3D stochastic inversion of borehole and surface gravity data using Geostatistics

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1 3D stochastic inversion of borehole and surface gravity data using Geostatistics P. Shamsipour ( 1 ), M. Chouteau ( 1 ), D. Marcotte( 1 ), P. Keating( 2 ), ( 1 ) Departement des Genies Civil, Geologique et des Mines, Ecole Polytechnique de Montreal, ( 2 ) Geological Survey of Canada, Continental Geoscience Division, Ottawa, Ontario, Canada Summary Geophysical inversion of potential field is impeded because of the intrinsic non-uniqueness of the solution. An inversion method based on a geostatistical approach (cokriging) is presented for threedimensional inversion of gravity borehole and surface data to limit the resulting solution space. Cokriging is a method of estimation that minimizes the error variance by applying cross-correlation between several variables. In this study the estimates are derived using gravity data (from boreholes and the surface) as a secondary variable and the density as the primary variable. The necessary gravity, density and gravity-density covariance matrices are estimated using the observed gravity data. The proposed method is applied to two different synthetic models: 1) a massive orebody; 2) a stochastic distribution of densities. Gravimeters are now becoming available (Gravilog from Scintrex), which can fit boreholes for mineral exploration. Using the borehole data, we can improve the depth resolution of surface inversion. The results show the ability of the method to invert surface and borehole data simultaneously. The results also clearly show the important role of borehole data to improve depth resolution. Introduction Many strategies can be used to deal with the non-uniqueness problem in gravity inversion. They all involve some kind of constraints or regularization to limit the solutions. Smoothness or roughness of density distribution which control gradients of parameters in spatial directions are used in magnetic inversion by Pilkington (1997). Li and Oldenburg (1998) counteract the decreasing sensitivities of cells with depth by weighting with an inverse function of depth. Another 3D inversion technique proposed by Fedi and Rapolla (1999) allows the definition of depth resolution. Geostatistical methods in geophysical inversion were applied by Asli et al (2000), Gloaguen et al (2005) and Hansen et al (2006). Bosch et al (2006) also applied geostatistical constraints to gravity inversion using Monte Carlo techniques. In fact, linear stochastic inversion was first described by Franklin (1970) and then popularized by Tarantola and Valette (1982). Chasseriau and Chouteau (2003) have done 3D inversion of gravity data using an a priori model of covariance. However, their method involved nonlinear constraints on density so they used an iterative approach. Shamsipour et al. (2009) proposed geostatistical techniques of cokriging and conditional simulation for the three-dimensional inversion of gravity data including geological constraints. Simulations allow identification of stable features of the inverted fields. We build on the approach of Shamsipour et al. (2009). We extend this approach to test using the borehole and surface gravity inversion at the same time. First, we present the method for 3D stochastic inversion of borehole and surface gravity data using cokriging. Then we use synthetic data sets from two models to illustrate the application of the method. The results prove the flexibility of the method to include borehole data.

2 Methodology The cokriging method gives weights to data so as to minimize the estimation variance (the cokriging variance). Here, the primary variable of cokriging system is density (ρ) and the secondary variable is gravity (g). Cgg is the gravity covariance matrix, C ρρ is the density covariance matrix and Cgρ is the cross covariance between gravity and density. The variables (ρ) and (g) are multidimensional * random variables and ρ is the estimated density defined on block support. Minimization of the estimation variance will yield the simple cokriging solution, Cgg Λ = Cgρ, and the estimate of * T densities is obtained from the gravity data using the optimal weights, ρ = Λ g. In this method, the model covariance is obtained using the V-V plot method (for details, see Shamsipour et al. (2008, 2009)). Results We present two different kinds of synthetic data. The first example is concerned with the detection of a compact ore body, which is common in mining exploration. The second example, a stochastic distribution of density, is common in reservoir modeling and oil exploration. Example 1): The 3D domain is divided into cubic prisms. The dimension of each elementary prism is meters resulting in a 3D domain of km. In this domain, we suppose a compact mass ( ) with uniform density contrast of 1000 kg/m 3 with respect to a homogeneous background at a depth of 500 m (Figure 1 (a)). We also assume a vertical borehole at (x, y)=(1000, 1000) with gravity data sampled every 10 m along it. The gravity data caused by this compact mass at ground surface and along the borehole is shown in Figure 1 (b) and (f) respectively. Using the gravity observations at ground surface, we can apply the inversion method based on cokriging to estimate the density distribution. The inverted densities at section y=1000 m are shown in Figure 1 (c). As we expect there is not good resolution at depth. Then we use the gravity observation along the borehole and perform the inversion to estimate the density distributions. The result is shown in Figure 1 (d). Finally, both the gravity observations at ground surface and along the borehole are used for inversion by cokriging (Figure 1 (e)). As we can see using the borehole data helps improving the density estimation especially at depth and recovering the exact shape and position of the compact mass. Example 2): Densities for km prisms were generated by non-conditional LU simulation using a spherical variogram model with C=60000 (kg/m), a h,45 =5 km, a h,135 =9 km, a vert =4 km where C is the variogram sill, and, a h,45,, a h,135, a vert are the variogram ranges along horizontal directions 45 o and 135 o and vertical direction respectively. We also assume a vertical borehole at (x, y)=(5000, 5000) such that the gravity response along this borehole is known at every 25 meters. The 3D domain is divided into cubic prisms. One section (y = 500 m) of this stochastic density distribution is shown in Figure 2 (a). The gravity caused by this distribution can be found across the surface and along the borehole. From now on, we assume the generated gravity data is known and we invert them in order to estimate the density distribution. Figure 2 (b) shows the inverted densities at section y = 500 m only using the surface gravity data. Again, the resolution at depth is not good. Figure 2 (c) shows the result of inversion by cokriging

