Gravity gradients and the interpretation

GtOPHYSICS. Vol.. 31. NO 6 (DECEMBER 1976). P. 1370-1376. 3 FIGS, 2 TABLtS GRAVITY GRADIENTS AND THE INTERPRETATION OF THE TRUNCATED PLATE JOHN M. STANLEY* AND RONALD GREE N* The truncated plate and geologic contact are commercially important structures which can be located by the gravity method. The interpretation can be improved if both the horizontal and verti- cal gradients are known. Vertical gradients are difficult to measure precisely, but with modern gravimeters the horizontal gradient can be meas- ured conveniently and accurately. This paper shows how the vertical gradient can be obtained from the horizontal gradient by the use of a Hil- bert transform. A procedure is then presented which easily enables the position, dip angle, depth. thickness, and density contrast of a postu- lated plate to be precisely and unambiguously derived from a plot of the horizontal gradient against the vertical gradient at each point meas- ured. The procedure is demonstrated using theo- retical data. INTRODCCTION In an important paper, Hammer and Anzo- leaga (1975) looked at the application of the grav- ity method to the problem confronting the petro- leum industry of finding stratigraphic traps. In particular, they point out that the horizontal and vertical gradients are better indicators of geologic contacts (such as occur in stratigraphic pinchout sections) than is the gravity anomaly measured with a gravimeter. This has also been discussed by Green (1975). For a number of reasons, the accu- rate measurement of the vertical gravity gradient due to a near-surface structure is quite imprac- tical. On the other hand, modern gravimeters can measure horizontal gradients conveniently and precisely. Hammer and Anzoleaga proposed to obtain the horizontal gradients by measuring the relative gravity between three closely spaced (100 m) stations located on the apex of a triangle. In this case, the gravity gradient in the .Y direction gI(s) would be: g,(X) = [g(X + AX) - g(x)]l~x. (1) A practical advantage of this strategy is that accu- rate elevations between sets of these gradient de- terminations is not required. Table I of Hammer and Anzoleaga’s paper shows that this method of determining the hori- zontal gradient works perfectly well. As their table 2 shows, however, the same method for di- rectly measuring the vertical gradient is not satis- factory. We propose to overcome this problem and by so doing make the gravity gradient method practical. It will be shown that for a gravity pro- tile measured over a geologic contact or a trun- cated plate, the vertical gradient can be readily calculated from the horizontal gradient. Just as was shown to be the case with the anomalous magnetic field over a contact (Green and Stanley, 1975, Stanley, 1975), the vertical gravity gradient g*(X) is related to the horizontal gradient gX(.u) by the Hilbert transform. This is represented by the expression g&) + H + g*(x). (2) Having established this relationship, we can de- termine accurately both the vertical and horizon- tal gravity gradients. A very simple method is then derived to determine unambiguously the parame- ters of position, dip angle, depth, thickness, and density contrast for a truncated plate or geologic contact. Manuscript received by the Editor February 9, 1976: revised manuscript received April 17, 1976 * University of New England. Armidale, N.S.W. 2351, Australia. @ 1976 Society of Exploration Geophysicists. All rights reserved. 1370 D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ Gravity Interpretation 1371 Nomenclature The following symbols have been used in this discussion: .Y= g(x) = gSx) = gz(.x) = G= P= N= A= B_ = 1c.4 = $E? = r A= r B= t= T= -H&l = 8= f(x) = K= P= Q= R= c’= M= N= X= Y= X’ = y’ = x; = X A= X H= distance along traverse from arbitrary origin. acceleration due to gravity. horizontal gravity gradient in direction of traverse. vertical gravity gradient. universal gravitational constant. density contrast across the structure. dip angle of the truncated face. top shoulder of the truncated plate. bottom shoulder of the truncated plate. angle from point x between ground surface and point A. angle from point x between ground surface and point B. radial distance from traverse point x and point A. radial distance from traverse point x and point B. depth to top of plate. depth to bottom of plate. Hilbert transform operator. phase angle of A”, Y’ polar plot. radius vector of A”, Y’ polar plot. constant = 2Gp sin LY. traverse point where $B = +A = T-N traverse point where rB = r, traverse point where (icB - #A) is a maximum. center of circle through points A, B, and R. distance from arbitrary origin to point P. distance from traverse point x to point P. horizontal g&x) axes. vertical g&y) axes. horizontal axes after rotating x through angle N. vertical axes after rotating Y through angle ~1. value of X’ at point P. the position along the traverse verti- cally above shoulder A. the position along the traverse verti- cally above shoulder B. Theory of the method In the present case of a truncated plate (or geologic contact), the plot is given in the polar form by the parametric equations (9) and (JO). These two equations define a symmetrical curve inclined to the horizontal axis at angle a, which is the dip of the truncated plate. Thus, the first pa- rameter defining the structure can be immediately deduced from the plot of the data. It is also im- portant to recognize that the x value of every data point on the curve is known. Consider the expressions given by Jung (1961) Our next operation is to rotate the axes through for the gravimetric anomalies arising from a trun- the angle N so that the major axis of symmetry of cated plate (Figure 1). The gravitational field is given by: g(x) = ZGp((x- M) sin a[sin (I ‘In rR/ra + cos 01 ‘(#B - $,,)I + (T$u - t$,)l. (3) The horizontal and vertical gradients of this ex- pression are: g,(x) = 2Gp sin ol[sin LY’ hi rN/r,4 + cm cr.(#,I - $“)I3 (4) g,(x) = 2Gp sin a[cos o(’ In r,jlrn - sinol.($,, - +A>]. (5) If we define an angle 8, a functionf; and a con- stant K such that tan P = (qSB - $,):/ln rIj, r,<, (6) [.f(x)l’ = (In rHIrn)’ + (Gil - +,4)‘, (7) K = 2Gp sin (Y, (8) then equations (4) and (5) may be written more simply as: g=(x) = K!(x) sin [a t- P(x>l, (9) g&) = K](x) cos [cu -t- Pb>l. (10) The important relationship between these two expressions is that they form a Hilbert transform pair (Bracewell, 1965). The vertical gradient g&x) is conveniently the Hilbert transform of the hori- zontal gradient g&), We can accurately measure the horizontal gradient and, hence, we can now also obtain from that data the vertical gradient with equal precision. Our procedure for interpretation simply in- volves the determination of the vertical and hori- zontal gradients at each value of x along the tra- verse and then preparing a plot of X = g*(x) versus Y = gX(s) as in Figure 2. D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ 1372 Stanley and Green r. FIG. I. The model used for a truncated plate. Note that when t is small relative to T the model represents a simple contact. The numeric values assigned to the parameters defining the plate refer to the specific worked example given in the text. the curve becomes parallel to the horizontal axis. The new axes are given by: X’ = g,(x) cos cy + g,(x) sin cy, (11) Y’ = -g,(x) sin Q + gJx) c0S Ly. (12) Multiplying equation (4) by cos N and equation (5) by sin N, we get g,(s) cos CY = K[cos CY. sin CY. In rH/rA + co? N ($B - 1C.4)lr and g>(s) sin (Y = K[cos N. sin cu.ln rB/ra ~ sir? N. (icB - &)I. Subtracting (14) from (I 3). we get ( 13) 14) g,(s) cos CY - g>(x) sin N = K(icR - GA). (15) Similarly, by multiplying (4) by sin N and (5) by cos N and subtracting. we obtain the result that: gx(s) sin N + g*(x) cos N = K In rs/ra. (16) Substituting (16) in (I I ) and (15) in (I 2). we get the following two equations defining the curve: X’ = K In r,( ‘r,,, (17) Y’ = K($,, - #I). (18) We now choose to define three important points on the curve which provide sufficient infor- mation to completely define the structure. The first of these is point P where Y’ = 0. At point P, ($” - I+,,) = 0, implying that if the face A6 of the truncated plate (Figure I) is extended. it will cut the ground surface at the point P. The x value of point P can be found from Figure 2 and, hence, P can be located on our traverse. Next we define the point Q where X’ = 0. At this point In rB/ra = 0 implying that Q is equidis- tant from both shoulders A and B. Q can thus also be located on our traverse. The third point of significance is R. where the angle (icH ~ GA) reaches its maximum value. This point corresponding to the maximum value of Y’ can also be located on the traverse. D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ Gravity Interpretation 1373 After considering Figure 3, it can be seen that and, similarly, the shoulders A and B of the plate can now be graphically determined. The angle (it/B - $A) must [sin $,]/[PA] = [sin (7~ - (Y ~ $,)]/[OP - x] be a maximum when it is the angle subtended by = [sin (CY - Is/A)]/[OP - x]. (20) chord AB of a circle drawn tangent to the ground. The point of tangent must necessarily be at R Substituting N for the distance (OP - x), we along the traverse. We may construct this circle derive from (19) the expression and determine shoulders A and B by its inter- section with the line through Pinciined at LY, the (sin $H)I(PB) = (sin ~5.