertical　derivatives　by　developing　an　evaluation　method,which　will　be　given　in　the　next　section.

GENERAL　DTSCUSSION　ON　THE　CHARACTERISTICS　OF　COMPUTING　EQUATION

　Since　the　given　gravity　is　an　analytical　function,if　g（x,y）is　written　as

　【数式】（18）

the　function　of　the　angular　wave　number　ω,φ　is　expressed　as

　【数式】（19）

Since　the　equation　calculating　the　vertical　gradient　is　generally　expressed　by　a　linear　combination　of　first　degree,noting　it　as　M（x,y）,we　get

　　　M（x,y）＝　Σc_jg（x+_jh_x+y+_jh_y）,　（20）

where　c_j　is　a　constant　and　its　suffix　j　represents　a　number　which　gives　each　point　relative　to　the　point（x,y）._jh_x,and　j_h_y,are　the　x-　and　y-component　of　distance　from　the　point　designated　by　j　to　the　point（x,y）respectively.From（17）and（19）,we　have

【数式】（21）

From　extension　of（3）we　get

【数式】（22）

and　if　M（x,y）is　an　approximate　equation　which　represents　the　n-th　vertical　gradient　of　g（x,y）,combining（20）with（21）and　making

【数式】（23）

|F（ω,φ）|　can　be　considered　as　a　characteristic　equation　of　the　n-th　vertical　gradient　of　gravity　with　the　component　of　wave　number（ω,φ）.Arg　|F（ω,φ）|　represents　the　phase　difference　of　the　component,and　c_j　is　taken　so　as　to　be　symmetrical　with　respect　to　the　point（x,y）.Thus　we　get

　　　Arg　|F（ω,φ）|＝0,　（24）

If　F（ω,φ）＝1,the　characteristic　equation　is　perfectly　satisfied,so　that　the　vertical　gradient　represents　the　actual　value　itself.If　F（ω,φ）＜1,the　vertical　gradient　becomes　smaller　than　the　actual　value,and　if　F（ω,φ）＞1,the　vertical　gradient　becomes　larger　than　the　actual　value.Therefore,if　the　following　equation　could　be　calculated:

【数式】（25）

（25）would　give　an　evaluation　of　the　computing　equation;but　as　a　matter　of　fact,it　is　difficult　to　carry　out　the　calculation　of（25）completely.Furthermore,all　the　values　of　ω,φ　are　not　necessary,so　that　it　would　be　better　to　show　graphically　F（ω,φ）from（23）.

NUMERICAL　EXAMPLES　OF　CHARACTERISTICS　OF　VERTICAL　GRADIENT

The　second　vertical　gradient
　Since　the　second　vertical　derivative　formula　by　Elkins　[3]　is　given　by

　【数式】（26）

the　characteristic　equation　|F（ω,φ）|　for　this　case　is　given　by

　【数式】（27）

in　which　s　is　taken　as　a　unit.Now　λ_x＝2π/ω　and　λ_y　＝　2π/φ,then　ω,φ　in（27）can　be　replaced　by　λ_x,λ_y,respectively.The　contour　lines　of　F（λ_r,λ_y）are　given　in　Fig.3　where　the　actual　wave　length　is　given　by　λ＝2π√（ω^2+φ^2）.As　seen　in　Fig.3,it　is　found　that　the　calculated　value　is　smaller　than　the　actual　value　and　F（ω,φ）never　exceeds　unity.Consequently,to　control　the　error　within　the　ranges　of　30　per　cent,it　is　necessary　to　take　the　wave　length　λ　over　6.However,the　Elkins　formula　has　an　advantage　of　being　indifferent　to　the　direction　of　grid　spacing　and　of　giving　approximately　the　same　magnification　for　all　directions　of　grid　spacing,provided　λ　is　equal　to　constant　for　all　values　of　λ_x,λ_y.This　formula　also　has　the　advantage　of　neglecting　a　structure　having　a　small　wave　length.
　The　characteristic　equation　for　the　Kato’s　formula　of（16）,taking　s　as　a　unit,is　given　as

