GEOPHYSICS, VOL. 72, NO. 4 共JULY-AUGUST 2007兲; P. I51–I60, 13 FIGS. 10.1190/1.2732557
Apparent-density mapping using entropic regularization
João B. C. Silva1, Francisco S. Oliveira2, Valéria C. F. Barbosa3, and Haroldo F. Campos Velho4
INTRODUCTION ABSTRACT
Geologic mapping is of the utmost importance in locating and delineating potential mineral targets. The production of a geologic map includes data collection in the field, aerial and satellite photo interpretation, and geologic interpretation. Therefore, geologic mapping uses information collected mainly at the earth’s surface. However, the elaboration of a reliable geologic model requires information about the structure and composition of deeper zones on the crust by means of geophysical measurements 共Maas et al., 2003兲. The gravity method has been used as an auxiliary tool in geologic mapping to locate and delineate both outcropping and buried geologic units and structures 共Gupta and Grant, 1985; Granser et al., 1989; Keating, 1992兲. Gravity interpretation as an auxiliary tool for geologic mapping consists of estimating the spatial distribution of the density contrasts. This is not a trivial task because gravity data are not sufficient to estimate the distribution of density in the subsurface in a unique and stable way 共Barbosa et al., 2002; Silva et al., 2002兲. Apparent-density mapping is therefore an ill-posed problem in the sense of Hadamard 共1902兲. The most effective way to transform an ill-posed problem into a well-posed one is to incorporate a priori information about the anomalous sources through optimizing a suitable stabilizing functional, subject to fitting the observations within the measurement errors by the computed anomaly produced by the estimated parameters of an assumed interpretation model 共Tikhonov and Arsenin, 1977兲. The choice of the particular functional to be optimized must take into account the geologic setting; the geologic setting determines the geophysical inversion method to be used in the geophysical interpretation. As a result, whatever the method selected to stabilize the solution of the geophysical inverse problem, the solution will be biased heavily toward the a priori information incorporated through the selected functional, subject to the gravity data being fitted within measurement precision. In the past, several authors developed different stabilizing functionals to locate and delineate geologic units and structures. To date,
We present a new apparent-density mapping method on the horizontal plane that combines the minimization of the first-order entropy with the maximization of the zeroth-order entropy of the estimated density contrasts. The interpretation model consists of a grid of vertical, juxtaposed prisms in both horizontal directions. We assume that the top and the bottom of the gravity sources are flat and horizontal and estimate the prisms’ density contrasts. The minimization of the first-order entropy favors solutions presenting sharp borders, and the maximization of the zeroth-order entropy prevents the tendency of the source estimate to become a single prism. Thus, a judicious combination of both constraints may lead to solutions characterized by regions with virtually constant estimated density contrasts separated by sharp discontinuities. We apply our method to synthetic data from simulated intrusive bodies in sediments that present flat and horizontal tops. By comparing our results with those obtained with the smoothness constraint, we show that both methods produce good and equivalent locations of the sources’ central positions. However, the entropic regularization delineates the boundaries of the bodies with greater resolution, even in the case of 100-m-wide bodies separated by a distance as small as 50 m. Both the proposed and the global smoothness constraints are applied to real anomalies from the eastern Alps and from the Matsitama intrusive complex, northeastern Botswana. In the first case, the entropic regularization delineates two sources, with a horizontal and nearly flat top being consistent with the known geologic information. In the second case, both constraints produce virtually the same estimate, indicating, in agreement with results of synthetic tests, that the tops of the sources are neither flat nor horizontal.
Manuscript received by the Editor November 14, 2006; revised manuscript received January 29, 2007; published online May 25, 2007. 1 Universidade Federal do Pará, Dep. Geofísica, CG, Belém, Pará, Brazil. E-mail:
[email protected]. 2 Universidade Federal do Pará, Belém, Pará, Brazil. E-mail:
[email protected]. 3 Observatório Nacional, Gal. José Cristino, São Cristóvão, Rio de Janeiro, Brazil. E-mail:
[email protected]. 4 INPE, LAC, Av. dos Astronautas, São José dos Campos, São Paulo, Brazil. E-mail:
[email protected]. © 2007 Society of Exploration Geophysicists. All rights reserved.
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most geophysicists maximize the smoothness of the spatial distribution of the physical property. In this procedure, the interpreter stabilizes the solution at the expense of a decrease in the source delineation resolution. Barbosa et al. 共2002, Figure 2兲 estimate apparent densities by minimizing the Euclidean norm of the first-order derivative of the density-contrast distribution along the x- and y-directions. They assume the gravity sources have known thicknesses and depths to the top as well as vertical sides and flat and horizontal tops. Ramos and Campos Velho 共1996兲 and Campos Velho and Ramos 共1997兲 have developed a regularizing technique known as minimum first-order entropy combined with the maximum zeroth-order entropy of the estimated spatial distribution of the physical distribution; they apply it to magnetotelluric data. This regularization method is substantially different from classic regularization methods, which consist of maximizing the smoothness of the spatial distribution of the physical properties and its slightly modified versions. Minimizing the first-order entropy leads to solutions presenting more abrupt limits, in this way better delineating the contacts between the estimated geologic units mapped by the proposed technique. The maximization of the zeroth-order entropy prevents the tendency of the estimated source to become an equivalent point source. We present a new apparent-density mapping method using the entropic regularization, defined by minimizing the first-order entropy combined with maximizing the zeroth-order entropy, as Campos Velho and Ramos 共1997兲 propose. We apply the method to estimate the spatial distribution of the density contrast as a function of the horizontal coordinates only. We assume the sources are homogeneous with flat and horizontal tops and bottoms and that the interpreter knows these depths beforehand. Minimization of the first-order entropy measure combined with maximization of the zeroth-order entropy measure restricts the solutions to a class of models comprising locally homogeneous regions confined by sharp bounds and embedded in host rocks, which are also mainly homogeneous, allowing a better delineation of the horizontal limits of the sources compared with the method that incorporates the smoothness constraint. We apply the proposed method to synthetic data produced by two groups of sources: those that satisfy the assumption of sources with flat and horizontal tops and bottoms and those that do not. In the first Observations y
Elementary cell x
case, the estimated density-contrast distribution delineates the sources with sharper borders and presents values closer to a constant value over the presumably homogeneous anomalous sources compared with the smoothness constraint inversion. In the second case, the estimates produced by both methods are poor and present no significant difference. We also apply the entropic regularization method to gravity data from the eastern Alps and Matsitama 共Botswana兲 regions. In the eastern Alps, the entropic regularization produced density-contrast estimates locally closer to constant values over the mapped anomalous sources, which is different from the smoothness constraint method. This indicates that at least portions of the anomalous source present approximately flat and horizontal tops and bottom and approximately constant density contrast. Conversely, in Matsitama, both methods present results very close to each other, indicating the source probably presents neither flat nor horizontal tops.
