Rapid gravity and gravity gradiometry terrain corrections via an adaptive quadtree mesh discretization

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1 CSIRO PUBLISHING Copublished paper, please cite all three: Exploration Geophysics, 211, 42, 88 97; Butsuri-Tansa, 211, 64, 88 97; Jigu-Mulli-wa-Mulli-Tasa, 211, 14, Rapid gravity and gravity gradioetry terrain corrections via an adaptive quadtree esh discretization Kristofer Davis 1,2 M. Andy Kass 1 Yaoguo Li 1 1 Center for Gravity, Electrical, and Magnetic Studies, Departent of Geophysics, Colorado School of Mines, 15 Illinois Street, Golden, CO 841, USA. 2 Corresponding author. Eail: kdavis@eos.ubc.ca Abstract. We present a ethod for odelling the terrain response of gravity gradioetry surveys utilising an adaptive quadtree esh discretization. The data- and terrain-dependent ethod is tailored to provide rapid and accurate terrain corrections for draped and baroetric airborne surveys. The surface used in the odelling of the terrain effect for each datu is discretized autoatically to the largest cell size that will yield the desired accuracy, resulting in uch faster odelling than full-resolution calculations. The largest cell sizes within the odel occur in areas of inial terrain variation and at large distances away fro the datu location. We show synthetic and field exaples for proof of concept. In the presented field exaple, the adaptive quadtree ethod reduces the coputational cost by perforing 351 ties fewer calculations than the full odel would require while retaining an accuracy of one Eötvös for the gradient data. The ethod is also used for the terrain correction of the gravity field and perfored 31 ties faster copared with a calculation of the full digital elevation odel. Key words: gravity, gravity gradioetry, processing, quadtree, terrain correction. Introduction The effect of terrain in gravity gradioetry surveys for petroleu and ining applications is often the predoinant signal easured, because the field decays as distance-cubed (Chen and Macnae, 1997) and the terrain variation is the closest source. Therefore accurate terrain corrections are critical to proper interpretation of the dataset. However, the accuracy of the terrain correction is ost often a function of the resolution of the available digital elevation odel (DEM). As a result, higher and higher resolution DEMs are acquired during gravity gradient surveys. While this results in ore accurate corrections, these high-resolution DEMs have the unfortunate consequence of creating extreely high coputational deands. Current ethods of terrain correction often forward odel the terrain effect with reasonable coarsening of terrain as distance fro the observation location increases. A ajor portion of the terrain is calculated at full resolution. Such an approach can take days and soeties weeks to coplete for large-scale surveys. Much faster algoriths are needed if interpretation of data is required in the field or if results are siply needed quickly. Terrain corrections for gravity data also require high resolution DEMs, especially for ground-based surveys. In addition, a uch larger spatial extent of the DEM is required for gravity surveys as copared to gravity gradient as the field decays as distance-squared. Thus terrain fro greater distances affects the calculation. While the iniu DEM extent and cell size requireents for ground-based gravity surveys are well defined (Haer, 1939), airborne gravity terrain corrections are ore troublesoe and less defined (Haer, 1974). Siilar issues exist with gravity as with airborne gravity gradioetry surveys. Although extreely high resolution DEMs are required in areas with large topographic relief, the resolution requireents can be relaxed as the distance fro the observation point increases. Also, if the characteristics of the terrain change over the survey, the resolution requireents of the DEM ay change as well. Rough or rugose terrain will require finer discretization to accurately represent the rapidly changing elevation. Furtherore, the required resolution is also a function of the flight height and acquisition filters (Reid, 198; Kass and Li, 28). Thus a data-adaptive discretization of the DEM can result in faster coputations and saller storage requireents while ensuring the desired accuracy. In this paper, it is shown that the adaptive quadtree ethod is appropriate for data-dependent discretization of DEMs for terrain corrections. The ethod allows for user-defined error bounds and parallel coputing capability, and reduces the nuber of required calculations by two orders of agnitude in the exaple. We start with the forward odelling calculation of the prisatic odel and then discuss the discretization ethod. Synthetic and field exaples are shown for illustration. Forward odel approach The terrain effect in gravity gradioetry is calculated as the response due to the region of ass bounded above by the terrain and below by a plane that passes through the lowest elevation of the survey area. We discretize this region into vertical priss, each with a constant density, and calculate their individual contributions. We have adopted a Cartesian coordinate syste 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. The forward odel of the gravity gradient follows a siilar construction as described in Li (21). As gravitational potential satisfies Laplace s equation, the potentials due to all sources superipose at each observation point and thus the contribution of all priss to the gravity gradient easured at a point is the su of their individual responses. For a ASEG/SEGJ/KSEG /EG /11/188