　【数式】（28）

F（λ_x,λ_y）is　graphically　given　in　Fig.4　where　ω,φ　are　replaced　by　λ_x,λ_y　respectively.As　shown　in　Fig.4,the　effect　on　the　direction　of　grid　spacing　is　great.For　instance,for　the　case　of　the　wave　length　equal　to　two　and　x-　or　y-axis　would　agree　with　the　direction　of　the　wave,F　is　approximately　equal　to　zero,while　if　each　of　the　two　axes　make　the　same　angle　with　the　direction　of　the　wave,the　amplitude　becomes　larger　than　two　times　the　actual　value,as　seen　in　Fig.4.If　there　is　a　wave　length　of　the　order　of　2,as　with　a　noise　that　is　a　small　negligible　gravity　anomaly,special　attention　must　be　paid.In　order　to　control　the　errors　within　30　per　cent　for　this　case,the　wave　length　must　be　λ＞　5～6.
　The　characteristic　equation　for　the　Henderson　and　Zietz’s　formula　is　now　required.Since　the　Henderson　and　Zietz’s　formula　[6]　is　of　the　form

△g_zz　＝1/3[21g（0）−　32g（s）＋12g（√2s）−g（2s）],　　（29）

the　characteristic　equation　is　given　by

【数式】（30）

Expressing　F　by　λ,and　λ_y　as　before,we　get　the　result　given　in　Fig.5.If　the　maximum　error　of　30　per　cent　is　granted,the　wave　length　must　be　λ＞　4,The　components　for　λ＞8　seem　to　be　equal　to　the　actual　values.
The　first　vertiral　gradient
　The　characteristic　equation　for　the　first　vertical　gradient　is　given　as

　【数式】（31）

In　this　case,if　we　take　the　grid　spacing　similar　to　the　second　vertical　gradient,（31）becomes

　【数式】（32）

in　which　x　＝　cosω　＋　cosφ,　y　＝　cosφcosω.
　If　we　use　the　Kato’s　formula　of（16）,the　characteristic　equation　becomes

　【数式】（33）

Therefore,in　the　same　way　as　before,expressing　F（ω,φ）by　λ_x　and　λ_y,we　get　the　result　as　shown　in　Fig.6.If　the　maximum　error　of　30　per　cent　is　admitted　the　wave　length　must　be　in　the　ranges　of　3　to　10.Mostly,the　com-ponents　for　λ＞10　seem　to　be　small　and　the　vertical　gradients　themselves　are　also　small,so　that　the　Kato’s　formula　is　found　to　be　available　for　λ＞3.In　this　case,as　seen　in　Fig.5,the　direction　of　grid　spacing　for　the　wave　of　λ＝2　has　a　large　influence.
　The　characteristic　equation　for（17）,representing　the　computation　formula　of　underground　structure　given　by　the　authors,is　expressed　as

　【数式】（34）

（34）is　graphically　shown　in　Fig.7　as　a　function　of　wave　lengths　of　λ_x,and　λ_y.From　this　we　can　see　that　F（ω,φ）＝1　for　the　components　for　λ　＝∞.This　leads　to　the　fact　that　the　underground　structure　is　represented　just　as　it　is,while　even　for　the　components　with　comparatively　long　waves,F　is　not　equal　to　unity　and　the　underground　structure　is　about　10　per　cent　smaller　than　the　actual　value.Further,it　is　found　that　F　has　a　tendency　to　become　large　over　unity　in　the　neighbourhood　of　λ_x,＝λ_y,＝　7.Therefore,as　a　whole,the　components　of　a　wave　length　longer　than　4　have　about　a　10　per　cent　error,while　the　components　of　a　wave　length　less　than　4　show　a　reduction　in　their　high　frequencies.If　λ_x　＝λ_y,the　characteristics　are　especially　good.This　is　due　to　the　fact　that　if　e^√（ω^2＋φ^2）is　made　approximately　equal　to　an　equation　of　second　degree,this　approximate　equation　becomes　less　than　e^√（ω^2＋φ^2）for　the　large　values　of　√（ω^2＋φ^2）while　F（ω,φ）takes　an　especially　large　value　when　λ_x＝λ_y＝3.This　is　also　characteristic　of　the　△g_z,and　△g_zz　given　by　Kato.At　all　events,the　opposite　results　above　mentioned　are　cancelled　with　each　other　and　consequently　F（ω,φ）is　nearly　equal　to　unity.This　is　the　reason　why　k＝0.5　is　selected.For　instance,if　λ_x＝λ_y＝-2　in　Fig.6,then　e^√（ω^2＋φ^2）F（ω、φ）＝29.839.In　other　words,if　there　is　a　component　with　the　wave　length　of　2　in　the　gravity　anomaly,it　is　magnified　by　about　30　times　the　actual　value.This　means　that　a　small　noise　or　error　of　measurement　in　the　gravity　anomaly　is　to　be　greatly　amplified.In　order　to　prevent　this　amplification　of　the　noise,the　values　which　are　considered　as　a　noise　have　to　be　omitted　beforehand.Thus,if　about　10　per　cent　error　is　admitted,it　is　possible　to　apply　the　equation　of（17）to　the　calculation　of　the　underground　mass　distribution　responsible　for　the　given　gravity　anomaly.