METHODOLOGY Consider a set of homogeneous gravity sources. We assume that the top and bottom of all sources are flat and horizontal and that their depths are known 共Figure 1兲. We approximate these sources by an interpretation model consisting of a grid of 3D rectangular, juxtaposed prisms. We presume that the grid encloses all anomalous sources. We assume also that each prism has a constant density contrast and that the tops and bottoms of all prisms are at the same depth, coinciding, respectively, with the known tops and bottoms of the true source 共Figure 1兲. From a set of N gravity anomaly observations g0共p兲 ⬅ 关g01, . . . ,gN0 兴T, we estimate a vector p of density contrasts of each prismatic cell of the interpretation model. If the information contained in the data were sufficient to estimate the parameters, we might obtain estimates of p by the minimization of
储g0 − g共p兲储2 ,
共1兲
where g共p兲 is an N-dimensional vector containing the computed anomaly and 储 . 储 is the Euclidean norm. Obtaining the minimizers of the functional given in equation 1 is an ill-posed problem because the solution of this problem is unstable; therefore, it is necessary to incorporate additional a priori information about the sources to transform this problem into a well-posed one. Traditionally, the a priori information incorporated in most geophysical inverse problems is the global smoothness of the spatial distribution of the physical property, which consists of requiring that the estimate of each parameter pˆi 共density contrast of the ith cell兲 be as close as possible to the estimate of parameter pˆ j 共density-contrast estimate of a neighboring cell either in the x- or the y-direction兲, subject to the observations being explained by the response of the interpretation model. Mathematically, we have
min兵pT RT Rp其,
共2兲
储g0 − g共p兲储2 = ␦ ,
共3兲
p
subject to z Gravity sources
Figure 1. Gravity sources, observations layout, and interpretation model consisting of a set of rectangular, 3D juxtaposed prisms with flat top and bottom.
where ␦ is associated with the experimental error, T is the transposition operator, and R is a matrix representing the first-order difference operator, whose lines present just two nonnull elements 共1 and ⫺1兲 at the columns associated with adjacent parameters pˆi and pˆ j.
Gravity entropic regularization We solve this problem by minimizing the functional
共p兲 = 储g0 − g共p兲储2 + pTWp,
共4兲
where W = R R, and is a nonnegative regularization parameter, selected according to a criterion described in the section on choosing ␥0 and ␥1. Instead of the overall smoothness regularization, we use a new regularization method called entropic regularization 共Campos Velho and Ramos, 1997; Ramos et al., 1999兲. It combines the minimization of the first-order entropy measure with the maximization of the zeroth-order entropy measure of p. We incorporate the maximum zeroth-order entropy constraint in the inverse problem by optimizing a functional Q0共p兲, based on the maximum entropy principle. The maximum entropy, as a criterion of inference, is proposed by Jaynes 共1957兲 using the concept of information entropy introduced by Shannon 共Shannon and Weaver, 1949兲. The minimum entropy method minimizes the entropy measure Q1共p兲 of the vector of first-order derivatives of p 共Campos Velho and Ramos, 1997; Ramos et al., 1999兲. We formulate the entropic regularization as follows:
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A兲. The reasons for using a quasi-Newton method are the nonlinearity of 共p兲 with respect to p and the nonexistence of simple analytic expressions for the Hessians of Q0共p兲 and Q1共p兲.
T
maximize
冋 册 Q0共p兲 Q0max
and minimize
冋 册
Q1共p兲 , Q1max
共5兲
Physical and geologic meaning of the entropic regularization The zeroth-order entropy measure 共equation 7 with ␣ = 0兲 may incorporate either the smoothness or the discontinuity constraint in the density-contrast-distribution mapping. The maximum and minimum of Q0 occur, respectively, when all Sk are equal and when just one of the Sk is different from zero, that is, when all prisms of the interpretation model have the same density-contrast estimate and when all but one of the prisms have a density-contrast estimate equal to zero. To understand the behavior of functional Q1, we combine equations 7–9 using ␣ = 1 and obtain M−1
Q1共p兲 = −
兺 k=1
冢
兩 pˆk + 1 − pˆk兩
兺
冣冢 log
M−1
兩 pˆi + 1 − pˆi兩
i=1
兩 pˆk + 1 − pˆk兩 M−1
兺
兩 pˆi + 1 − pˆi兩
i=1
subject to
储g0 − g共p兲储2 = ␦ ,
共6兲
Q␣共p兲 = −
兺
Sk log共Sk兲,
D
␣ = 0 or 1,
共7兲
k=1
are entropy measures of zeroth-order if ␣ = 0 and first-order if ␣ = 1, with
rk
Sk =
L
共8兲
,
兺 ri
i=1
where
rk =
再
pˆk +
if ␣ = 0
兩pˆk+1 − pˆk兩 + ⑀ if ␣ = 1
.
