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Gravity inversion of 2D basement relief using entropic regularization


GEOPHYSICS, VOL. 75, NO. 3 MAY-JUNE 2010 ; P. I29–I35, 6 FIGS. 10.1190/1.3374358

Gravity inversion of 2D basement relief using entropic regularization

Jo?o B. C. Silva1, Alexandre S. Oliveira1, and Valéria C. F. Barbosa2

ABSTRACT
We have developed a gravity interpretation method for estimating the discontinuous basement relief of a sedimentary basin. The density contrast between the basement and the sediments is assumed to be known, and it could be either constant or vary monotonically with depth. The interpretation model consists of a set of vertical, juxtaposed prisms, whose thicknesses are the parameters to be estimated. We used the entropic regularization that combines the minimization of the ?rst-order entropy measure with the maximization of the zeroth-order entropy measure of the solution vector. We validated the method by applying it to synthetic data produced by a simulated basin bordered by high-angle step faults; we obtained a good de?nition of the relief, particularly of the discontinuities. We also applied the method to a pro?le across the Büyük Menderes Valley in West Turkey and obtained a solution exhibiting a gravity fault with large slip on the northern border of the valley. When applied to the interpretation of a discontinuous basement relief, the method has a better performance than the global smoothness method. It is comparable to the weighted smoothness method, but it does not require the a priori knowledge about the maximum basin depth.

INTRODUCTION
The knowledge of the basement relief of a sedimentary basin is important in oil prospecting because it could indicate probable locations of stratigraphic and structural traps. Stratigraphic traps such as pinch-outs might be formed at the ?anks of a smooth positive relief feature associated with oscillations of the basement relief, which can be mapped by gravity inversion methods that stabilize the solution by assuming it is smooth Le?o et al., 1996; Barbosa et al., 1997; Chakravarthi and Sundararajan, 2007 . On the other hand, structural traps might be associated with faults that displaced the sedimentary layers and basement. In this case, the knowledge of the discontinu-

ous relief could help to locate oil structural traps, which might be associated with the discontinuity. Applying the above-mentioned methods to map a discontinuous relief has yielded unsatisfactory results. Barbosa et al. 1999 developed a method, named weighted smoothness, which manages to map high-angle faults in the basement topography, but it requires the knowledge of the maximum basin depth. We present a gravity inversion method suited to map a discontinuous basement relief based on entropic regularization, which consists in minimizing the ?rst-order entropy measure of the vector containing the depth-to-basement estimates at discrete points, inhibiting at the same time any excessive minimization of the zeroth-order entropy. The latter imposition is necessary to prevent the collapse of the solution into an unrealistic basin estimate consisting of an excessively narrow and deep relief. The proposed method has been tested on synthetic data produced by a simulated extensional basin whose density contrast between the sediments and the basement decreases with depth according to a hyperbolic law. The modeled basin presents an overall smooth relief de?ned by terraces and local sharp discontinuities separating them. The results demonstrate the good performance of the method in delineating the sharp basement discontinuities, and reinforce con?dence in the interpretation of real data, which consist of two gravity pro?les. The ?rst one is a pro?le across the Poema Bridge at the Federal University of Pará campus. At ebb tide, the river channel is exposed, and its bottom topography might therefore be estimated from a gravity pro?le across it similar to the procedure used to interpret the basement relief of a sedimentary basin with constant density contrast between the sediments and the basement. The overall features of the channel were detected using the proposed method. The second pro?le runs across the Büyük Menderes Valley in West Turkey. An interpretation model, assuming a density contrast decreasing with depth, led to a solution exhibiting a gravity fault with large slip on the northern border of the valley, in accordance with the geology of the area. For comparison, we applied the global and weighted smoothness methods to the same sets of synthetic and real data.

Manuscript received by the Editor 17 July 2009; revised manuscript received 16 November 2009; published online 13 May 2010. 1 Federal University of Pará, Geophysics Department, Belém, Brazil. E-mail: joaobcsy@yahoo.com.br; alexandrecn8@yahoo.com.br. 2 Observatório Nacional, Geophysics Department, Rio de Janeiro, Brazil. E-mail: valcris@on.br. ? 2010 Society of Exploration Geophysicists. All rights reserved.