GRAVITY　MEASUREMENTS　IN　THE　NOBI　PLAIN　AND　ITS　VICINITY

Gravity　measurements　and　reductions
　The　gravity　measurements　were　carried　out　at　539　points　in　an　area　of　about　2,900　km2　in　the　Nobi　plain　and　the　adjacent　area　to　its　south.In　this　number　are　included　not　only　all　bench　marks　laid　at　an　average　interval　of　2km　along　the　lines　of　the　first　and　the　second　order　precise　levels　belonging　to　the　Geographical　Survey　Institute,but　also　some　triangular　points,local　bench　marks　of　Nagoya　city,and　other　identifiable　points,of　which　the　heights　are　known　with　sufficient　accuracy　for　the　purpose　of　gravity　reductions.The　distributions　of　points　at　which　the　gravity　values　were　determined　are　particularly　dense　in　the　city　of　Nagoya　and　amount　to　150　points.The　Nobi　plain　and　its　surrounding　region　occupy　the　south　central　part　of　Honshu,the　main　island　of　Japan;the　area　surveyed　and　the　topographical　features　are　shown　in　Fig.8.
　Each　gravimeter　reading　at　all　the　stations　was　corrected　for　the　effects　of　the　drift　of　the　gravimeter,the　earth　tides,and　the　height　of　the　instrument　above　the　stations　at　which　the　measurement　was　made.In　calculating　the　gravity　anomalies,the　free　air　reduction　was　made　by　the　formula　0.308600　×　h（m）.The　Bouguer　reductions　and　the　terrain　corrections　were　made　by　assuming　that　the　density　of　the　subsurface　rock　was　2.00.The　normal　gravity　r_0　was　calculated　according　to　the　international　gravity　formula

　　　r_0　＝　978.049（1+0.0052884sin^2φ−0.000059sin^2　2φ）.