共9兲
Integer L is equal to the number of unknown parameters M if ␣ is equal to 0 or is equal to M − 1 if ␣ is equal to 1, and is a small positive constant 共smaller than 10−8兲 that guarantees the entropy measures will always be defined. We solve the constrained optimization problem defined in equations 5 and 6 by minimizing the functional
共p兲 = 储g0 − g共p兲储2 −
␥0Q0共p兲 ␥1Q1共p兲 + , Q0max Q1max
共10兲
where ␥0 and ␥1 are positive numbers called regularizing parameters. Note that the negative sign imposed to ␥0 leads to a maximization of the zeroth-order entropy measure. We discuss the choice of values for ␥0 and ␥1 later in this article. We minimize functional 共p兲 via a quasi-Newton method 共Gill et al., 1981兲, using the Broyden-Fletcher-Goldfarb-Shanno implementation to compute the Hessian update at each iteration 共see Appendix
共11兲
Now assume that nonnull differences 兩pˆk + 1 − pˆk兩 are approximately constant and equal to c. Then equation 11 becomes
where Q0max and Q1max are normalizing constants, L
冣
.
Q1共p兲 = −
c
冉 冊
兺1 Dc log
c = log共D兲, Dc
共12兲
where D is the number of adjacent cells for which 兩pˆk + 1 − pˆk兩 ⫽ 0, which corresponds to the number of discontinuities between adjacent estimates. Equation 12 shows that as the nonnull differences 兩pˆk + 1 − pˆk兩 approach a constant value, the minimization of Q1 tends to minimize the number of discontinuities in the apparent-densitycontrast distribution. This tendency evolves gradually throughout the iterative process. From the properties of Q0 and Q1, we infer that minimizing Q1, subject to an acceptable anomaly misfit, tends to concentrate the density-contrast estimates either about zero or about a nonzero constant value, as shown in the simulated 1D apparent-density-contrast mappings of Figure 2a-c. For this sequence of density-contrast mappings, minimizing Q1 implies minimizing Q0 also, as seen in Figure 2g. However, distributions M3–M6 共Figure 2c-f兲 show that a very small decrease in Q1 implies a considerable decrease in Q0 共see Figure 2g兲. For shallow sources, the gravity anomaly has enough resolution to deter the minimization of Q0 and, consequently, to prevent the estimation of unrealistic, minimum-volume sources; so the minimization of Q1 subject to an acceptable misfit is just enough to produce sharply bounded solutions. For deep sources, however, it is necessary to prevent any spurious minimization of Q0 by assigning a value different from zero to ␥0. Because the transition between shallow and deep sources is gradual and hard to pinpoint, the maximization of Q0 is always recommended, particularly when the minimization of Q1 alone does not produce the expected results. Finally, note that because the minimization of the first-order entropy measure tends to minimize the number of discontinuities in the estimated density distribution, the resulting solution will tend to favor solutions presenting no holes in its interior. This occurs because the presence of holes substantially increases the number of disconti-
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nuities 共and consequently the first-order entropy measure兲. In this way, the entropic regularization favors solutions displaying compact sources.
Choice of parameters ␥0 and ␥1 We select parameters 共equation 4兲, ␥0, and ␥1 共equation 10兲 in the following way. Parameter controls the solution stability, which is obtained at the expense of a decrease in solution resolution. As a result, it must be the smallest positive value still producing stable solutions. Values of larger than this optimum value increase the solution smoothness beyond that necessary to produce a stable estimate of the sources limits, whereas smaller values produce unstable solutions. We characterize an unstable solution by perturbing the observations with different sequences of pseudorandom noise realizations for fixed value and obtaining the corresponding solutions, which we consider to be stable if they are sufficiently close to each other, according to an established criterion. Conversely, ␥1 allows discontinuities in the solution 共physical properties兲. A small value of ␥1 produces a solution with undefined borders in the same way that the global smoothness method does, whereas a large value of ␥1 produces solutions exhibiting spurious
M1
0
6 10 15 Horizontal distance (km)
20
M3
0
6 10 15 Horizontal distance (km)
20
M2
0
6 10 15 Horizontal distance (km)
20
1.0 0.8 0.6 0.4 0.2 0.0
Stopping criterion We interrupt the iterative process according to the following mathematical criterion. Consider the numerical values of the objective function 共p兲 along the iterations 共Figure 3兲. A measure of the decay rate of 共p兲 at the kth iteration is dk共1兲 = 共pˆk兲 − 共pˆk−1兲. The decrease of this measure along successive iterations indicates the presence of a flat area in the behavior of 共p兲 displayed in Figure 3. We detect the right-hand-side limit of this flat area by an increase in the absolute value of dk共1兲. We interrupt the iterative process at the iteration k, where the first 共1兲 − dk共1兲 from positive to negative occurs. change of signal of dk共2兲 ⬅ dk+1 This mathematical criterion aims at stopping the iteration at the end of the first flat area of the behavior of function 共p兲 along the iteration. We select the first flat area because the solutions associated with subsequent flat areas tend to produce smaller values of Q1共p兲 and,
M4 5.5
0
6 10 15 Horizontal distance (km)
20 5.0
f) 1.0 0.8 0.6 0.4 0.2 0.0
M5
0
6 10 15 Horizontal distance (km)
20
1.0
M6
0.8
4.5
0.6 0.4
τ (p)
e) Density contrast (g/cm3)
1.0 0.8 0.6 0.4 0.2 0.0
d) 1.0 0.8 0.6 0.4 0.2 0.0
Density contrast (g/cm3)
Density contrast (g/cm3)
c)
Density contrast (g/cm3)
b) 1.0 0.8 0.6 0.4 0.2 0.0
Density contrast (g/cm3)
Density contrast (g/cm3)
a)
oscillations and discontinuities in the estimated density contrasts of the cells. These oscillations are not related to instability; rather, they characterize attempts to estimate sharp source limits. Their presence demonstrates that we have overestimated the number of discontinuities in the physical property. In this way, ␥1 must be the largest positive value leading to no more oscillations or discontinuities than those expected for the geologic body being interpreted. As shown in the previous section, the entropy measures Q0 and Q1 are not independent from each other, and the minimization of Q1 implies, under certain circumstances, the minimization of Q0, as well. As a result, we must assign provisionally to ␥0 a small value 共including zero兲. 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.