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Silva et al. vector ri produced by the jth prism with thickness p j. We want to estimate the vector p p1, . . . ,pM T from the vector 0 g0 p g0, . . . ,gN T, containing the gravity observations. The solu1 tion to this problem must honor the observations, and this might be achieved by the minimization of the nonlinear functional

The results obtained with the proposed method using synthetic and real data were compared with the estimates using the global and weighted smoothness constraints. The entropic regularization and the weighted smoothness produced, in general, better estimates in the presence of a discontinuous relief, as compared with the global smoothness estimator. The main advantage of the proposed method relative to the weighted smoothness is that it does not need the a priori information about the basin maximum depth.

p

g0

gp

2 2,

4

METHODOLOGY
Let g0 p be a set of N Bouguer anomaly observations produced by the basement relief of a sedimentary basin S in Figure 1 . From now on, we assume that possible regional effects were removed previously. We refer to the residual Bouguer anomaly simply as Bouguer anomaly. The basement is assumed to be homogeneous, and the density contrast z between the sedimentary section and the basement is assumed to be known and constant or to decrease monotonically with depth according to the hyperbolic law Litinsky, 1989

where g p is an N-dimensional vector containing the anomaly produced by the interpretation model functions 2 or 3 , computed at the same observation points, and · 2 is the Euclidean norm. Estimating the vector p that minimizes the functional 4 is an ill-posed problem because the solutions are unstable. Hence, it is necessary to incorporate geologic or physical constraints to transform this problem into a well-posed one, that is, a problem having stable solutions. The usual technique used to stabilize the solutions is global smoothness, which ? imposes that the estimate p j thicknesses of the ith prism be as close ? as possible to the estimate of the adjacent parameter p j, subject to the data being ?tted by the anomaly produced by the interpretation model within the experimental precision Barbosa et al., 1997 . This trade-off is achieved by minimizing the function

z

0

2

z

2,

1

p

g0

gp

2 2

Rp 2, 2

5

where 0 is the density contrast at the earth’s surface, and is a parameter controlling the decrease of the density contrast with depth z. We estimate the basement relief S from the gravity observations by assuming an interpretation model consisting of M 2D vertical, juxtaposed prisms Figure 1 with a density contrast either constant or decreasing with depth according to equation 1. The prisms’thicknesses are the parameters to be estimated, and they are related to the gravity anomaly gi through the nonlinear relationships
M

gi
j 1

A p j,

,ri ,

i

1,2, . . . N,

2

where R is an L M matrix representing the ?rst-order discrete differential operator whose lines contain just two nonnull elements, equal to 1 and 1, located at the columns i and j, respectively. The integer L M 1 is the number of adjacent parameters, and is a nonnegative coef?cient, chosen in accordance to a criterion described later. This method estimates a smooth basement, which might be satisfactory for most intracratonic basins presenting a smooth basement relief, but it is not capable of correctly delineating a basement relief containing abrupt discontinuities. Barbosa et al. 1999 developed a gravity interpretation method directed to the interpretation of a discontinuous basement relief, named weighted smoothness. This method minimizes the functional

for a constant density contrast Telford et al., 1991 , or
M

p

g0

gp

2 2

s

WRp

2 2

r

p

pmax 2, 2 6

gi
j 1

F p j,

0,

,ri ,

i

1,2, . . . N,

3

for a variable density contrast decreasing with depth according to the hyperbolic law given in equation 1 Rao et al., 1994 , where A p j, ,ri and F p j, 0, ,ri are nonlinear functions yielding the gravity anomaly at the ith observation point de?ned by the position

S

Figure 1. Gravity anomaly above and interpretation model below consisting of M vertical, juxtaposed prisms, whose depths p j are the parameters to be determined.

where s and r are stabilizing coef?cients, W is an L L diagonal matrix of weights, and pmax is a vector of maximum basement depths, presumably known a priori. The th diagonal element w of matrix W weights the th line of matrix R, which imposes proximity between the th pair of adjacent parameters: The larger the value of w , the closer will be the estimates of these parameters. The value assigned to w is inversely proportional to the difference between the th pair of estimates, so it is de?ned iteratively. The choices of coef?cients in equation 5, and s and r, in equation 6, are made as follows. Coef?cients and s impose solution stability at the expense of solution resolution. This approach follows from the well-known trade-off between resolution and stability Backus and Gilbert, 1968; Parker, 1977 , and to understand it, we must distinguish between “true resolution” and “demanded resolution.” The former is the smallest distance between two pointwise physical property distributions, so that the geophysical observation can still recognize them as separated. The demanded resolution, on the other hand, is the resolution that the geophysicist expects from the data by assuming a given value for a stabilizing parameter. A small value of increases the demanded resolution but also substantially increases the solution instability. A large value of , on the other hand, stabilizes the solution but reduces the demanded resolution.