The　Bouguer　anomalies　thus　determined　are　given　in　Appendix　together　with　the　gravity　values,the　free　air　correction,the　terrain　correction.and　the　normal　gravity　at　all　the　stations.The　latitudes（φ）and　longitudes（λ）of　gravity　stations　were　read　from　maps　of　the　scale　1/50,000.The　distributions　of　the　Bouguer　anomalies　are　shown　in　Fig.9　where　the　lines　of　equal　Bouguer　anomalies　were　drawn　at　2　milligal　intervals.Furthermore　the　Bouguer　anomaly　at　the　bench　marks　of　No.177　was　determined　as　−4.60,the　value　of　which　was　required　to　be　the　same　as　that　obtained　by　Tsuboi　and　others　[14].This　was　used　as　a　standard　value　in　making　the　present　Bouguer　anomaly　map.
　In　order　to　determine　the　underground　mass　distribution　in　this　district,the　gravity　anomalies　which　were　considered　as　noises　were　excluded　in　such　a　way　that　the　gravity　anomalies　were　graphically　smoothed.The　gravity　values　were　determined　by　computing　the　average　reading　for　four　points　in　succession,computing　the　average　of　the　next　four　points,etc.Thus,poises　having　a　wave　length　under　2　were　excluded　as　much　as　possible.Since　this　method　of　smoothing　tends　to　drop　the　signal　of　gravity　representing　the　underground　features　as　well　as　to　diminish　the　noise,the　interval　of　a　unit　grid　spacing　must　be　appropriately　selected.
　With　regard　to　the　characteristic　equation　of（34）,since　F（ω,φ）≒1　in　the　neighbourhood　of　λ＝9,the　main　wave　length　of　gravity　anomaly　must　coincide　with　the　grid　spacing　interval.As　a　consequence,the　depth　to　the　plane　of　condensed　mass　is　inevitably　determined.The　wave　length　of　gravity　anomaly　for　the　Nobi　plain　is　almost　equal　to　8-10　km,which　is　deduced　from　about　4-5　km　of　the　undulation　of　the　signal,as　seen　in　the　isoanomaly　line　map.In　this　case,if　the　components　under　the　wave　length　of　8-10　km　would　be　excluded,the　components　for　the　signal　of　gravity　anomaly　might　be　also　omitted.
　On　the　other　hand,as　the　unit　grid　spacing　is　assumed　to　be　1km,the　underground　structure　at　the　average　depth　of　1km　is　inevitably　considered.In　Nagoya　city,the　y-axis　of　grid　spacing　is　the　direction　of　N　15-　W　and　the　x-axis　perpendicular　to　it.The　underground　structure　was　determined　by　using　136　points　in　number.In　the　Nobi　plain　the　y-axis　is　taken　in　the　N-S　direction　and　only　the　graphical　smoothing　of　gravity　anomaly　was　done,as　the　number　of　measurement　points　is　less　than　that　of　Nagoya　city.At　any　rate　about　1,300　points　were　used　for　the　determination　of　the　underground　structure　in　the　Nobi　plain　including　the　city　of　Nagoya.In　this　way,by　means　of　the　present　formula　of（17）the　underground　relief　was　determined,as　shown　in　Fig.10.
Bouguer　anomaly　and　underground　structure
　With　regard　to　the　distributions　of　these　Bouguer　anomalies,the　following　are　some　of　the　more　interesting　features　which　are　pointed　out　in　Fig.9.
　（1）The　line　of　gravity　anomaly　runs　in　about　NNW-SSE　direction　at　the　north　and　west　of　Nagoya　city,while　it　runs　in　about　NNE-SSW　direction　at　the　south　of　Nagoya　city,indicating　the　tendency　of　geologic　features　in　this　district.
　（2）The　Bouguer　anomaly　is　negative　in　the　most　part　of　the　Nobi　plain　where　thick,young　formations　are　considered　to　be　well　developed.The　general　features　of　anomalies　are　similar　to　those　obtained　by　Tsuboi　and　others　[14].The　greatest　negative　anomaly　amounts　to　more　than　−44　milligals　and　is　located　on　the　eastern　side　of　the　Yoro（養老）mountains　on　the　western　margin　of　the　Nobi　plain,from　the　vicinity　of　Imao（今尾）to　north　of　Komano（駒野）.At　the　west　of　this　locale　the　steep　gradient　of　gravity　anomaly　is　seen,so　that　an　abrupt　downward　change　in　the　underground　structure　may　be　deduced　east　of　the　Yoro　mountains,suggesting　a　big　fault.In　the　Nobi　plain　the　basin　structure　may　be　interpreted　and　the　basement　may　be　considered　to　be　shallower　eastward　and　deeper　westward.
　（3）At　the　northwestern　margin　of　this　district　there　is　a　relative　high　of　gravity　anomaly　near　the　north　of　Akasaka（赤坂）town,suggesting　the　direct　effect　of　the　older　formations　in　geological　structure　at　Kinshozan（金山田）.
　（4）There　is　a　zone　of　densely　grouped　isoanomaly　lines　in　Nagoya　city,suggesting　a　fault-like　underground　structure.This　structural　line　is　believed　to　extend　from　Atsuta（熱田）to　Nishibiwajima（西枇杷島）parallel　to　the　railroad　of　the　Tokaido　line.The　northern　extension　of　this　fault-like　structure　line　becomes　obscure　north　of　the　Shona（庄内）river,while　its　southern　extension　is　still　unknown,being　either　coincident　with　or　cut　by　the　Kasadera（笠寺）-Narumi（鳴海）line.
　（5）The　Kasadera-Narumi　line　runs　in　NNE-SSW　direction　and　seems　to　Separate　this　district　into　two　main　basins,one　of　which　is　the　Nobi　plain　including　Nagoya　city　and　the　other　the　Mikawa（三河）plain,which　may　be　assumed　to　be　a　fault.The　line　seems　to　correspond　to　an　extension　of　a　fault,called　the　Sanagel（猿投）fault.which　is　located　just　outside　this　district　on　its　eastern　side.
　（6）The　Boguer　anomaly　is　positive　in　the　Mikawa　plain　in　the　southeastern　part　of　tins　district;　the　young　formation　seems　to　thin　out　southward.
　（7）There　is　a　bulge　of　the　gravity　low　at　the　location　of　the　lower　reaches　of　the　rivers,Shonaigawa（庄内川）and　Kisogawa（木曾川）,suggesting　a　geosynclinal　underground　structure.
　（8）With　regard　to　the　Chita　Peninsula　the　isoanomaly　lines　are　more　or　less　disturbed　from　their　mutual　parallelism.In　the　northern　part　the　wide　interval　of　isoanomaly　lines　suggests　a　rather　flattened　underground　structure,and　in　the　middle　part　near　Handa（半田）the　relative　low　of　gravity　anomaly　suggests　a　geosynclinal　structure.In　the　southern　part　the　positive　gravity　anomaly　suggests　a　shallow-seated　harder　rock,because　of　the　approach　to　the　tectonic　line,called　the　Median　Dislocation　Line.This　is　also　considered　from　the　point　of　view　of　the　relative　high　of　gravity　anomaly　which　increases　gradually　southward　in　an　area　south　of　the　cities　of　Anjo（安城）and　Kariya（刈谷）.
　As　seen　in　Fig.10,the　underground　basement　determined　from　our　formula（17）shows　the　undulation　at　the　average　depth　of　1km　beneath　the　earth’s　surface.The　general　tendency　of　the　structure　supposed　from　Fig.10　is　similar　to　that　deduced　from　the　Bouguer　anomalies　except　that　equal　contour　line　is　rather　complicated.This　complicated　contour　line　seems　to　be　due　to　the　existence　of　a　noise　which　is　either　of　the　same　order　of　the　signal　or　greater　than　it.At　any　rate,it　may　be　said　that　a　large　underground　structure　falls　down　the　west　side,because　the　layer,located　at　a　depth　of　several　hundred　meters　beneath　the　surface　on　the　east　of　this　district,corresponds　to　that　at　about　3,000m　deep　on　the　west　side.This　seems　to　be　somewhat　deeper,so　that　some　correction　may　be　necessary.We　must,therefore,reconsider　the　factors　influencing　the　calculation　of　underground　structure.
　First,the　value　of　the　density　used　must　be　taken　into　reconsideration.In　calculating　the　underground　structure　we　assumed　the　density　as　2.00　in　the　upper　layer　of　the　earth’s　crust,but　variations　in　this　density,according　to　the　locality,may　be　expected,especially　under　the　rivers　running　through　the　plain.Because　at　the　turning　point　of　the　running　direction　of　a　river　the　gravity　anomaly　is　generally　noticed　to　be　great,whether　the　anomaly　is　positive　or　negative,even　though　it　is　unknown　whether　the　assumption　of　the　density　causes　the　large　anomaly　or　not.This　variation　of　density　is,however,unknown　yet.　In　any　way,it　may　be　concluded　that　the　effect　seems　to　be　not　so　great　as　to　change　the　underground　structure.