2 Rapid terrain corrections Exploration Geophysics 89 source at location r with density r, the gravity gradient at an observation point (denoted by vector r o ) is a tensor T with individual coponents represented by ð q q 1 T kl ¼ gr qk ql jr r o j dv for k; l ¼ x; y; z; ð1þ V where V is the volue of the source, g is the gravitational constant, and k and l represent the three orthogonal directions described above. The integral for a rectangular prisatic volue follows the agnetic derivation by Shara (1966) or Zhang et al. (2), with treatent to avoid singularities at observation locations coplanar with the faces of the priss. As the equations are lengthy and not required for understanding of the ethod, we direct the reader to Shara (1966) for a detailed discussion of the integral solution. For source-free regions, this tensor is syetric and traceless, and coonly is written in the for: 2 3 T ¼ 6 4 j¼1 T xx T xy T xz T yy DV j T yz T zz 7 5: ð2þ We can write the expression for the gravity gradient at a datu point, d i,as 8 9 >< ð d i T kl ðr oi Þ¼ XM q q 1 >= g r j >: qx k qx l jr r oi j dv >; ¼ XM j¼1 r j G ij : In this construction, we are suing the response due to M priss, DV j, of constant density r j, having vertical extents fro the surface to the botto plane. The forward odel of the gravity field follows a siilar approach. The field due to a volue source is represented as j¼1 ð g z ðr o Þ¼gr DV j V z z o jr r o j 3 dv; leading to the gravity field at one data point d i due to the su of responses: 8 9 >< ð d i ¼ XM z z >= o g r j >: jr r o j 3 dv >; ¼ XM r j z ij : ð5þ It is iportant to note that the atrix (G or ) can be quite large. However, the forward odelling algorith does not require the foration and storage of the coplete atrix at one tie, but rather only the coputation for one datu at a tie. Furtherore, the discretization of priss varies fro datu to datu and thus the value of M and lengths of the rows of the atrix are constantly changing. This is fundaental to providing rapid calculations with inial storage. It is also iportant to note that the varying length of M and the principle of superposition are the key properties allowing for parallelisation of the coputation using ultiple threads. Algorith Traditional approaches to terrain correction rely on prescribed coarsening of the eleentary approxiation to the terrain as a function of horizontal distance fro the observation location. To extend this approach and reove the change of prescriptions that j¼1 ð3þ ð4þ ay be sub-optial, we propose an adaptive coarsening of the syste based on a quadtree esh discretization of terrain for each observation location. The discretization algorith considers both distance and terrain relief. Quadtrees are a hierarchical representation of inforation. They are priarily used in data structures and data ining (Harrington, 2) to relate details with general headings or inforation in order to save coputation tie or storage space. Quadtree esh discretization has been priarily used in geophysics for reote sensing applications for the storage of iages (e.g. Gerstner, 1999). The iage decreases in size by assebling siilar pixels into groups, yet areas of interest or rapid change retain the original pixels to aintain high resolution. Recently, quadtrees have also found application in 2D DC resistivity inversion (Eso and Oldenburg, 27). DC resistivity odelling incorporates a siilar idea with saller cells grouped into larger cells based on distances between source or observation locations and esh eleents. The quadtree ethod is particularly useful in large scale probles by iniising the nuber of required odel paraeters (Frey and Marechal, 1998; Ascher and Haber, 21). The ethod discretizes the odel ore finely at locations of desired high resolution areas and discretizes coarsely where low resolution is peritted. This type of local grid refineent is used in our application of terrain effect calculation. We adopt an algorith that starts with a low resolution rediscretization of the terrain and iteratively splits paraeters to increase resolution up to the original resolution of the digital elevation odel based on the gravity or gravity gradient response. This is done for each observation location. The esh begins as large prisatic cells (i.e. a esh) that have grouped terrain values of the sallest cell size (e.g. the DEM grid spacing) within the. The forward-odel response of the large cell is calculated first using an average height. Next, the cell is split into quarters and new average heights are used to calculate the cobined response of those cells. If the values differ by ore than a defined local error level, the four cells are kept for further analysis. Otherwise, the large cell first odelled is used. This process continues iteratively until all responses are within the given error level or the resolution has reached the full resolution of the DEM. In this application, we define error as the data difference between the full resolution and quadtree odels. It should be noted that this defined error does not include the inherent errors in the DEM. Thus, the accuracy of the ethod is bounded by both the DEM resolution and the respective errors in the acquisition of the terrain. The error level for the quadtree algorith is noralised by the nuber of cells in the odel creating a local error level for each group of cells. Thus, the user-specified error level is the su of the axiu local error of each odel eleent. Hence, if a cell is split (into four saller cells), the local error threshold for that cell is divided by four in order to preserve the upper bound for the grouped cells. For exaple, if the initial odel is split into 256 cells, then the first local error level is the user-defined error divided by 256. If the initial cell is split into quadrants, the local error for those quadrants is 1/(256 * 4) or 1/124 of the usergiven error. The local error level is a terination echanis for the quadtree esh discretization. Although the error is an upper bound, in practice the true error is uch saller, as shown in the exaples. Selection of the initial discretization has no effect on final accuracy. However, the two end ebers where the discretization is close to the initial DEM interval and where the cell size is half the DEM (e.g. 2 2) should be avoided. We have epirically deterined that a initial discretization is a good starting point in ost cases.