0.2 0.0
0
6 10 15 Horizontal distance (km)
20
4.0
g) Entropies Q0 and Q1
3 3.5 2
1
Q0 Q1
3.0 0
4
8
12
16
20
Iteration 0
M1 M2 M3 M4 M5 M6 Apparent-density-contrast distribution
Figure 2. One-dimensional density-contrast distributions M1–M6 共a–f兲 and the respective entropy measures of orders zero and one 共g兲.
Figure 3. Schematic decay of the objective function 共p兲 along the iterations. The iterative process has been interrupted at the thirteenth iteration 共vertical arrow兲, corresponding to the right-hand-side limit of the first flat area in the evaluation of 共p兲.
Gravity entropic regularization consequently, smaller values of Q0共p兲. As a result, these solutions present more discontinuities or smaller sources than could be expected for the geologic bodies we are interpreting.
I55
way. We can verify this result by comparing the horizontal projection of the true body with the gradient of the contour lines of the density-contrast distribution estimated by both methods. In addition, the entropic regularization provides a better characterization of the sources’homogeneity, indicated by the flat area occurring in the estimated density contrast, close to 0.3 g/cm3 共see the perspective view of Figure 4d兲. The entropic regularization has a superior performance in this test because it concentrates the nonnull estimated density contrast in a smaller number of blocks, permitting a larger number of cells with estimated density contrasts close to zero between the two sources, and leading to an estimate with better resolution of the lateral limits of the sources. Conversely, the global smoothness presents a limited resolution in the region between the sources. We can visualize the gain in resolution obtained with the entropic regularization by the development of a deeper trough separating the density-contrast estimates of each body 共Figure 4d兲 compared with the corresponding estimates of the density contrast produced by the global smoothness 共Figure 4b兲.
APPLICATION TO SYNTHETIC DATA In this section, we compare the solutions of the inverse gravity problem as a function of the x- and y-variables using the classic global smoothness method 共minimizer of equation 4 with ⫽ 0兲 with the solutions obtained with the entropic regularization 共minimizer of equation 10 with ␥1 ⫽ 0 and ␥0 = 0 or ␥0 ⫽ 0兲. The interpretation model consists of an nx ⫻ ny grid of rectangular vertical prisms with dimensions dx and dy, juxtaposed along the x- and y-directions, respectively. The synthetic models simulate intrusive rocks in sediments or metasediments, presenting a constant density contrast. We present two tests with synthetic data, simulating the gravity anomaly produced by the intrusion of a granitic stock into sedimentary rocks. In the first test, the simulated source honors the condition that the source must have a flat and horizontal top; in the second test, it does not.
Source with flat bottom and irregular top Consider an isolated prismatic body presenting a plane and horizontal bottom located at 4 km depth. The source’s top presents a variable depth with an overall downward concavity 共Figure 6兲. This body simulates an igneous intrusion with uniform density contrast of
Source with flat top and flat bottom
x (km)
0.25
0.25
x (km)
0.10 0.15 0.25
0.50
0.10 0.15 0.25
Figure 4a shows the Bouguer anomaly 共solid green line兲 produced by two prismatic bodies with horizontal sections shown in Figure 4b in thick white lines; tops and bottom are at 0.0105 and 0.2105 km, respectively, and density contrast a) c) is 0.3 g/cm3. We corrupt the theoretical anomaly 0.35 0.35 0.65 with zero-mean Gaussian pseudorandom noise 0.30 0.30 0.25 0.25 with a standard deviation of 0.01 mGal. The in0.20 0.20 terpretation model consists of a 16⫻ 24 grid of 0.15 0.15 0.70 0.65 vertical prisms in the x- and y-directions, respec0.10 0.10 0.25 0.25 tively, with the same horizontal dimensions of 0.05 0.05 0.025 km. We impose that the top and base of the 0.05 0.15 0.25 0.35 0.45 0.55 0.05 0.15 0.25 0.35 0.45 0.55 prisms coincide with the corresponding top and y (km) y (km) base of the simulated anomalous sources and presume their depths are known. b) d) Figure 4b displays a contour map and a perg/cm3 Density Density spective view of the estimated density-contrast contrast contrast distribution, stabilized with the global smooth0.32 (g/cm3) (g/cm3) ness using = 0.02. The fitted anomaly appears 0.28 0.20 0.20 in Figure 4a in dashed black lines. Figure 4c pre0.00 0.00 0.20 sents the same anomaly 共in solid green lines兲 as Figure 4a. Figure 4d shows in a contour map and 0.14 in perspective the estimated distribution of the 0.08 density contrast using the entropic regularization with ␥0 = 1.2 and ␥1 = 2.1. The fitted anomaly 0.50 0.50 0.02 appears in Figure 4c as dashed black lines. Figure 0.35 0.35 –0.04 5 shows the variation of the objective function y (km) y (km) 0.20 0.20 共p兲 along the iterations. We indicate the stop0.05 0.05 0.35 0.25 0.35 0.25 ping iteration by a vertical arrow. Note that the 0.15 0.05 0.15 0.05 stopping iteration locates an inflection of 共p兲 x (km) x (km) with a flat area before it. The results of Figure 4b and d show that both Figure 4. Test with synthetic data produced by two flat-topped sources. 