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1 Rapid gravity and gravity gradiometry terrain correction via adaptive quadtree mesh discretization Kristofer Davis, M. Andy Kass, and Yaoguo Li, Center for Gravity, Electrical and Magnetic Studies, Colorado School of Mines SUMMARY We present a method for modeling the terrain response in gravity and gravity gradiometry surveys utilizing an adaptive quadtree mesh discretization. The data-dependent method is tailored to provide rapid and accurate spatial terrain corrections for ground or draped airborne surveys. The surface used in the modeling of terrain effect at each datum is discretized automatically to the largest cell size that will yield the desired accuracy, resulting in much faster modeling than fullresolution calculations. The largest cell sizes within the model occur in areas of minimal correction and at large distances away from the data location. We show synthetic and field examples for proof of concept. The adaptive quadtree method reduces the computational cost of the field example by performing 99.7% fewer calculations than the full model would require while retaining an accuracy of one Eötvös for the gradient data, and runs in.9% of the time for the gravity field terrain corrections compared with a commercial software package. INTRODUCTION The effect of terrain in gravity gradiometry surveys for petroleum and mining applications is often the predominant signal measured, as the field decays as distance-cubed (Chen and Macnae, 1997). Therefore accurate terrain corrections are critical to proper interpretation of the dataset. However, the accuracy of the terrain correction is most often a function of the resolution of the available digital elevation model (DEM). As a result, higher and higher resolution DEMs are acquired during gravity gradient surveys. While this results in more accurate terrain corrections, these high-resolution DEMs have the unfortunate consequence of creating extremely computationally intensive terrain corrections. Current methods of terrain correction often forward model the terrain effect at full resolution across the dataset, taking days and sometimes weeks to complete. Much faster algorithms are needed if field interpretation of data is required or if results are simply needed quickly. Gravity data also require high resolution DEMs, especially for ground-based surveys. In addition, a much larger spatial extent of the DEM is required as the field decays as distance-squared. While the minimum DEM extent and cell size for a groundbased gravity survey is well defined (Hammer, 1939), airborne gravity terrain corrections are more troublesome and less defined (Hammer, 1974). While extremely high resolution digital elevation models are required in areas with high relief and/or near observation locations, the resolution requirements can be relaxed as the distance from the observation point increases. Also, if the characteristics of the terrain change over the survey, the resolution requirements of the DEM may change as well. Moreover, the required resolution is also a function of the flight height (Reid, 198; Kass and Li, 28). Thus a data-adaptive discretization of the DEM can result in faster computations and smaller storage requirements while ensuring desired accuracy. In this abstract, we show that the adaptive quadtree method is appropriate for data-dependent discretization of DEMs for terrain corrections. The method allows for user-defined error bounds and parallelization, and reduces the number of required calculations by three to five orders of magnitude. We start with the forward modeling calculation of the prismatic model and then discuss the discretization method. We show a synthetic case for an illustration as well as a field example. FORWARD MODEL The terrain effect in gravity gradiometry is calculated as the response due to the prism contribution bounded above by the terrain and below by a plane that passes through the lowest terrain point in the survey area. The forward model of the gravity gradient follows a similar construction as described in Li (21). As gravitational potential satisfies Laplace s equation, the potentials due to each source superimpose at each observation point and thus the contribution of all prisms to the gravity gradient measured at a point is the sum of their individual responses. For a volume source at location r with density ρ, the gravity gradient at an observation point (denoted by vector r ) is a tensor T with individual components represented by T kl = γρ 1 dv V x k x l r r o (1) where V is the volume of the source. The integration follows the magnetic derivation by Sharma (1966) or Zhang et al. (2), with treatment to avoid singularities at observation locations coplanar with the faces of the prisms. As the equations are lengthy and not required for understanding of the method, we direct the reader to Sharma s paper for a detailed discussion of the discretization of the integral equations. For source-free regions, this tensor is symmetric and traceless, and commonly is written in the form: [ Txx T xy T xz ] T = T yy T yz T zz Here we have adopted a Cartesian coordinate system having its origin at the Earth s surface, x-axis pointing towards grid north, y-axis pointing towards grid east, and z-axis pointing vertically downward. Thus we can write the expression for the gravity gradient at one data point, d i, as d i T kl (r oi ) { M } 1 = γρ j V j x k x l r r oi dv (2) 1789