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Basement depth entropic regularization Instability shows up when the demanded resolution becomes greater than the true resolution can attempt to extract more information from the data than is physically possible . Consequently, the demanded resolution will coincide with the true resolution for the largest parameter still producing stable solutions. We adopted the following practical procedure to assess if a solution is stable. Starting with a very small tentative value of , we obtain a set of k estimated solutions, pi, i 1, . . . K, from observations contaminated by k different pseudorandom noise sequences. Then ? ? we compute vectors pi p j, i 1, . . . K, j i 1 . . . K, representing the differences between all possible pairs of estimated solution vectors, and calculate the Chebyshev norm of each one. Finally, the solutions are assumed to be unstable when at least one of the Chebyshev norms is greater than a value W, which depends on the geologic setting and on the purpose of the interpretation. If the solution is unstable, the value of is increased and the process repeated. In this way, and s must be the largest value still producing a stable solution. Coef?cient r, on the other hand, stabilizes the solution by imposing the constraint that the parameter estimates be closest to the maximum expected depths. Therefore, it must be the largest positive value that produces an estimated relief presenting no more discontinuities than those expected for the basement in the study area. In this study, we use the entropic regularization method presented by Ramos et al. 1999 , which involves two entropy measures. The ?rst one is the zeroth-order entropy measure given by
M

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Q0 p
k 1 i

pk
M

log
i

pk
M

,

7

pi
1 1

pi

and the second one is the ?rst-order entropy measure given by
L

Q1 p
k 1 i

tk
L

log ti
i

tk
L

, ti

8

1

1

Entropy measures Q0 and Q1

Depth (km)

where is a small positive constant smaller than 10 8 used to guarantee the de?nition of the entropy measures, and tj is an element of vector t Rp. Mathematically, the maximization of Q0 p equation 7 occurs when all M prisms of the interpretation model have the same thickness estimates. On the other hand, the minimization of Q0 p occurs when all but one of the M prisms of the interpretation model have a null thickness estimate. Silva et al. 2007, equation 12 show that the minimization of Q1 p equation 8 occurs when the number of discontinuities between adjacent parameter estimates is minimum regardless of the number of null thickness estimates. Then the ?rst-order entropy measure tends to minimize the number of discontinuities in the basement relief. In this study, we solved the nonlinear inverse problem of minimizing the functional

sociated with 0 leads to a maximization of the zeroth-order entropy measure. Actually, we do not in fact want to maximize Q0 p , but instead to prevent its excessive minimization as will be explained below. To interpret the physical and geologic meaning of functionals Q1 p and Q0 p , we simulated three simple examples of possible obtainable solutions not necessarily having geologic meaning for 2D sedimentary basins B1, B2, and B3 in Figure 2a-c . We want to interpret geologically meaningful discontinuities in the basement such as B2.As shown by the values of Q0 p and Q1 p associated with solutions B1 – B3 Figure 2d , the minimization of Q1 p alone discards the solution B1. However, this minimization cannot distinguish between solutions B2 and B3 because they present about the same number of discontinuities, and therefore they present values of Q1 p that are close to each other see Silva et al., 2007, equation 12 . As a result, the minimization of Q1 p might lead to sources with extremely low values of Q0 p , associated with unrealistic estimates. For shallow solutions, the gravity anomaly has enough resolution to deter the excessive minimization of Q0 p otherwise, the anomaly produced by solution B3 will not honor the observations and, consequently, to prevent the estimation of unrealistic, minimum-volume solutions. As a result, the minimization of Q1 p subject to an acceptable mis?t is just enough to produce geologically reasonable sharp-bounded basement relief estimates. For deep solutions, however, basement topography like B2 and B3 might produce similar anomalies, so solution B3 could be accepted as a geophysical solution. In this case, it is necessary to prevent any spurious minimization of Q0 p by assigning a small positive value to 0. Coef?cients 0 and 1 should be selected as follows. Coef?cient 1 allows discontinuities in the solution depth to the basement . A small value of 1 produces a solution with unde?ned discontinuities in the same way that the global smoothness method does , whereas a large value of 1 produces solutions exhibiting spurious oscillations and discontinuities in the estimated relief. These oscillations are not related to instability; instead, their presence demonstrates that we have overestimated the number of basement discontinuities. In this way, 1 must be the largest positive value leading to no more oscillations or discontinuities than those expected for the basement