共a兲 Observed 共solid green lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the global the global smoothness and the entropic regularsmoothness. 共b兲 Density-contrast estimate via global smoothness with = 0.02. 共c兲 Obization recover the positions of the center of the served 共solid green lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the ensource, its overall shapes, and its density contropic regularization. 共d兲 Density-contrast estimate via entropic regularization with ␥0 trasts. However, the entropic regularization de= 1.2 and ␥1 = 2.1. The solid white lines represent the outlines in plan view of the prislineates the abrupt sources’ borders in a better matic sources. 0 0.2 0 0.1
0 0.2 0 0.1
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0.3 g/cm3 relative to the circumjacent sedimentary rocks. Figure 7a and c shows in solid green lines the Bouguer anomaly produced by this simulated igneous intrusion. We corrupt the theoretical anomaly with zero-mean Gaussian pseudorandom noise with a standard deviation of 0.1 mGal. The interpretation model consists of a 20⫻ 25 grid of vertical prisms in the x- and y-directions, respectively, with the same horizontal dimensions of 2 km and with tops and bottoms located at depths of 0.1 and 4 km, respectively. Figure 7b and d displays the contour maps and perspective views of the estimated density-contrast distributions, stabilized, respectively, with the global smoothness using = 0.02 and with the entropic regularization using ␥0 = 0 and ␥1 = 3.0. The corresponding
fitted anomalies appear in Figure 7a and c as dashed black lines. The most striking feature of Figure 7b and d is that neither the global smoothness nor the entropic regularization presents noticeable difference with respect to the source delineation. This is a consequence of the irregular top presented by the true source. Also note that the density-contrast estimates in both methods present variable values, even though the maximum estimated density contrast is about the true value of 0.3 g/cm3 共see the perspective views of Figure 7b and d兲. This is important in a practical problem. Similar estimates produced by the global smoothness and the entropic regularization may show that the source does not present a flat and horizontal top.
APPLICATION TO REAL ANOMALIES
5.2
To evaluate the performance of the new apparent-density mapping method proposed in this paper, we apply it to two data sets of real gravity anomalies. The first is from the eastern Alps, and the second is from the region of Matsitama, northeastern Botswana.
4.8
Eastern Alps 4.4
τ (p)
Geologic setting
4.0
3.6
3.2 0
4
8
12
16
Iteration
Figure 5. Test with synthetic data produced by a source with a flat top. Decay of the objective function 共p兲 along the iterations. We interrupted the iterative process at the tenth iteration 共vertical arrow兲. 40
3.5
30
2
x (km)
2.5
20 1.0 1.5 0.1 10 3.0
The Alps encompass an extreme rock variety. They consist predominantly of sedimentary and metamorphic rocks with minor occurrences of volcanic rocks. Figure 8 shows an overview of the major rock zones occurring in the eastern Alps 共Krainer, 2002兲. The Northern Calcareous Alps zone consists mainly of Permian to Lower Tertiary limestones and dolomites, with minor deposits of sandstones, conglomerates, and breccias. The Drau Range contains the Gailtal crystalline basement rocks in addition to a sedimentary sequence similar to that of the Northern Calcareous Alps. The Permomesozoic Radstätter Tauern, at the southern rim of the northern graywacke zone, contains quartz phyllite and Mesozoic sedimentary rocks with some degree of metamorphic overprinting. The Engadin Window is one of the few places in the eastern Alps where the Penninic zone emerges through tectonic windows in the Austroalpine nappes. It consists of very old metamorphic rocks intruded by granitic plutons and Permian to Lower Cretaceous volcanosedimentary rocks deposited in the Penninic Ocean 共a shallow sea that formed the southern shelf of the European continent at the time of the breakup of Pangaea兲. The Gurktal Nappe is made up of an Old Palaeozoic group of sedimentary rocks with metamorphic overprinting consisting of slates, quartzites, siliceous schists, and limestones and of Permomesozoic sequences of conglomerates, sandstones, and carbonate deposits. According to Schuster and Kurz 共2005兲, within the Gurktal Nappe, the metamorphic grade decreases upward from amphibolite facies at the base to diagenetic conditions at the top of the nappe pile. The Palaeozoic Graz Group consists of marine sedimentary rocks, mainly slates and fossil-bearing limestones, deposited between the Ordovician and the Carboniferous and locally intercalated with volcanic rocks. The crystalline basement rocks, generally referred to as Altkristallin, are made up mainly by gneisses and mica schists.
Apparent-density mapping results
0 0
10
20
30
40
50
y (km)
Figure 6. Contoured depths 共in kilometers兲 to the top of the prismatic source producing the anomaly displayed in Figure 7a and c.