~0s $1j angle of dip. The center of the circle is located by + cos ol’sin fiII)/N the intersection of a perpendicular to the ground (21) surface from point R with a line inclined at (cu - .*. (tan #,3)/(PB) = (sin (v 7r/2) to the surface and passing through Q. The radius of the circle must be the distance RC. The + cos (Y. tan tiB),:/iV, shoulders A and B are thus determined unambig- uously. which reduces to: The geometric construction just presented may tan 1+5~ = (PB’sin oi)/(N - PB’cos 0). (22) be trigonometrically justified. In so doing, we ar- rive at mathematical expressions from which the parameters defining the plate may be automati- Similarly. cally computed. tan GA = (PA .sin cu)/(N - PA ,COS a). (23) With reference to Figure I, the following trigo- nometric identities exist: Subtracting (23) from (22), we get [sin $B]/[PB] = [sin (r - N - IcB)]/[OP - X] (1c’B - #A) = [sin (a + $B)]/[OP - x], (19) = arctan (PB.sin a)/( N -~ PB. COs Ly) FIG. 2. A plot of gX(x) versus g*(x) for the truncated plate model of Figure 1. Note that the principle axis of symmetry is inclined to the horizontal axis at the dip angle N. Also shown are the X’ and Y’ axes obtained after performing a rotation through angle cy. P then defines the traverse point where the angle term equals zero implying that the line AB in Figure 1 intersects the surface at this point. Q defines the point where the log term is zero implying that Q is equidistant from the plate shoulders A and B. Point R defines the traverse point where the angle term is maximum. The measured positions of P. Q, and R plus the determination of N are adequate to unambiguously define the truncated plate structure. D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ 1374 Stanley and Green R FIG. 3. R is the point where (qtl - $A) is a maximum. This condition is met when (gB - .$A) is the angle subtended by chord AB of a circle tangent to the ground surface at R. The center C‘ of this circle is located by the intersection of a vertical line from point R with a line inclined at (m - x/2) to the surface and passmg through point Q. Shoulders A and B of the truncated plate are, thus. unambiguously determined. This process may be carried out automatically. - arctan (PA.sin a)/(N - PA.cos a). C = (PR. RQ. cot w). (24) Note the sign of lengths PR and PQ. They are Differentiation, with respect to N, yields defined as the length from P to R or Q, relative to the positive direction of traverse. The radius of d($n - +/J/GIN = [(PB - PA) this circle is RQ.cot N and, hence. its equation is given by: .(PA. PB - N’).sin a]/[(N’ - 2PB .N.cos cr + PB’)(N” - 2PA . N,cos (Y + PA”)]. (x - PR)’ + (y ~ RQ.cot a) = (RQ.cot a)‘. (28) (25) The equation of the line PAB along the truncated At maximum and minimum values of (GB - $A), face is’ d($,1 - $A)/d.W = 0. (26) b’ = x.tan a. (29) Consequently, ($8 - $‘A) takes a maximum at A’ Solution of equations (28) and (29) yields the = - 1 m and a minimum at ,V = \ PA;PB. coordinates of the shoulders A and B of the trun- From this we conclude that point R is distant - cated plate relative to an origin at P. It can be 1 m from point P. Hence, determined, consequently, that the traverse point which is vertically above shoulder 4 of the plate is PR”~ = PA. PB. (27) given by: xA = M - cos’ a This expression is the standard identity relating to a circle passing through points A, B, and R and .[PQ + fi’ - PR’/cos”a]. (30) a point P such that PR is a tangent and PAB is a straight line. The equation of this circle can be and the point above shoulder B by readily determined. Its center C must be located XLI = M _ cos2 a where the vertical from R intersects the line through Q inclined at (a/Z - x) so as to be /;;----- ‘) . [PQ~ - y rQ~ - PR~-/cusL~]. (51) perpendicular to line PAB. The coordinates of point C relative to the origin relocated at P are, The depth to the top of the plate is given by therefore, f = sinff.coscu D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ Gravity Interpretation 1375 Table 1. Model gravity gradient data corresponding to a traverse made across a truncated plate. This data was employed in the worked example demonstrating the interpretation procedure. x Y = g&u) distance IO @secz fO =, $1 2 Lll0Ilg measured computed irxvene i1orizontill vertical Cm) gradient gradient 0 0.6 ~2.6 400 I.1 -3.5 800 2.6 -4.6 IO00 4.2 -5.1 1200 6.4 -4.1 1400 8.9 ~3.2 1600 Il.0 PO.2 1800 9.5 6.9 2000 3.6 6.1 2200 1.9 5.1 2600 0.X 3.4 3000 0.5 2.5 ~___. . [PQ + 6’ - PR’;lcos2a], (321 and the depth to the bottom of the plate is T = sino!.cosa . [PQ _ fi”--p$,‘],/[2G sin ol.ln (PQ --T,-.