a) 0.0
Depth (km)
0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 B1

b)
Depth (km)

0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 B2

Horizontal distance (km)

Horizontal distance (km)

d)
3 2 1 0 Q0 Q1 B1 B2 B3

c) 0.0
0.2 0.4 0.6 0.8 1.0 B3

p

g

0

gp

2

0Q0 p /Q0max

1Q1 p /Q1max,

9
where Q0max and Q1max are normalizing constants. The regularizing coef?cients 0 and 1 are positive numbers. The coef?cient 1 is selected so as to impose a minimization of Q1 p . The negative sign as-

0

5

10

15

20

Horizontal distance (km)

Basin type

Figure 2. a-c Two-dimensional basins B1 – B3, and d the respective entropy measures of orders zero and one.

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Silva et al. given in equation 1 with 0 0.5 g / cm3, and 3.0 km. The theoretical anomaly was corrupted with pseudorandom zero-mean Gaussian noise with a standard deviation of 0.1 mGal. We assumed an interpretation model consisting of 60 juxtaposed, 1-km-wide prisms and presumed that the true density contrast between the basement and the sediments is known. The noise-corrupted anomaly was inverted using the entropic regularization with 0 1.75 and 1 0.45. The depth-to-basement estimate is shown by the dashed line in Figure 3a, indicating the potential of the method to delineate the relief of discontinuous basements. The ?tted gravity anomaly is shown by the solid line in Figure 3a upper panel . Figure 3d shows the distributions of the zeroth- and ?rst-order entropy measures along the iterations. Convergence occurred at iteration 58 according to the criterion established in inequality 10. Figure 3b presents the solution produced by the global smoothness with 1.35. We note the presence of small spurious oscillations on the estimated relief dashed line in the lower panel impairing the correct estimation of discontinuities and terraces. These oscillations might be reduced by increasing the value of , but the solution will become smoother and the delineation of the discontinuities will worsen. The ?tted gravity anomaly is shown by the solid line in Figure 3b upper panel . In Figure 3c, we present the solution produced by the weighted smoothness with s 1.45 and r 0.000001, and a maximum assumed depth of 1.5 km. The solution produced a good delineation of the basement relief, comparable to the entropic regularization estimate Figure 3a , but in the latter case no assumption on basin depth was required. The ?tted gravity anomaly is shown by the solid line in Figure 3c upper panel .

relief being interpreted. As shown in the previous section, the entropy measures Q0 and Q1 are not independent of each other, and the minimization of Q1 implies, under certain circumstances, the minimization of Q0 as well. As a result, we must provisionally assign to 0 a small value including zero . If the solution collapses into a source whose horizontal dimensions are substantially smaller than those expected for the true source, we must increase the value assigned to 0. To solve the nonlinear inverse problem of minimizing p equation 9 , we used the iterative quasi-Newton method Gill et al., 1981 with the Broyden-Fletcher-Goldfarb-Shanno BFGS implementation. The iteration stops when the condition

Q1k

1

Q1k
1

Q1k

0.005

10

is satis?ed for ?ve consecutive iterations, where Q1k is the value of the ?rst-order entropy measure at the kth iteration. This criterion ensures that the distribution of Q1 along the iterations has attained an almost constant level.