Figure 9 shows the Bouguer anomaly from an area in the eastern Alps after subtracting a regional component, presumably caused by irregularities on the Moho topography 共Granser et al., 1989兲. Granser et al. 共1989兲 point out the correlation of the residual gravity
Gravity entropic regularization
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the x- and y-directions, respectively, and tops and bottoms located at depths of 4 and 15 km, respectively. We choose the depths to the top and bottom of the elementary prisms to allow an acceptable fit to the observed anomaly. Shallower depths to the top produce fitted anomalies with steeper gradients than the observed anomaly. Figure 10b and d displays contour maps and perspective views of the estimated density-contrast distributions using, respectively, the global smoothness with a) c) = 1 and the entropic regularization with ␥0 = 30 and ␥1 = 30. Figure 10a and c shows the observed 35 35 and corresponding fitted anomalies in solid green and dashed black lines, respectively. Compared 25 25 10 10 with the global smoothness, the entropic regular20 20 30 30 ization produces a sharper image of the apparent15 15 20 20 density mapping, denoted in Figure 10b and d by 10 10 more abrupt gradients, by the presence of sharper 5 5 and more linear borders and by the occurrence of 5 15 25 35 45 15 25 35 45 5 flatter areas in the map, particularly close to both y (km) y (km) maxima and at the valley between them. Figures 8 and 9 show that the positive and negb) d) ative gravity anomalies correlate with the surface Density Density expression of the Gurktal Nappe and of the cryscontrast contrast g/cm3 3 (g/cm ) (g/cm3) talline basement rocks, respectively. Granser et 0.20 0.20 al. 共1989兲 report a total apparent density contrast 0.34 0.00 0.00 between these areas of approximately 0.15 g/cm3 0.28 by assuming that the upper crust has a thickness 0.22 of 10 km. These estimated density contrasts agree with reported rock sample density measure0.16 ments 共Granser et al., 1989兲. We assume a similar 0.10 45 45 thickness 共11 km兲 but place the sources’ tops at 35 35 35 35 4 km to obtain an acceptable anomaly fit. As a re25 25 25 25 0.04 y (km) y (km) 15 15 15 15 sult, we obtain a total apparent density contrast of 5 5 5 5 x (km) x (km) 0.02 approximately 0.45 g/cm3 共Figure 10a and b兲. This estimated density contrast might appear Figure 7. Test with synthetic data from a source with an irregular top. 共a兲 Observed 共solid overestimated when considering the density meagreen lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the global smoothsurements reported by Granser et al. 共1989兲. ness. 共b兲 Density-contrast estimate via global smoothness with = 0.02. 共c兲 Observed However, according to Schuster and Kurz 共2005兲, 共solid green lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the entropic regularization. 共d兲 Density-contrast estimate via entropic regularization with ␥0 = 0 and at the base of Gurktal Nappe, the metamorphic ␥1 = 3.0. grade corresponds to amphibolite facies, decreasing to diagenetic conditions at the top. Considering that amphibolites may have densities as high as 3.20 g/cm3, the obtained contrast of 0.45 g/cm3 may be consistent with the premise that the observed anomalies are caused by the density contrast between amphibolites at the base of the Gurktal Nappe and gneisses 47° N and mica schists of the crystalline basement. Finally, note that the 10
x (km)
10
x (km)
anomalies with the major surface geologic units in the eastern Alps. They obtain an apparent-density map where the gravity high presents an estimated density between 2.75 and 2.80 g/cm3 and the gravity low leads to an apparent density between 2.625 and 2.675 g/cm3. We assume an interpretation model consisting of a 20⫻ 20 grid of vertical prisms with horizontal dimensions of 0.87 and 0.78 km in
28 22
8 –2
8 14° E
15° E
Paleozoic of Graz Northern calcareous Alps, Drau Range Southern Alps Crystalline basement rocks
Major faults
Figure 8. Simplified geologic map of part of the eastern Alps. After Krainer 共2002兲.
47° N
16
–12
18
10 8
Engadin Window Radstatter Tauern Tertiary basins Gurktal Nappe
–2
4
–2
–2 –8 14° E
15° E
–14 mGal
Figure 9. Bouguer anomaly of a selected area in the eastern Alps, after removing a regional component. After Granser et al. 共1989兲.
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Silva et al. negative anomaly, which correlates with the crystalline rocks, leads to a flat-bottom valley in the density-contrast map 共Figure 10d兲, whereas the positive anomaly, correlated with the Gurktal Nappe, leads to a hill with more irregular top, presenting, just locally, areas with flat tops. This difference may be related to the greater variety of rocks and structures in the Gurktal Nappe compared with the crystalline rocks.
c) 2
14
0 –1
14
0 –1 10
10
14
26
47° N
2
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47° N
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2
a)
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2
14° E
15° E
14° E
b)
15° E
d) Density contrast (g/cm3)
Density contrast (g/cm3)
g/cm3
0.2 0.0
0.2 0.0
Matsitama igneous complex, Botswana
0.32 0.24
Geologic setting
Botswana is located in an area of high mineral potential where the geologic mapping is a diffi0.08 cult task because of a thin layer of sedimentary 0 0 rocks 共mainly sandstones兲 deriving from the 0.00 0.12 Kalahari and Karoo units. A few geologic cam0.12 –0.08 0 paigns have indicated the existence of buried Pre0.12 0 47° N cambrian metamorphic terrains. In this kind of 47° N –0.16 15° E 0.12 0 15° E 0.04 terrain around the world, geophysicists have discovered economically important deposits of base 14° E 14° E metal in volcanosedimentary rocks of the greenstone type 共Reeves, 1985兲. Figure 10. Eastern Alps. 共a兲 Observed 共solid green lines兲 and fitted 共dashed black lines兲 Figure 11 is a geologic map of northeastern Bouguer anomalies using the global smoothness. 共b兲 Density-contrast estimate via global smoothness with = 1. 共c兲 Observed 共solid green lines兲 and fitted 共dashed black lines兲 Botswana; Figure 12 is the corresponding map of Bouguer anomalies using the entropic regularization. 共d兲 Density-contrast estimate via the Bouguer anomaly in the same scale, indicatentropic regularization with ␥0 = 30 and ␥1 = 30. ing that the high percentage of mafic minerals in the greenstone belts produces a positive gravity Kalahari cover 20° S anomaly. In this figure, the most extensive greenstone belt from Karoo cover Granite-gneiss Matsitama 共B兲 becomes progressively masked to the west because of Greenstones the overlying Kalahari and Karoo sediments. The positive gravity anomaly over this complex indicates that dense rocks extend themselves westward. In this area, a shallow borehole directed to coal exploration in the sedimentary cover confirms the presence of PrecamB 21° S brian ultramafic rocks. –0.04
Apparent-density mapping results
50 km
25° E
0.2
0.04
–0.04
0.16
26° E
27° E
Figure 11. Simplified geologic map of Matsitama, northeastern Botswana, containing igneous rocks 共B兲 and its extension, inferred 共open circles兲 by the inspection of the Bouguer anomaly map shown in Figure 12. After Reeves 共1985兲. –110 –120
–100
–100
–90
–90
–120
–110
–100
–90
B –70
–110 –110
–100
–80
–100
–90 –90
50 km –120
Figure 12. Matsitama. Bouguer gravity anomaly of the area shown in Figure 11. Contour intervals of 10 mGal. After Reeves 共1985兲. The dashed rectangle indicates the study area.
