Inversion of Gravity Data Using a Binary Formulation
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Geophys. J. Int. (2006) 167, 543–556 doi: 10.1111/j.1365-246X.2006.03179.x Inversion of gravity data using a binary formulation Richard A. Krahenbuhl and Yaoguo Li Center for Gravity, Electrical & Magnetic Studies; Department of Geophysics, Colorado School of Mines, USA. E-mail: [email protected] Accepted 2006 August 16. Received 2006 August 16; in original form 2005 February 24 SUMMARY We present a binary inversion algorithm for inverting gravity data in salt imaging. The density contrast is restricted to being one of two possibilities: either zero or one, where one represents the value expected at a given depth. The algorithm is designed to easily incorporate known density contrast information, and to overcome difficulties in salt imaging associated with nil zones. The problem of salt imaging may be formulated as a general inverse problem in which a piecewise constant density contrast is constructed as an indirect means of identifying the salt boundary. Difficulty arises when the salt body crosses the nil zone in depth. As a result, part of the salt structure is invisible to the surface data and many inversion algorithms have difficulties GJI Geodesy, potential field and applied geophysics in recovering the salt structure correctly. The binary condition places a strong restriction on the admissible models so that the non-uniqueness caused by nil zones might be resolved. In this paper, we will present the binary formulation for inversion of gravity data, develop the solution strategy, illustrate it with numerical examples, and discuss limitations of the technique. Key words: genetic algorithm, gravity, inversion, optimization, potential field. a negative gravity anomaly on the surface. In the final scenario, the 1 INTRODUCTION salt body straddles a depth at which the sediment density is equal Conservative estimates indicate that at least 15 per cent of US to the salt density. This region of equal density between salt and domestic oil and 17 per cent of its natural gas production come from sediment is referred to as a nil zone. fields along the continental shelf margin off the shores of Louisiana In the last scenario described above, the portion of salt within (Gibson & Millegan 1998). To explore for future reserves, industry the nil zone does not contribute to surface gravity anomaly. This has expanded towards exploration of the deeper water regions of the is a natural consequence of having zero density contrast with the continental slope. While the industry target is obviously hydrocar- surrounding sediment. Likewise, portions of the salt body above the bon traps, the geophysical targets are the geologic features in the sed- nil zone will have positive density contrast, producing a positive imentary section that are responsible for these accumulations of oil anomaly in surface gravity data. Salt below the nil zone, in con- and gas. Some of these features include reefs, faults, anticlines and trast, generates a negative gravity anomaly because it has a negative variations in thickness of horizontal salt beds. In particular, the salt, density contrast with respect to the surrounding medium. The net including domes, ridges and pillows, are relatively incompressible result is that the positive and negative anomalies from the top and and, therefore, remain fairly constant in density throughout. This bottom portions of salt cancel out in parts of the surface gravity incompressibility likewise allows for abundant traps beneath salt data (Gibson & Millegan 1998). This effect is referred to as an an- bodies throughout the Gulf of Mexico (Gibson & Millegan 1998). nihilator. Depending on how an inversion algorithm is formulated As a result, they have become major targets in oil and gas explo- to reconstruct such a salt body, the combination of nil zones with ration. Gravity inversion is one of the tools available to geophysicists annihilators in the salt body can cause major difficulties. for imaging the base of salt as a means for exploring these targets. Inversion methods for imaging salt structure using gravity data The gravity inverse problem for salt body imaging is one of find- fall under two general categories. The first is interface inversions. ing the position and shape of an anomalous constant density em- These methods assume a simple topology for the salt body and bedded in a sedimentary background whose density increases with known density contrast and construct the base of the salt (e.g. depth due to compaction. Depending on the depth and depth extent Jorgensen & Kisabeth 2000; Cheng 2003). Similar method has also of the salt body, three scenarios can occur. In the first scenario, the been used extensively in other applications of gravity inversion, salt is shallow enough so that its density is greater than that of the such as in basin