We present a threedimensional (3D) gravity modeling and inversion approach and its application to complex geological settings characterized by several allochthonous salt bodies embedded in terrigenous sediments. Synthetic gravity data were computed for 3D forward modeling of salt bodies interpreted from Prestack Depth Migration (PSDM) seismic images. Density contrasts for the salt bodies surrounded by sedimentary units are derived from densitycompaction curves for the northern Gulf of Mexico’s oil exploration surveys. By integrating results from different shape and depthsource estimation algorithms, we built an initial model for the gravity anomaly inversion. We then applied a numerically optimized 3D simulated annealing gravity inversion method. The inverted 3D density model successfully retrieves the synthetic salt body ensemble. Results highlight the significance of integrating highresolution potential field data for salt and subsalt imaging in oil exploration.
Hydrocarbon exploration is largely based on geophysical methods among which seismic reflection is the most intensely employed. Increased interest in subsalt related plays in the Gulf of Mexico and in other sedimentary basins around the world has turned oil and gas prospecting within these regions into a major challenge. Physical property contrasts of salt features such as highly contrasting seismic velocities relative to the surrounding media lead to complex wave diffraction patterns and lack of illumination near and below them.
In this context, gravity methods are well suited to support seismic prospection and improve subsalt imaging by taking full advantage of the density contrasts between salt bodies and surrounding sedimentary targets. Salt bodies retain low densities, whereas upon burial sediments compact increasing the density contrast. Table
Pwave velocity, density, and permeability of the rock salt bodies placed in oil and gas prospecting zones (SI units).
Property  Range of values  Reference 

Seismic velocity (Pwave)  4270 to 5190 [m/s]  Grant and West, 1967 [ 
Density  2,100 to 2,200 [kg/m^{3}]  Gardner et al., 1974 [ 
Permeability  <10^{−20} [m^{2}]  Carter et al., 1993 [ 
Complex geological imaging using modeling and inversion of potential field anomalies has been examined in recent studies. OrtizAlemán and UrrutiaFucugauchi [
In this work we built a 3D gravity model including several allochthonous salt bodies as interpreted from a Prestack Depth Migration seismic volume, integrating results from different potential field techniques such as edgesource detection, depthtosource estimation, and 3D gravity inversion.
The whole computational domain, including several salt features, was discretized into an ensemble of rectangular prismatic elements. Its gravity response, that is,
(a) Discretized media formed by
The total gravity response calculated at some observation point was the sum of the gravity contributions generated by the
Now, the vertical component
As illustrated in Figure
Taking into account the above and the fact that salt bodies embedded in terrigenous sediments are geometrically irregular, they can be modeled as ensembles of regular rectangular prisms, formed by discrete points. Analyzed salt bodies (Figure
Salt bodies interpreted from a 3D PSDM velocity model. Each body is composed of many discrete points with a regular volume grid interval.
On computing the gravity response of the salt bodies illustrated in Figure
Density versus depth curve, representative of sediments of Gulf of Mexico and rock salt density. Based on Nelson and Fairchild [
The gravity response is therefore calculated by first obtaining a density contrast between the salt bodies and the surrounding sediments, in the position of each point source (prism), subtracting (
Figure
Gravity anomaly caused by the salt bodies depicted in Figure
Several methods especially suited to enhance anomalies and estimate depth to source are commonly applied to potential field data. While there are methods that use systematic search algorithms to find a solution of the distribution of the densities of the model [
Figure
Lateral extent of source bodies from a set of depthtosource estimation methods. (a) Horizontal Gradient, (b) Analytic Signal Amplitude, (c) 1storder EAS, and (d) 2ndorder Enhanced Analytic Signal.
(a) Maxima location estimated from the methods depicted in Figure
After estimation of projected source location on a horizontal plane (plantview of sources), the corresponding distribution of the sources with depth was inferred in order to build an initial 3D structural model. For this purpose, we applied the 3DED algorithm to the gravity anomaly grid, considering a structural index
Distribution with depth of gravity anomaly sources estimated by the 3DED algorithm.
We built a 3D structural model including two huge salt diapirs surrounded by sedimentary rocks with relative density contrast assigned as a function of depth (
Figure
3D initial model built from 3D Euler deconvolution solutions and the maxima of the lateral extent estimation methods previously applied to the gravity anomaly.
Gravity anomaly caused by the 3D initial model shown in Figure
Table
3D initial model main characteristics.
Model size  Ensemble discretization  Density range 