Figure 13a and c shows with a solid green line the observed Bouguer anomaly in the study area 共dashed rectangle in Figure 12兲 presumably produced by an intrusive body that, according to information from a borehole, is located at a few tens of meters 共locally, the borehole intercepted the body at a depth of about 100 m兲. We assume that the depth to the bottom, required by our proposed method, is 4 km because this value produces estimates for the density-contrast distribution of about 0.3 g/cm3, which is in accordance with a density contrast between greenstone rocks and sediments or granites. The interpretation model consists of a 46⫻ 36 grid of vertical prisms in the x- and y-directions, respectively with the same horizontal dimensions of 3.85 km and with tops and bottoms located at depths of 0.1 and 4 km, respectively. Figure 13b and d displays contour maps and perspective views of the estimated density-contrast distributions using, respectively, the global smoothness with = 0.01 and the entropic regularization with ␥0 = 0.001 and ␥1 = 3.0. Figure 13a and c shows the corresponding fitted anomalies in dashed black lines. Figure 13b and d shows that both methods produce similar estimates of the densitycontrast distribution. On the grounds of similar results obtained with
Gravity entropic regularization
a)
Figure 13. Matsitama. 共a兲 Observed 共solid green lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the global smoothness. 共b兲 Density-contrast estimate via global smoothness with = 0.01. 共c兲 Observed 共solid green lines兲 and fitted 共dashed black lines兲 Bouguer anomalies using the entropic regularization. 共d兲 Density-contrast estimate via entropic regularization with ␥0 = 0.01 and ␥1 = 3.0.
c) 160
160 20
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20
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20
20 20
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100
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100 80
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b)
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d) Density contrast (g/cm3)
Density contrast (g/cm3) 0.2
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g/cm3 0.38 0.30
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4 0.1 100 80 60 40 20 x (km)
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120 100 80 60 40 20 y (km)
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synthetic data, we infer that the top of the source of the Matsitama anomaly is not flat. Available geologic information supports this conclusion. At point B on Figure 12, the source either outcrops or is very close to the earth’s surface, whereas along the circle path 共Figure 12兲, the greenstone rocks are covered by sediments; therefore, its top probably is neither flat nor horizontal.
CONCLUSIONS We have presented a new apparent-density-contrast mapping method whose novelty consists of combining the maximum zerothorder entropy and minimum first-order entropy measures, favoring an estimated density-contrast distribution presenting locally smooth regions separated by abrupt discontinuities. This judicious combination of entropy measures is necessary to prevent the solution’s horizontal dimensions from being reduced to nonrealistic values. Compared with a standard global smoothness method, our method has the disadvantage of computing the apparent-density-contrast map through the solution of a nonlinear optimization problem, which requires an iterative algorithm and demands a larger computational load. However, the results produced by the entropic regularization method are superior to those produced by the global smoothness in the case of geologic units separated by abrupt contacts. Tests with synthetic data, produced by simulated intrusive bodies into sedimentary or metamorphic rocks, demonstrate this advantage. When the sources present flat tops, both the global smoothness and the entropic regularization methods determine the position of the center of the sources equally well, but the proposed method has a superior performance in delineating the sources limits. If the true sources do not
120 100 80 60 40 20 y (km)
0.12 0.06 0.00 –0.06
present flat tops, neither method is able to define a sharp boundary for the source. In this case, they produce virtually the same result. Applications to the real data from the eastern Alps and from intrusive rocks in Matsitama, Botswana, have confirmed these conclusions. In the eastern Alps, the results produced by both methods were substantially different, in contrast to the intrusive body in Matsitama 共known to have an irregular top兲, which produced virtually the same result when interpreted with the global smoothness and the entropic regularization methods. The stopping criterion of the iterative entropic regularization method uses the discrete second difference of the objective function along the iterations. The iterative process stops when the objective function evaluation, along the iterations, reaches the limit of the first approximately flat and horizontal region. The position where the discrete second difference of the objective function along the iterations changes its sign mathematically determines this limit. We developed this criterion in our work to be applied to minimizing an objective function involving entropy measures. It differs substantially from the usual criterion employed to minimize the nonlinear functionals commonly used in geophysical inversion. We can extend this method in a straightforward way to apparent magnetization mapping and to the interpretation of a discontinuous relief of the basement of a sedimentary basin.
ACKNOWLEDGMENTS Conselho Nacional de Desenvolvimento Científico e Tecnológico 共CNPq兲, Brazil, supported the authors in this research. CNPq provided additional support for J. B. C. Silva and V. C. F. Barbosa under
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contracts 504419/2004-8, 505265/2004-4, and 471168/2004-1. FAPERJ 共contract E-26/170.733/2004兲 and CNPq 共contract 474878/2006-6兲 supported V. C. F. Barbosa, and FAPESP supported H. F. Campos Velho. We thank Steven Arcone, Xiong Li, and reviewers Pierre Keating and Olivier Boulanger for helpful comments on this manuscript. We greatly appreciate constructive and thoughtful comments of Colin Farquharson. We thank Chrysa Cullather for improving the readability.