3 using only the gravity data along the borehole and Figure 2 (d) shows the results using gravity data from both the surface and the borehole. Again, adding the information along the borehole can improve the results of inversion and increase depth resolution. In Figure 3, we compare the initial data with the estimated data. From Figure 3 (c), as we expect the initial gravity matches very well to the calculated gravity. However, this is not the case for density. From Figures 3 (a) and (b), we observe that including the borehole information in the stochastic inversion results in better match between model and estimated densities. Figure 1) Results for compact mass example

4 Figure 3) Comparison of estimated data and initial data for the stochastic example Conclusions We presented an inversion method based on a geostatistical approach for three-dimensional inversion of gravity data. The proposed non-iterative inversion method based on cokriging is computationally efficient, as practical solutions exist for large problems. This enables easy incorporation of known gravity values in all locations. We showed on synthetic models the value of using borehole gravity in cokriging to improve the accuracy of density estimation. The application of borehole gravity can be useful in exploration wherever information is available from boreholes. This method can be extended to other 2D and 3D geophysical linear inversion methods as it has been done by authors of this article to magnetic data and to joint inversion of gravity and magnetic data. References Asli, M., D. Marcotte, and M. Chouteau, 2000, Direct inversion of gravity data by cokriging, in Kleingeld and Krige, eds., 6th International Geostatistics Congress, Cape town, South Africa, Bosch, M., R. Meza, R. Jimenez, and A. Honig, 2006, Joint gravity and magnetic inversion in 3D using Monte Carlo methods: Geophysics, 71, G153. Chasseriau, P. and M. Chouteau, 2003, 3D gravity inversion using a model of parameter covariance: Journal of Applied Geophysics, 52, Fedi, M. and A. Rapolla, 1999, 3-D inversion of gravity and magnetic data with depth resolution: Geophysics, 64, Franklin, J., 1970, Well posed stochastic extensions of ill posed linear problems: J. Math, 31, Gloaguen, E., D. Marcotte, M. Chouteau, and H. Perroud, 2005, Borehole radar velocity inversion using cokriging and cosimulation: Journal of Applied Geophysics, 57, Hansen, T., A. Journel, A. Tarantola, and K. Mosegaard, 2006, Linear inverse Gaussian theory and geostatistics: Geophysics, 71, R101. Li, Y. and D. Oldenburg, 1998, 3-D inversion of gravity data: Geophysics, 63, Shamsipour, p. (2008). 3D inverison of gravity data using cokriging and cosimulation. M.A.Sc Thesis, Ecole Polytechnique de Montreal. Shamsipour, p., Marcotte, D., Chouteau, M., and Keating, P. (2009). 3D stochastic inversion of gravity data using cokriging and cosimulation. Accepted for publication in Geophysics journal. Pilkington, M., 1997, 3-D magnetic imaging using conjugate gradients: Geophysics, 62,

5 Tarantola, A. and B. Valette, 1982, Generalized nonlinear inverse problems solved using the least squares criterion: Rev. Geophys. Space Phys, 20,