1,900 to 2,590 [kg/m^{3}] 
The computed gravity anomaly for the 3D initial model qualitatively resembles the shape of the anomaly produced by the salt bodies interpreted from a PSDM volume (Figure
According to (
This is a linear system of equations, where
Here,
To solve the inverse problem, we chose the simulated annealing global optimization method. A main drawback of global optimization lies in the excessive amount of forward problem computations required to solve the inverse problem. In the past decades, global optimization has been successfully applied to several geophysical exploration issues, where dimensionality of the inverse problem did not represent a bottleneck [
The simulated annealing method was conceived as a mathematical analogy with the natural optimization process of crystal formation from a mineral fluid at high temperature. Its basic concepts were taken from the statistical mechanics.
The simulated annealing optimization process emulates the evolution of a physical system as it slowly cools down and crystallizes at a state of minimum energy. If temperature,
Following Kirkpatrick et al. [
This acceptancerejection procedure is repeated several times for a fixed temperature,
To compute the energy level in each stage, we used a normalized
The cooling schedule we choose reduces the temperature in an exponential fashion by multiplying the actual temperature by some parameter
Finally, this process is repeated until reaching the limit
One first improvement made to the basic simulated annealing method in this work was to accelerate the product
The final improvement made to the SA method consisted in applying an auto adjustment to the amplitude control parameter,
The final SA inversion algorithm is summarized in the diagram shown in Figure
Flux diagram representation of the simulated annealing algorithm applied in this work (modified from OrtizAlemán and Martin [
This modified SA algorithm was applied to the gravity anomaly data generated by the postulated set of salt bodies, with the following restrictions:
The lateral extent of all models generated by the inversion procedure was restricted to the interpreted source borders (Figure
The model space was bounded according to the salt and sediments density contrast (Figure
Parameter values related with the 3D gravity data inversion procedure.
Parameters  Values 

Number of inverted parameters  27,000 
Number of observed data points  2,601 
Initial temperature  1.0 
Final temperature 

Energy of the initial model  33.484543 
Energy of the final model  0.0178259 
Number of temperature reduction steps  1,000 
Reduction temperature factor  0.98 
Previous cycles to the 
10 
Cycles of thermal equilibrium  5 
Total number of tried models  149,400,000 
Number of accepted models  115,281,390 
Number of rejected models  34,118,610 
The 3D density model resulting from the inversion procedure (Figure
3D Inverted Model generated from the 3D gravity anomaly inversion.
The gravity anomaly grid generated by the 3DInvM shows that, despite the apparent differences in the central part of the grid corresponding to the gravity minimum, the amplitudes are similar to the observed gravity (Figure
Gravity anomaly caused by the 3DInvM shown in Figure
In order to quantify the quality of the 3DInvM, we calculated the differences between the gridded gravity anomaly data of the salt bodies (Figure
Spatial distribution of residuals along the grid and their amplitudes, computed as the absolute differences between gravity data before and after inversion.
Relative frequency distribution of residuals shown in Figure
The misfit curve, representing the relationship between temperature and energy parameters along the inversion process, exhibits three different kinds of convergence rates: a gradual decay in the beginning of the inversion, an intermediate region of sharp decreasing misfit, and a zone of progressively slower convergence rates until final entrapment (Figure
Temperature versus energy curve in the 3D gravity inversion process.
In this study we applied 3D gravity modeling and inversion in a complex geological setting involving several allochthonous salt features embedded in terrigenous sediments, representing a challenging and quite realistic scenario commonly found in the southern Gulf of Mexico.
Several methods especially suited to enhance anomalies and estimate depth to source are used in this work to determine an initial 3D density distribution for inverse modeling. These methods include the Horizontal Gradient (HG), the 3D Analytic Signal Amplitude (AS), the Enhanced Analytic Signal (EAS), and the 3D Euler deconvolution (3DED). We built an initial density model by integrating results from this set of shape and depthsource estimation algorithms. We finally applied a numerically optimized threedimensional simulated annealing gravity inversion approach. As the total amount of evaluated forward models in this study case was quite large (~150 million), application to other realistic gravity modeling efforts should consider the use of high performance computing for the forward and inverse problems, as recently introduced by CouderCastañeda et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors of this work acknowledge the financial support provided by SENERCONACyT Project 128376 and Mexican Institute of Petroleum Projects D.00475 and H.61006.