APPLICATION TO SYNTHETIC DATA Graben bordered by step faults
We evaluate the performance of the proposed method by applying it to the Bouguer anomaly gray dots in Figure 3a-c, upper panels produced by a simulated 2D graben limited by step faults solid gray line in Figure 3a-c, lower panels . The density contrast between the sediments is assumed to decrease with depth according to the law
Bouguer anomaly (mGal)
0 –5 –10 –15 –20 –25

Bouguer anomaly (mGal)

a)

b)

0

10

20

30

40

50

60

0 –5 –10 –15 –20 –25

The validity region
0 10 20 30 40 50 60

Horizontal distance (km)
0 0.5 1.0 1.5 0 10 20 30 40 50 60

Horizontal distance (km)
0 0.5 1.0 1.5 0 10 20 30 40 50 60

Horizontal distance (km) Bouguer anomaly (mGal)

Horizontal distance (km)

c)

Entropy measure

0 –5 –10 –15 –20 –25

d)
4.5 4.0 3.5 3.0 2.5 2.0 10 20 30 40 50 60 1.5 0 10 20 30 40 50 60 First-order entropy Zeroth-order entropy

0

10

20

30

40

50

60

Horizontal distance (km)
0 0.5 1.0 1.5 0

Horizontal distance (km)

Iteration number

Figure 3. Application to synthetic data. a-c True basement relief and the corresponding gravity anomaly shown, respectively, by the solid lines lower panels and gray dots upper panels . Estimated relief dashed lines in the lower panels and the corresponding ?tted anomaly solid lines in the upper panels produced by a the entropic regularization with 0 1.75 and 1 0.45, b the global smoothness with 1.35, c the weighted smoothness with s 1.45 and r 0.000001, and a maximum assumed depth of 1.5 km. d Distribution of the zeroth- and ?rst-order entropy measures along the iterations.

We conducted a numerical analysis to investigate the ability of the entropic regularization to detect and locate faults with small or large vertical throws as a function of depth. To this end, we ?rst simulated a 2D graben limited by vertical gravity faults solid thick line in Figure 4a , with fault throw d, depth H, to the midpoint of the fault displacement, and horizontal distance D between the faults. The density contrast is assumed to be constant and equal to 0.3 g / cm3. Then the gravity anomaly of the faulted basement de?ned by the pair d,H is inverted using an M-prism interpretation model as described before. This procedure is repeated by designing other true basement topographies characterized by several values of d and H, spaced by, respectively, 0.2 km and 0.5 km, using the entropic regularization. If the M-prism solution detects and locates the gravity fault using 0 0 and 1 0, the corresponding point d,H on plane d-H is assigned to the shallow region R1. If the normal fault is only detected and located, using 0 0 and 1 0, the corresponding point d,H on plane d-H is assigned to the intermediate region R2. Otherwise, if the fault is neither detected nor delineated, the corresponding point d,h on plane d-H is assigned to the deep region R3. The upper limits of

Depth (km)

Depth (km)

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Depth (km)

Basement depth entropic regularization regions R1 and R2 along the H-axis are the most important limits to be mapped with this procedure. Figure 4b shows the fault-detectable regions R1 and R2 dark and light gray areas, respectively on the plane d-H generated by applying the above procedure in the intervals d 0.5 km, 5 km and H 0.25 km, 8.5 km . The boundary R2-R3 converges to constant values of H as d approaches either large or small values and presents a steep gradient at about d 1.5 km. For large values of d, the existence of a bounding value for H follows from the limiting ability of the gravity anomaly to resolve a very deep structure. For small values of d, a bounding value of H close to zero follows from the fact that the enhancement in the gravity signal caused by a decrease in H overcomes the signal decrease produced by a decrease in d. The boundary R1-R2 is roughly parallel to the boundary R2-R3 up to d 2 km, and from this point on it converges to increasingly smaller values of H. This decrease follows from the limited ability of the gravity anomaly to detect separate sources the two gravity faults limiting the graben at a large distance resolution limit . This decrease is more accentuated for a large H / D ratio.

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Horizontal distance

a)
Depth (km) H

d D

b)
Fault midpoint displacement H (km)
8

R3

6

R2

4

R1
2

APPLICATION TO REAL DATA
To illustrate the applicability of the proposed method, it was applied to two gravity anomalies associated with a discontinuous interface separating an upper, less dense medium from a lower, more dense medium.