depth determination (e.g. Oldenburg 1974; Pedersen immediate sedimentary host. This leads to a positive density con- 1977; Chai & Hinze 1988; Reamer & Ferguson 1989; Barbosa et al. trast in the salt, and a positive gravity anomaly in surface data. In 1999). Methods in the second category are generalized density in- the second scenario, the salt is positioned at depth so that its density versions. These methods construct a density contrast distribution as is less than the density of the surrounding sediments. This leads to a function of spatial position and image the base of salt by the transi- an entirely negative density contrast for the salt body and, therefore, tion in density contrast (Li 2001). Similar approaches have also been C 2006 The Author 543 Journal compilation C 2006 RAS 544 R. A. Krahenbuhl and Y. Li used widely in mineral exploration problems (Green 1975; Last & objective function φ m and data misfit φ d: Kubik 1983; Guillen & Menichetti 1984; Oldenburg et al. 1998). min.φ= φ (ρ) + βφ (τ), The interface inversion has the advantage that it directly inputs d m the known density contrast at each location in the subsurface and subject to ρ ∈{0,ρ(z)}. (1) provides a direct image of the base of salt. However, the drawbacks Assuming that we know the standard deviation of each datum σ i , are that the problem is non-linear and can be more difficult compu- we can define the data misfit function as tationally. In addition, the assumed simple topology of salt creates N obs − pre 2 difficulties when either regional field or small-scale residuals due φ = di di , d σ (2) to shallow sources are not completely removed. The inconsistency i=1 i between the assumed model and data can lead to large errors, or where dobs and dpreare the observed and predicted data, respectively. even failure of inversion. i i Assuming a Gaussian statistics, the expected value of the data misfit The density inversion has the flexibility of handling multiple is equal to the number of data, N. We would like to construct a anomalies, more complex shapes and the solution is easier to obtain compact model that is also structurally simple. Therefore, we use because the relationship between observations and density contrast the following generic model objective function that measures the is linear. However, as they are currently formulated, these methods length of the model and its flatness in different spatial directions are not well suited for cases where nil zones are present. Because (e.g. Li & Oldenburg 1998). For 2-D problems, the model objective of the presence of a nil zone and annihilators as discussed above, a function is given by, portion of the salt body is invisible to the surface gravity data. Den- sity inversion methods allowing continuous density values (e.g. Li ∂τ 2 ∂τ 2 φ = α τ 2 v + α v + α v, & Oldenburg 1998) will in general produce a model that has little m s d x ∂ d z ∂ d (3) V V x V z resemblance to the true structure. The data are satisfied by interme- where τ is the binary model, V is the subsurface region over which diate density values and distributions that only image a portion of the model is defined, and α , α and α are relative weights of the the salt body. s x z individual components of the model objective function. We note that To overcome difficulties associated with both methods, we use model values are discontinuous and, therefore, the derivative terms a binary formulation that enables one to incorporate the density in the model objective function in eq. (3) are only defined in the contrast values, a strength of non-linear interface inversion, while sense of generalized functions. We present in such a form to provide retaining the flexibility and linearity of density inversion. The diffi- conceptual understanding. However, the numerical algorithm works culty of the binary formulation, however, lies in the discrete nature of directly with the discretized form of eq. (3), and the derivative terms the density contrast. Because the variable can only take on discrete therefore represent differences between adjacent elements. values, 0 or 1 for sediment or salt, respectively, derivative-based Requiring φ to achieve an expected value ensures all models that minimization techniques are no longer applicable. In the follow- d do not adequately fit the observed data, or overfit them, be eliminated ing, we will first present the methodology of binary inversion and from the set of possible solutions. The binary formulation places a discuss the solution by genetic algorithm (GA). We then present limit on the types of models that fit the data. However, the problem results from synthetic tests of a 1-D mathematical problem and two is still non-unique. Therefore, we use the model objective function, 2-D gravity problems with finite strike length (often referred to as φ , to further narrow our solution set to only geologically reasonable 2.5-D problems).