-- - JpG’- PR-/ cos’ a)/( PQ + firP2ma>]. (35) Equations (30), (3l), (32), (33). and (35) com- pletely and unambiguously define the parameters of the truncated plate. PRACTICAL APPLICATIOh’ OF THE METHOD-A WORKED EXAMPLE The method requires the measurement of the horizontal gravity gradient in the direction per- pendicular to the strike of the structure. The gradient is determined by two gravity measure- ments of known separation. The only correction that need be applied to the field data is that of compensating for the elevation and latitude differ- ence between the two stations from which each gradient is determined. Gradient measurements will typically be required to attain an accuracy of IO 9 set’. To fulfill this specification, gravity measurements of resolution 0.01 mgal require a measurement base of 100 m. Precision require- ments determined by Hammer and Anzoleaga (1975) allow for an error of * 5 m over this interval and an elevation determination to within 5 cm. With some modern gravimeters. measure- ments to a wgal can be made. giving option to either greater resolution or the same resolution at- tainable using a measurement base of 10 f 0.5 m. Gravity gradient surveying thus becomes more practical than the conventional measurement of a gravity profile, especially where topographical surveys are lacking. The measured gradient data arc entered into a computer programmed to perform a Hilbert transform of the horizontal gradient profile. The Hilbert transform may be executed by performing a cubic spline interpolation between field meas- urements and then using Simpson‘s rule to eval- uatc numerically the integral defining the Hilbert transform (Bracewell. 1965). A fast Hilbert trans- form based upon the fast Fourier algorithm is being investigated. The result of this computation is a value for both the horizonral and vertical gradients at each point along the traverse. Table I lists such gradient data corresponding to a tra- verse made across a truncated plate. The next operation is to plot the horizontal gradient on the Y axis and the vertical gradient on the X axis. If the assumption of a truncated plate (or a contact) is valid. this plot will appear as a curve with two orthogonal axes ol‘symmetry. The data of Table 1 have been plotted in Figure 2. The curve will be tangent to X at the origin, and the major axis of symmetry will be inclined to the X axis at the angle of dip (cu) of the truncated face. Note that the origin corresponds to traverse points an infinite distance from the truncated face and that most of the curve is defined by measure- ments made relatively close to this contact. If advantage is made of the symmetry of the curve, then field measurements are onl! required very close to the contact. Thus, the method is ex- ceedingly free from interference from neighboring sources. The X and Y axes are now rotated through angle cr. The new horizontal X’ axis then intersects the curve at the point P and the vertical Y’ axis D ow nl oa de d 03 /0 9/ 13 to 2 10 .7 6. 19 5. 89 . R ed ist rib ut io n su bje ct to SE G lic en se or co py rig ht; se e T erm s o f U se at htt p:/ /lib rar y.s eg .or g/ 1376 Stanley and Green Table 2. The parameters of the truncated plate in the *rorked example. 1973 m I280 “I 1526111 30” I 800 111 shoulder B Depth to top or platr I IIepth to bottom of plate I I)ensit) contrast of plate,, 1107m 100 m 500 171 0.075 gmicm ’ Intersects the curve at Q. The point where Y’ reaches a maximum value occurs at the point R (Figure 2). The distance from P to Q (having regard to sign relative to the positive .Y direction) and P to R are then determined and. together tiith the previous11 obtained value of C\ substituted in equations (30). (31 ). (32). (33). and (35). to yield the parameters uniquely defining the structure. These results have been listed in Table 2. The Hilbert transform method of determining the vertical gradient from horizontal gradient data has facilitated ;1 new approach to regional gravity surveying and interpretation. Field pro- cedures have been greatly simplified, resulting