0 0 1 2 3 4 5

Fault throw d (km)

Poema Bridge, Pará, Brazil
A major dif?culty in validating a new interpretation method with real data is that the true source geometry usually is not known. In attempting to lessen this dif?culty, we applied the proposed method to a pro?le along a bridge located at the Federal University of Pará campus, Brazil, known as Poema Bridge. The anomaly source is the density contrast between the air and the material located on either side of the bridge. The bridge is located close to the river mouth, so at ebb tide the streambed is exposed, allowing a precise knowledge of its geometry. Estimating the relief of the streambed has the same mathematical formulation as the estimation of the basement relief of a sedimentary basin, so the conclusions obtained in this application could be extended immediately to the case of sedimentary basins. The geologic material on either side of the bridge consists of compacted clay, sand, and gravels. The abutments of the bridge are made of concrete, and their geometries differ. The one at the western end presents a larger vertical extent as compared with the abutment on the eastern end Figure 5a . Because the concrete abutments contribute to the gravity anomaly, the relief to be estimated is not limited to the topography of the streambed and the land?ll on each side of the bridge. It should include the relief formed by the union of the abutments, the streambed and the land?ll. We note that this relief presents a large discontinuity on the western end. Moreover, the

Figure 4. a Synthetic 2D faulted basement relief solid line with vertical fault throw d, midpoint depth of the fault displacement at H, and fault separation D. b Regions on the plane d H, where it is suf?cient to set 0 0 and 1 0 region R1 , and 0 0 and 1 0 region R2 , to obtain normal fault-detecting solutions. In region R3, normal faults are not detected by the method.
Bouguer anomaly (mGal)

a)
0

b)

0 –0.1 –0.2

1.58 m

W
0 1 2 3 4

0

4

8

12

16

Horizontal distance (m)

20 E

g1 W

g2

Depth (m)

A

B

0

4

8

12

16

20

Horizontal distance (m) Bouguer anomaly (mGal) Bouguer anomaly (mGal)

c)

0 –0.1 –0.2

d)

0 –0.1 –0.2

W
0 1 2 3 4

0

4

8

12

16

Horizontal distance (m)

20 E

W
0 1 2 3 4

0

4

8

12

16

Horizontal distance (m)

20 E

Depth (m)

0

4

8

12

16

20

Depth (m)

0

4

8

12

16

20

Horizontal distance (m)

Horizontal distance (m)

Figure 5. Poema Bridge. a View of the bridge at ebb tide. The bridge abutments are indicated by A and B. Note the asymmetrical gradients g1 and g2 on both sides of the stream channel. b-d The Bouguer anomaly is shown by the gray dots upper panels , and the measured topography below the bridge is displayed in dashed lines lower panels . Estimated relief solid line in the lower panels and the corresponding ?tted anomaly solid line in the upper panels produced by b the entropic regularization with 0 0.00005 7, and d the weighted smoothand 1 0.0000219, c the global smoothness with ness with s 0.1, r 0.0001, and a maximum assumed depth of 3.7 m.

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Silva et al. and x 15 m. Furthermore, it requires the approximate knowledge of the maximum valley depth. Finally, the weighted smoothness favors solution displaying a wide and ?at bottom, which is not factual information in this example. The ?tted anomalies produced by the global and weighted smoothness estimators are shown by the solid lines in the upper panels of Figure 5c and d, respectively. We stress that some features in the solutions of Figure 5b and d are apparently spurious, such as the oscillations at about x 11 m. The gravity data alone cannot resolve these features in fact, the gravity alone resolves only a feature of the size of the whole bridge . The de?nition of features of this size can be obtained only by the introduction of a priori information. The oscillations, in this case, are caused by insuf?cient solution stabilization. An increase in the stabilizing parameter might suppress these oscillations, but only at the expense of a decrease in the resolution of important solution features such as the one associated with the concrete abutment A .