3 9 Exploration Geophysics K. Davis et al. Exaples Gaussian hill We first use a synthetic exaple based on the Gaussian hill odel, presented in Figure 1a, to illustrate the accuracy of the ethod. The function used to generate the DEM is given by: Tðx; yþ ¼3 3 exp x2 3 2 þ y2 3 2 : ð6þ The hill extends fro at ground level to 3 above the ground at its peak and has a positive density contrast of 2.65 g/cc. The total nuber of observation locations was 625. We first calculated the gravity gradient responses of this terrain at 35 above ground level for each coponent and allowed a 1-Eötvös error. The starting cell size is 16 by 16 based on the longest radial wavelength. We show the adaptive discretization of a Gaussian hill for a single observation location directly above the peak of the hill in Figure 1b used in gravity gradioetry calculation. The asyetry seen in the discretization about the x and y axes is a result of the centre of the initial cells before quadtree-splitting not being coincident with the peak of the odel. However, we do observe the appropriate axis of syetry which is rotated 45 degrees above the positive abscissa and through the origin. We copared the gradient response of the Gaussian hill calculated with our adaptive quadtree discretized DEM (Figure 2) to the full resolution calculation based on a regular grid. Although we allowed the algorith as uch as 1-Eötvös error, the axiu difference between the results fro these two approaches was.17 Eötvös (Figure 3). (a) (b) Fig. 1. (a) 3 high Gaussian hill with a density contrast of 2.65 g/cc used as a synthetic test dataset. (b) Autoated adaptive quadtree discretization deterined by the algorith that achieves a iniu of 1-Eötvös accuracy 5 iediately above the peak for all tensor coponents.