2 M ρ j G i j. In this construction, we are summing the response due to many prisms V j with constant density ρ j within the prism. In a compact notation for a linear system, we can show Equation 2 as Gρ = d, (3) where d is the data response from the prisms based on their density, ρ. It is important to note that the matrix, G, can be quite large. However, the forward modeling algorithm does not require the formation of the complete matrix at one time, but rather only the computation for one datum at a time. The forward model of the gravity field follows a similar approach. The field due to a volume source is represented as z z o g z (r o ) = γρ dv, (4) V r r o 3 leading to the gravity field at one data point d i due to the sum of responses: d i = { M } z z oi γρ j V j r r oi 3 dv (5) M ρ j G i j. DISCRETIZATION APPROACH In order to reduce the number of computations for the terrain correction, we employ an adaptive quadtree mesh discretization of the supplied digital elevation model. This results in fewer prisms for which to calculate the gravity gradient response. The derivation of the discretization process for the gravity field is identical to that of the gravity gradient. The adaptive quadtree mesh design (Frey and Marechal, 1998) is one that places larger cells where no or little signal is present (in this case response from topographic relief) and smaller cells to increase resolution where sources are located. The quadtree method is useful in large scale problems by reducing the total number of model cells (Ascher and Haber, 21). Quadtree mesh discretization has been used primarily in geophysics in remote sensing applications (e.g. Gerstner, 1999), but recently has found application in use with mesh padding for DC resistivity (Eso and Oldenburg, 27) and magnetic processing (Davis and Li, 27). The mesh design is adaptive due to the data dependence from dataset to dataset as well as region to region within the same data set, as opposed to a quadtree mesh pre-defined by a known function or distribution. In general, the mesh starts with cells of a large, user-defined width; if a property has a value higher than a given threshold, it is split into quarters. This process continues until all cell properties are within the threshold. The quadtree mesh threshold is chosen as a function of a userdefined error level (usually at or below the aggregate survey noise). The error is defined by the difference between the examined large prism and the smaller prisms (e.g. higher resolution) within the same area. A quadtree algorithm is then used to discretize the terrain for each data point. At each stage, the binned prisms have a height of average terrain within the respective cell boundaries. The gravity gradient response of each cell is calculated and then the cell is split into four. The terrain effect of each new cell with a new average height is calculated again (five total forward calculations). The four cells are kept if the difference between the gravity gradient components of the four cells is more than the error threshold. Those are then examined and split again, if necessary. As the process approaches the optimal resolution, splitting the cells will provide less and less benefit until further splitting no longer has an effect greater than the accuracy threshold or the split cell size is equal to the DEM resolution. The error level for the quadtree algorithm is normalized by the number of cells in the model at each iteration. Thus if the maximum error is allowed in each cell it will superimpose and the upper bound of the error between the full and quadtree model is the defined error level. Then, if a cell is split (into four smaller cells), the error threshold for that cell is divided by four in order to preserve the upper bound. For example, if the initial model is split into 256 cells, then the first error level is the user-defined error divided by 256. Though the error is an upper bound, in practice the true error is much smaller as shown in the examples. Selection of the initial discretization has no effect on final accuracy, but is critical for the overall computation speed. In order to quantitatively determine a data-dependent initial discretization we examine the largest radial wavelength of the digital elevation model. The radial wavelength is used because of the nature of the two-dimensional DEM. Moreover, this simple step is fast to calculate because it depends on the number of data points in each direction and their respective spacing and the Fourier transform is unnecessary. Once the largest wavelength is calculated, it becomes the width of the initial discretized prisms for modeling. The splitting of these cells into smaller ones then occurs to retain resolution of the DEM. Further discretization though the adaptive mesh is analogous to adding higher frequencies in the Fourier domain. SYNTHETIC EXAMPLE As an example for illustrating the accuracy of the method, we use the Gaussian hill topography model presented in Figure 1. The hill extends from meters at ground level to 3 meters above the ground at its peak. The total number of observations locations was 625. We calculated the gravity gradient response of this terrain at 35 meters above ground level for each component and allowed a 1-Eötvös error. The starting cell size is 16 m by 16 m based on the longest radial wavelength. We show the adaptive discretization of a Gaussian hill for a single observation location in Figure 1. The asymmetry seen of the discretization is a function of the cen- 179