SUMMARY

　A　formula　for　calculating　the　subterranean　mass　distribution　from　the　gravity　anomalies　was　theoretically　established.The　formula　is　expressed　by　a　linear　combination　of　gravity　values　and　their　first　and　second　vertical　derivatives　at　the　earth’s　surface.The　evaluation　of　the　vertical　derivatives　involved　was　checked　by　a　characteristic　equation　connected　with　the　wave　lengths,with　special　reference　to　the　formulas　given　by　Elkins,Henderson　and　Zietz,and　Kato.
　The　present　method　was　applied　to　the　gravity　values　which　were　measured　in　the　Nobi　plain　Bouguer　anomaly　distributions　were　obtained,based　on　the　international　gravily　formula.The　intervals　of　lines　of　equal　anomaly　are　2milligals　The　calculated　subterranean　mass　distribution　for　average　depth　of　1km　beneath　the　surface　of　the　earth　was　obtained.From　these　data,the　following　main　underground　structures　were　deduced　in　this　district:
　1）A　big　fault　and　an　abrupt　downward　change　in　the　underground　structure　east　of　the　Yoro　mountains.
　2）The　Nobi　basin　structure　and　the　Mikawa　basin　structure.
　3）The　basement　being　deeper　westward　and　shallower　eastward　and　southward.
　4）The　geosynclinal　structure　under　the　lower　reaches　of　the　rivers,Shonai-gawa　and　Kisogawa　and　the　middle　part　near　Handa,Chita　peninsula.
　5）A　fault-like　structure　at　the　Atsuta-Nishibiwajima　line　in　Nagoya　city.

ACKNOWLEDGEMENTS

　In　conclusion,the　writers　wish　to　thank　a　number　of　individuals　and　officials　in　Aichi　and　Gifu　Prefecture　and　Geological　Survey　of　Japan,from　whom　they　received　assistance　and　supports　in　executing　the　gravity　measurements　mentioned　above.We　particularly　wish　to　thank　Messrs.T.Matsuda　and　K.Ogawa,Geological　Survey　of　Japan,for　their　assistance　in　carrying　out　the　gravity　survey　in　Nagoya　city　or　the　Nobi　plain　and　to　also　express　our　gratitude　to　Messrs.T.Wada,J.Nakai,M.Kumazawa,K.Baba,H.Ohta,H.Ujho　who　have　greatly　helped　us　in　this　gravity　survey　in　a　part　of　this　district.We　wish　to　acknowledge　the　Imperial　Oil　Company,for　permission　to　use　the　gravity　values　measured　at　several　points　on　the　east　side　of　the　Yoro　mountains.

REFERENCES

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APPENDIX