Rapid terrain corrections Exploration Geophysics 91 3 15 15 3 4 18 4 26 48 7 92 114 136 158 18 22 224 246 268 29 312 334 356 378 4 3 15 1 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 4 18 4 26 48 7 92 114

4 Rapid terrain corrections Exploration Geophysics Fig. 2. Gravity gradient response for each coponent on a plane 5 above the peak of the 3 -high Gaussian hill odel. Units are in Eötvös

5 92 Exploration Geophysics K. Davis et al Fig. 3. Difference between the gravity gradient terrain effects calculated with the adaptive quadtree ethod versus a full-resolution surface-based ethod. The axiu error threshold between these two odels was 1 Eötvös. Units are in 1/1 of an Eötvös ( 1 to 1 Eötvös colour bar).

Rapid terrain corrections Exploration Geophysics 93 We exaine the nuber of calculations perfored in order to evaluate the efficiency of the adaptive algorith.

Using the 1-Eötvös error level exaple, our ethod ade 1 468 486 calculations, 31 ties less than (or.3% of) the full odel.

Figure 4 is the odel generated by the adaptive quadtree discretization for a single data location, siilar to the gradioetry odel. The accuracy was set to a axiu of.5 Gal.

6 Rapid terrain corrections Exploration Geophysics 93 We exaine the nuber of calculations perfored in order to evaluate the efficiency of the adaptive algorith. The full resolution odel contains cells, which eans that there are required calculations. Using the 1-Eötvös error level exaple, our ethod ade calculations, 31 ties less than (or.3% of) the full odel. We also use this terrain to deonstrate the ethod for gravity terrain correction. Figure 4 is the odel generated by the adaptive quadtree discretization for a single data location, siilar to the gradioetry odel. The accuracy was set to a axiu of.5 Gal. It took 129 ties fewer calculations than the full odel required ( versus calculations). The data generated fro the odel is shown in Figure 5. The difference between the full odel and quadtree discretized odel is siilar to that of the gravity gradient data. Field dataset To deonstrate the efficiency of the algorith using a realistic scenario, we calculated the terrain corrections for a draped airborne gravity gradient and gravity survey fro an Airborne Gravity Gradioetry test site in Australia (Figure 6). There are 8379 observations locations within a 5 by 5 area. The terrain data is sapled at 1 intervals in a 4 by 4 area. We first exaine the gravity gradioetry response of the terrain, and then the vertical gravity field. The gravity gradient terrain corrections used a density contrast of 2.67 g/cc and allowed a axiu of 1-Eötvös error while the gravity corrections used the sae density contrast and.5 Gal error tolerance. Figure 7a shows the discretization of the DEM for one observation location. The iediate area surrounding the datu is presented in Figure 7b for clarity. The discretization Gal Fig. 4. The quadtree odel discretization for one gravity observation. The quadtree-based calculation resulted in a speed increase of two orders of agnitude over the full resolution calculation. Fig. 5. The odelled vertical gravity terrain correction for synthetic Gaussian hill in Gal. The difference between full and quadtree odels is less than.5 Gal. 1 2 Elevation () Elevation () Fig. 6. Vertically-exaggerated terrain with survey locations shown in white. There are 8379 observation locations and the DEM is 4 4 with 1 cell size.

This exeplifies the utility of the ethod to take into account the response of the terrain, rather than having full dependency on the distance away fro the datu.

7 94 Exploration Geophysics K. Davis et al. requiring the iniu cell width (e.g. the DEM interval) is asyetric. This is due to the hill to the east of the datu, which requires finer discretization than the flatter terrain to the west. This exeplifies the utility of the ethod to take into account the response of the terrain, rather than having full dependency on the distance away fro the datu. The gradioetry correction took 351 ties fewer calculations, eaning there were ~.3% of the calculations necessary to produce the corrections as copared to using full-resolution. Figure 8 shows the calculated coponents of the gravity gradient. The terrain correction for an airborne survey of the vertical coponent of the gravity field was calculated in the sae anner. Figure 9 shows the corrections in Gal, which have the expected features. These appear soothed due to the observation elevation and the less decay rate than the gradioetry data. The full resolution odel would require calculations, (a) 6.53e e e e e e e e (b) e Fig. 7. (a) Quadtree odel discretization used for the gravity gradient terrain effect calculation at one observation location. (b) Close up of the quadtree discretization around the data location.