3 meters Northing (m) Easting (m) Figure 1: Gaussian hill with height equal to 3 meters. Automated adaptive quadtree discretization of terrain to achieve a maximum of 1 Eo tvo s accuracy 5 meters immediately above the peak for all tensor components. ter of the initial cells not being coincident with the peak of the model. We compared the response of the Gaussian hill calculated with an adaptive quadtree discretized DEM (Figure 2) to the regularlygridded, surface-based forward modeling. Though we allowed Figure 2: Gravity gradient response for each component at the algorithm as much as 1-Eo tvo s error, the maximum differa plane 5 meters above the maximum height of a 3 meterence between the two datasets was.17 Eo tvo s (Figure 2). high Gaussian hill. Units in Eo tvo s. Difference between We next examine the number of calculations performed in orthe terrain effect calculated with the adaptive quadtree method der to evaluate the efficiency of the adaptive algorithm. The versus a full-resolution surface-based method. The maximum full model contains 729,88 cells which means that there are error threshold given to the adaptive mesh was 1 Eo tvo s. Units 455,68, calculations. Using the 1 Eo tvo s error level exare in 1/1 of an Eo tvo s (-1 to 1 Eo tvo s color bar). ample, our method made 1,468,486 calculations, 31 times less (or.3%) of the full model. This is an extreme case of efficiency because of the nature of the of the terrain model itself. FIELD EXAMPLE To demonstrate the efficiency of the algorithm, we calculated the terrain corrections for a draped airborne gravity gradient and gravity survey from an AGG test site in Australia. The gravity gradient terrain corrections used a density contrast of 2.67 g/cc and allowed a 1 Eo tvo s error while the gravity corrections used the same density contrast and 1 mgal error tolerance. Figure?? shows the discretization of the DEM for one observation location. The terrain corrections calculated in this manner had a compression ratio of approximately 351, Figure 3: Vertically-exaggerated terrain with survey locations shown in white. 1791

4 meaning there were approximately.3% of the calculations necessary to produce the corrections as compared to using fullresolution. Figure 4 shows the calculated Tzz component of the gravity gradient. The airborne terrain correction for the vertical component of the gravity field was calculated in the same manner. Figure?? shows the corrections which have the expected features; these appear smoothed due to the observation elevation (Davis et al., 28). In comparison with an industry-standard commercial software package, the terrain correction took less than.9% of the time to run. CONCLUSIONS Terrain calculations with adaptive DEM gridding are faster than full-resolution calculations and allow user-specified accuracy. Depending on the complexity and relative relief of the terrain model, the adaptive quadtree correction method can reduce the number of model cell calculations by at least a magnitude which, depending on the problem, results in significant computational savings. The algorithm begins with a starting model with cell discretized the largest wavelength of observed terrain. It then proceeds to split cells to increase resolution and accuracy until a pre-defined error threshold is achieved. ACKNOWLEDGMENTS The authors would like to thank the Korea Resources Corporation and the sponsors of the Gravity and Magnetics Research Consortium: Anadarko, BGP, BP, ConocoPhillips, Marathon, and Vale for funding support. We thank Theo Aravanis and Rio Tinto for providing the Lidar data, which were acquired by Fugro Spatial. We also thank Trevor Irons for helpful discussions. (c) Figure 4: Discretization for one gravity gradient observation location for the calculation of terrain effect. Calculated Tzz component of the calculated terrain correction. (c) Calculated terrain correction for the vertical component of the gravity field. Units in mgal. 1792

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 21 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Ascher, U. M., and E. Haber, 21, Grid refinement and scaling for distributed parameter estimation problems: Inverse Problems, 17, no. 3, , doi:1.188/ /17/3/314. Chen, J., and J. Macnae, 1997, Terrain corrections are critical for airborne gravity gradiometer data: Exploration Geophysics, 28, no. 2, 21 25, doi:1.171/eg Davis, K., M. Kass, R. Krahenbuhl, and Y. Li, 28, Survey design and model appraisal based on resolution analysis for 4D gravity monitoring: Presented at the 78th Annual International Meeting, SEG. Davis, K., and Y. Li, 27, A fast approach to magnetic equivalent source processing using an adaptive quadtree mesh discretization: Presented at the 19th Annual Conference and Exhibition, Australian Society of Exploration Geophysicists. Eso, R., and D. Oldenburg, 27, Efficient 2.5D resistivity modeling using a quadtree discretization: Conference Proceedings, Symposium of Applied Geophysics on Engineering and Environmental Problems, Frey, P. J., and L. Marechal, 1998, Fast adaptive quadtree mesh generation: 7th International Meshing Roundtable, Sandia National Laboratories, Gerstner, T., 1999, Adaptive hierarchical methods for landscape representation and analysis : Lecture Notes in Earth Sciences, 78, 75 92, doi:1.17/bfb972. Hammer, S., 1939, Terrain corrections for gravimeter stations: Geophysics, 4, 184., 1974, Topographic and terrain correction for airborne gravity: Geophysics, 39, 537. Kass, M., and Y. Li, 28, Practical aspects of terrain correction in airborne gravity gradiometry surveys: Exploration Geophysics, 39, no. 4, , doi:1.171/eg821. Li, Y., 21, 3D inversion of gravity gradiometry data: Presented at the 71st Annual International Meeting, SEG. Reid, A. B., 198, Aeromagnetic survey design: Geophysics, 45, , doi:1.119/ Sharma, P. V., 1966, Rapid computation of magnetic anomalies and demagnetization effects caused by bodies of arbitrary shape: Pure and Applied Geophysics, 64, no. 1, 89 19, doi:1.17/bf Zhang, C., M. F. Mushayandebvu, A. B. Reid, J. D. Fairhead, and M. E. Odegard, 2, Euler deconvolution of gravity tensor gradient data : Geophysics, 65, , doi:1.119/