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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

Array 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.

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【数式】(1)
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【数式】(2)
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【数式】(3)
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【数式】(4)
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【数式】(5)
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【数式】(6)
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【数式】(7)
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【数式・1】
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【数式・2】
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【数式】(8)
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TABLE1.VALUES OF a,b,c FOR VARIOUS VALUES OF k
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FIG.1.Relation between the values of coefficients a,b,c in a+bx+cx^2 and k in(8).
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FIG.2.Showing the values of D for various values of k.
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【数式・3】
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【数式】(13)
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【数式】(14)
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【数式】(15)
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【数式】(16)
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【数式】(17)
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【数式・4】
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【数式・5】

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

Array 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).

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【数式】(18)
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【数式】(19)
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【数式】(21)
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【数式】(22)
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【数式】(23)
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【数式】(25)

NUMERICAL EXAMPLES OF CHARACTERISTICS OF VERTICAL GRADIENT

ArrayThe 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.

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【数式】(26)
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【数式】(27)
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【数式】(28)
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FIG.3.F(λ_x,λ_y)for △g_zz by Elkins.
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FIG.4.F(λ_x,λ_y)for △g_zz by Kato.
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【数式】(30)
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FIG.5.F(λ_x,λ_y)for △_gzz by Henderson and Zietz.
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【数式】(31)
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【数式】(32)
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【数式】(33)
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【数式】(34)
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FIG.6.F(λ_x,λ_y)for △gz by Kato.
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FIG.7.F(λ_x,λ_y)for △h(x,y)in the present formula(17).

GRAVITY MEASUREMENTS IN THE NOBI PLAIN AND ITS VICINITY

ArrayGravity 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.

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FIG.8.Map showing gravity surveyed area and simplified topography of the Nobi plain and its adjacent area.
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FIG.9.Map of Bouguer anomalies in the Nobi plain and its vicinity based on the international gravity formula(unit:milligal).
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FIG.10.Map showing the undulation of basement in the Niibi plain.

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

[1] BARANOV,V.,Calcul du gradient vertical du champ de gravity ou du champ magnetique mesure a la surface du sol:Geophys.prosp.,1,171-191(1953).
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[4] —,The effect of random errors in gravity data on second vertical derivatives:Geophysics,17,70-88(1952).
[5]EVJEN,H.M.,The place of the vertical gradient in gravitational interpretations:Geophysics,1,127-136(1936).
[6]HENDERSON,R.A.,and ZIETZ,I.,The computation of second vertical derivatives of geomagnetic fields:Geophysics,14,508-516(1949).
[7]IIDA,K.,and AOKI,H.,An approximate formula of determining subterranean mass distribution from gravity anomalies(in Japanese):J.Geodetic Soc.Japan,5,1-6(1958).
[8]KATO,M.,On the vertical gradients of gravity and their computing formulas(in Japanese):Geophys.Exploration(Butsuritanko),7,128-139(1954).
[9]NETTLETON,L.L.,Regionals,residuals,and structures:Geophysics,19,1-22(1954).
[10]ROSENBACH,0.,A contribution to the computation of the second derivative from gravity data:Geophysics,18,894-912(1953).
[11]TOMODA,Y.,and SENSHU,T.,A simplified method for deducing subterranean mass distribution by the use of the response for unit gravity(in Japanese):J.Geodetic.Soc.Japan,3,41-50,95-101(1957);4,87-94(1958).
[12]TSUBOI,C.,and FUCHIDA,T.,Relations between gravity values and corresponding subterranean mass distribution:Bull.Earthquake Research Inst.,Tokyo Univ.,15,636-649(1937);16,273-284(1938).
[13]—,and KATO,M.,The first and second vertical derivatives of gravity:J.Phys.Earth,1,95-96(1952).
[14]—,JITSUKAWA,A.,and TAJIMA,H.,Gravity survey along the lines of precise levels throughout Japan by means of a Worden gravimeter,Part VI Chubu district:Bull.Earthquake Resear-ch Inst.,Tokyo Univ.,Suppl.Vol.4,Part 5,200-310(1955).

APPENDIX

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GRAVITY VALUES AND BOUGUER ANOMALIES FOR ALL THE STATIONS SURVEYED