6兲
where bk = pˆk + 1 − pˆk and uk = ⵜp兵 共p兲其兩p = pˆk+1 − ⵜp兵 共p兲其兩p = pˆk. Test for algorithm convergence following the stopping criterion described in the “Methodology” section. If convergence is achieved, the iteration is stopped; otherwise increment k and go to step 1.
We initialize the iteration by making pˆ o equal to the global smoothness solution and H0 equal to the identity matrix I.
APPENDIX A
REFERENCES
QUASI-NEWTON OPTIMIZATION ALGORITHM BY UPDATING THE HESSIAN MATRIX VIA THE BROYDEN-FLETCHER-GOLDFARB-SHANNO METHOD
Barbosa, V. C. F., J. B. C. Silva, and W. E. Medeiros, 2002, Practical applications of uniqueness theorems in gravimetry: Part II — Pragmatic incorporation of concrete geologic information: Geophysics, 67, 795–800. Campos Velho, H. F., and F. M. Ramos, 1997, Numerical inversion of twodimensional geoelectric conductivity distributions from electromagnetic ground data: Revista Brasileira de Geofísica, 15, 133–144. Gill, P. E., W. Murray, and M. H. Wright, 1981, Practical optimization: Academic Press Inc. Granser, H., B. Meurers, and P. Steinhauser, 1989, Apparent density mapping and 3D gravity inversion in the eastern Alps: Geophysical Prospecting, 37, 279–292. Gupta, V. K., and F. S. Grant, 1985, Mineral-exploration aspects of gravity and aeromagnetic surveys in the Sudbury-Cobalt area, Ontario, in W. J. Hinze, ed. The utility of regional gravity and magnetic anomaly maps: SEG, 393–412. Hadamard, J., 1902, Sur les problèmes aux dérivées partielles et leur signification physique: Bulletin of Princeton University, 13, 1–20. Jaynes, E. T., 1957, Information theory and statistical mechanics: Physical Review, 106, 620–630. Keating, P., 1992, Density mapping from gravity data using the Walsh transform: Geophysics, 57, 637–642. Krainer, K., 2002, The Alps — Structural history of a mountain chain, in I. Ackerl, ed. The wonderful world of mountains: TheAustrianAlps: Federal Press Service, 10–22. Maas, M. V. R., C. G. Oliveira, A. C. B. Pires, and R. A. V. Moraes, 2003, Airborne geophysics applied to mineral exploration and geologic mapping in the southwest sector of Orós-Jaguaribe copper belt, northeastern Brazil: Revista Brasileira de Geociências 共in Portuguese兲, 33, 279–288. Marquardt, D. W., 1963, An algorithm for least-squares estimation of nonlinear parameters: Journal of the Society of Industrial and Applied Mathematics, 2, 601–612. Ramos, F. M., and H. F. Campos Velho, 1996, Reconstruction of geoelectric conductivity distributions using a minimum first-order entropy technique: 2nd International Conference on Inverse Problems on Engineering, Proceedings, 199–206. 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. Reeves, C. V., 1985, The Kalahari Desert, central southern Africa: A case history of regional gravity and magnetic exploration, in W. J. Hinze, eds., The utility of regional gravity and magnetic anomaly maps: SEG, 144–153. Schuster, R., and W. Kurz, 2005, Eclogites in the eastern alps: High-pressure metamorphism in the context of the alpine orogeny: Mitteilungsblatt der Österreichische Mineralogische Gesellschaft, 150, 173–188. Shannon, C. E., and W. Weaver, 1949, The mathematical theory of communication: University. of Illinois Press. Silva, J. B. C., W. E. Medeiros, and V. C. F. Barbosa, 2002, Practical applications of uniqueness theorems in gravimetry: Part I — Constructing sound interpretation methods: Geophysics, 67, 788–794. Tikhonov, A. N., and V. Y. Arsenin, 1977, Solutions of ill-posed problems: V. H. Winston & Sons.
We used the quasi-Newton method with the Broyden-FletcherGoldfarb-Shanno implementation to minimize the objective function 共p兲 given by equation 10. At the kth iteration, the quasi-Newton algorithm consists of the following steps: 1兲 2兲
3兲
Solve the direct problem for pˆk and compute the objective function 共pˆk兲. Compute by finite differences the M-dimensional gradient vector ⵜp兵 共p兲其兩p = pˆk of the function 共p兲 with respect to vector p, evaluated at p = pˆk. Estimate the step ⌬pˆk by solving the linear equation
Hk⌬pˆk = − ⵜp兵 共p兲其兩p = pˆk ,
共A-1兲
where Hk is the quasi-Newton approximation to the M ⫻ M Hessian matrix of the function 共p兲. The linear equationA-1 is obtained by minimizing the second-order functional given by
共⌬pk兲 = ⵜpT兵 共p兲其兩p = pˆk⌬pk +
1 T ⌬p Hk⌬pk 2 k
with respect to the search direction vector ⌬pk. To guarantee the Hessian matrix is positive-definite, we use Marquardt’s 共1963兲 strategy at each iteration. Then we continue the steps. 4兲 5兲
Obtain pˆk + 1 = pˆk + ⌬pˆk. Update the Hessian matrix using the Broyden-Fletcher-Goldfarb-Shanno implementation by
Hk+1 = Hk +
bkbTk bTk uk
−
Hk共ukuTk 兲Hk uTk Hkuk
,
共A-2兲