topography of the streambed and the land?ll presents a lower gradient on the western end g1 in Figure 5a as compared with the gradient on the eastern end g2 in Figure 5a . On the basis of the heterogeneous composition of the gravity sources, we assumed an average density of 2.3 g / cm3 for the material on both sides of the bridge. Figure 5b shows the Bouguer anomaly gray dots along the Poema Bridge, measured during ebb tide. We assumed an interpretation model consisting of 43 juxtaposed prisms with a density contrast of 2.3 g / cm3, referred to the average density of the materials on either side of the bridge. Figure 5b shows that the solution lower panel of the entropic regularization with 0 0.00005 and 1 0.0000219 delineates the overall valley topography with its asymmetrical gradients and the discontinuity located close to x 3 m, produced by the contact between the concrete abutment and the air A in Figure 5a . The maximum estimated depth of 3.5 m agrees reasonably with the measured 3.88 m, and the low gradient area g1 in Figure 5a is well delineated. Figure 5b shows in dashed lines the valley topography measured from the bridge with a tape measure. The differences between the measured and inverted pro?les are primarily due to the fact that the gravity is in?uenced by the concrete abutments, whereas the topography measured from the bridge consists exclusively of unconsolidated soil. The ?tted anomaly is shown in Figure 5b upper panel by the solid line. Figure 5c and d presents the estimates of the global and weighted smoothness, respectively, using 7, s 0.1, r 0.0001, and a maximum assumed depth of 3.7 m. The maximum estimated depths in both cases are slightly underestimated, but the discontinuity close to x 3 m is not well delineated by the global smoothness estimate. The weighted smoothness option produced a reasonable estimate of the discontinuity at x 3 m and of the gradient area g1 in Figure 5a , but introduced large spurious discontinuities close to x 13 m
Bouguer anomaly (mGal)

Büyük Menderes Graben, Turkey

The valley of the Büyük Menderes River is located in West Turkey, a region undergoing active deformation, and being affected by several earthquakes caused by extension. A summary of the area geology is given by Paton 1992 . The valley structure consists of a graben 150 km long and between 10 to 20 km wide. We consider only the central part of the graben, located between the cities of Ortaklar on the west and Cubukdag on the east, and take a gravity pro?le X-Y in Figure 6a across the valley. The grabens in this region including the Büyük Menderes are asymmetrical, presenting small antithetic faults. According to Paton 1992 , the principal high-angle fault of Büyük Menderes Graben is located on the northern side. Sari and Salk 2002 , based on borehole data, and assuming a monotonic decrease of the density cona) b) 0 trast with depth according to the hyperbolic law –10 of equation 1, estimate the values of 0.98 g/cm3 –20 Y –30 38° and 2.597 km, respectively, for parameters 0 –40 Cubukdag and . 0 5 10 15 20 25 30 35 40 N SSE Horizontal distance (km) NNW B. Menderes Ortaklar The observed Bouguer anomaly according to B. Menderes Paton, 1992 across pro?le X-Y see location in 0 Karacabu 0 10 km 0.5 Figure 6a is shown in gray dots in Figure 6b. We Olne 27°30’ Rivers 1.0 Major towns assumed an interpretation model consisting of 35 Major faults Minor faults 1.5 juxtaposed prisms with a density contrast deX 28°40’ 2.0 0 5 10 15 20 25 30 35 40 creasing with depth according to the hyperbolic Horizontal distance (km) law of equation 1 with 0.98 g / cm3 and 0 2.597 km. The lower panel in Figure 6b c) d) 0 0 shows the solution of the entropic regularization –10 –10 –20 –20 with 0 1.5 and 1 1.475. We note, in this so–30 –30 lution, the presence of a high-angle fault on the –40 –40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 northern side, as anticipated by the geologic a priSSE SSE NNW Horizontal distance (km) Horizontal distance (km) NNW ori information. The ?tted anomaly is shown by the solid line in Figure 6b upper panel . 0 0 0.5 0.5 Figure 6c shows the solution of the global 1.0 1.0 smoothness estimator with 0.75. We note 1.5 1.5 that, in this case, the estimated relief the lower 2.0 2.0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 panel in Figure 6c does not present any large verHorizontal distance (km) Horizontal distance (km) tical displacement on the southern side, but it does present two subdued faults on the northern Figure 6. Büyük Menderes Graben. a Map displaying the major and minor faults de?nside. The ?tted anomaly is shown by the solid line ing the graben structure after Paton, 1992 . b-d The Bouguer anomaly pro?le along X-Y is shown by the gray dots. Estimated relief solid line and the corresponding ?tted in Figure 6c upper panel . anomaly solid line produced by b the entropic regularization with 0 1.5 and 1 Figure 6d shows the solution of the weighted 1.475, c the global smoothness with 0.75, and d the weighted smoothness with smoothness estimator with s 0.75, r 0.75, 0.000001, and a maximum assumed depth of 1.78 km.
Bouguer anomaly (mGal) Depth (km)
s r

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Depth (km)

Bouguer anomaly (mGal)

Depth (km)

Basement depth entropic regularization 0.000001, and a maximum assumed depth of 1.78 km. Although the obtained solution the lower panel in Figure 6d presents high-angle faults on both sides of the valley, the discontinuity on the south has a larger displacement than the one on the north, in contrast with the geologic information about the graben.

I35

scholarship A. S. Oliveira from Coordena??o do Aperfei?oamento de Pessoal de Nível Superior CAPES , Brazil. Additional support for V. C. F. Barbosa was provided by CNPq grant 471913/2007-3 and by Funda??o Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro FAPERJ grant E-26/100.688/2007 .

CONCLUSIONS
We have presented a gravity inversion method to interpret the basement relief of a sedimentary basin presenting high-angle discontinuities. The solution is stabilized by the minimization of the ?rst-order entropy measure combined with the maximization of the zeroth-order entropy measure of the vector containing the basement depth estimates at discrete points. When applied to the gravity anomaly of a sedimentary basin whose basement consists of step faults separated by nearly horizontal terraces, the present method has a superior performance as compared with the classical global smoothness estimator. The solutions of the proposed method are generally similar to the ones obtained by the weighted smoothness method, but the proposed method does not require the knowledge of the maximum basin depth as the weighted smoothness does. The price paid for this better performance is the extended computational load. The inversion of a gravity pro?le, similar to the one shown in the application to synthetic data, requires about 30 s on a computer with 4 GB of RAM memory and a 3.7 -GHz processor. The present method could be extended to map the magnetic basement of a sedimentary basin.

REFERENCES
Backus, G. E., and F. Gilbert, 1968, The resolving power of gross earth data: Geophysical Journal of the Royal Astronomical Society, 16, 169–205. Barbosa, V. C. F., J. B. C. Silva, and W. E. Medeiros, 1997, Gravity inversion of a basement relief using approximate equality constraints on depth: Geophysics, 62, 1745–1757. ——–, 1999, Gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth: Geophysics, 64, 1429–1437. Chakravarthi, V., and N. Sundararajan, 2007, 3D gravity inversion of basement relief — A depth-dependent density approach: Geophysics, 72, no. 2, I23–I32. Gill, P. E., W. Murray, and M. H. Wright, 1981, Practical optimization: Academic Press. Le?o, J. W. D., P. T. L. Menezes, J. F. Beltr?o, and J. B. C. Silva, 1996, Gravity inversion of basement relief constrained by the knowledge of depth at isolated points: Geophysics, 61, 1702–1714. Litinsky, V. A., 1989, Concept of effective density: Key to gravity determinations for sedimentary basins: Geophysics, 54, 1474–1482. Parker, R. L., 1977, Understanding inverse theory: Annual Review of Earth and Planetary Sciences, 5, 35–64. Paton, S., 1992, Active normal faulting, drainage patterns and sedimentation in southwestern Turkey: Geological Society of London Journal, 149, 1031–1044. Ramos, F. M., H. F. Campos Velho, J. C. Carvalho, and N. J. Ferreira, 1999, Novel approaches on entropic regularization: Inverse Problems, 15, 1139– 1148. Rao, V. C., V. Chakravarthi, and M. L. Raju, 1994, Forward modelling: Gravity anomalies of two-dimensional bodies of arbitrary shape with hyperbolic and parabolic density functions: Computers and Geosciences, 20, 873–880. Sari, C., and M. Salk, 2002, Analysis of gravity anomalies with hyperbolic density contrast: An application to the gravity data of Western Anatolia: Journal of the Balkan Geophysical Society, 5, 87–96. Silva, J. B. C., F. S. Oliveira, V. C. F. Barbosa, and H. F. Campos Velho, 2007, Apparent-density mapping using entropic regularization: Geophysics, 72, no. 4, I51–I60. Telford, W. M., L. P. Geldart, and R. E. Sheriff, 1991, Applied geophysics: Cambridge University Press.

ACKNOWLEDGMENTS
We thank J. W. Peirce for his detailed and encouraging criticism of this paper and other papers as the associate editor of GEOPHYSICS. We also thank two anonymous reviewers for their valuable criticism. The authors received support for this research through fellowships V. C. F. Barbosa and J. B. C. Silva from Conselho Nacional de Desenvolvimento Cientí?co e Tecnológico CNPq and through a

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