Rapid terrain corrections Exploration Geophysics 95 6.5145e+6 14 85 265 18 69 227 76 6.513e+6 44 53 37 189 151 12 21 113 6.5115e+6 2 5 75 52 11 37 6.51e+6 84 27 1 116 43 39 6.

8 Rapid terrain corrections Exploration Geophysics e e e e e e e e e e e e e e e Fig. 8. Gravity gradient tensor terrain effect at the 8179 observation locations. The density used was 2.67 g/cc allowing a axiu of 1-Eötvös error. The quadtree ethod required 351 ties fewer calculations than for the full resolution odel.

96 Exploration Geophysics K. Davis et al. 6 514 5 6 513 6 511 5 6 51 6 58 5 73 5 732 733 5 Gal 2.1 19.39 18.69 17.98 17.28 16.57 15.86 15.16 14.45 13.75 13.4 12.34 11.63 1.92 1.22 9.51 8.81 8.1 7.4 6.

9 96 Exploration Geophysics K. Davis et al Gal Fig. 9. Calculated terrain correction for the vertical coponent of the gravity field (units in Gal). The axiu error per datu is.5 Gal. The quadtree ethod required 318 ties fewer calculations than for the full resolution odel. whereas the quadtree discretization only needed to be within the.5 Gal error tolerance. The correction took 318 ties fewer calculations than the full odel, requiring 2 h on a 4.4 GHz dual-core PC. The sae PC ran an industry-standard coercial package for the terrain correction. The calculation was aborted after 4 days, and projected to take 26 days to finish. We understand that the coercial package used in this coparison ay not be optiised for the particular set-up for an airborne gravity survey; however, the draatic difference in coputing ties does highlight the efficiency of the adaptive quadtree ethod. Thus, the new approach offers a valuable alternative to the calculation of terrain correction, which is a requisite step in any gravity processing. Conclusions An efficient algorith has been developed for rapid calculation of terrain effects in airborne gravity and gravity gradioetry surveys. The ethod eploys quadtree esh discretization to represent the DEM adaptively for each datu based on the distance fro the observation point, local ruggedness of the terrain, and a user-specified error tolerance. The algorith achieves a uch higher coputational efficiency than the traditional full calculation of the DEM, and allows for ore accuracy than ethods that use prescribed discretization based on distance alone. Depending on the coplexity and relative relief of the terrain odel, the adaptive quadtree correction ethod can speed up the calculation by two orders of agnitude over the full-resolution calculation and at least one order of agnitude over currently available algoriths. The new algorith is well suited for application in odern airborne surveys that are characterised by a large nuber of observations, highly variable altitude, and accopanying highresolution DEMs such as those generated by light detection and ranging (LiDAR) surveys. Thus, this algorith provides a valuable and efficient alternative for calculating gravity and gravity gradient terrain corrections in odern airborne surveys. Acknowledgeents The authors thank the Korea Resources Corporation and the sponsors of the Gravity and Magnetics Research Consortiu: Anadarko, BGP, BP, Chevron, ConocoPhillips, Marathon, and Vale for funding support. We thank Theo Aravanis, Bob Sith, and Rio Tinto for providing the terrain data. We also thank Trevor Irons and Richard Lane for helpful discussions. References Ascher, U. M., and Haber, E., 21, Grid refineent and scaling for distributed paraeter estiation probles: Inverse Probles, 17, doi:1.188/ /17/3/314 Chen, J., and Macnae, J., 1997, Terrain corrections are critical for airborne gravity gradioeter data: Exploration Geophysics, 28, doi:1.171/eg99721 Eso, R., and Oldenburg, D. W., 27, Efficient 2.5D resistivity odeling using a quadtree discretization: Conference Proceedings, , Syposiu of Applied Geophysics on Engineering and Environental Probles. Frey, P. J., and Marechal, L., 1998, Fast adaptive quadtree esh generation, 7th International Meshing Roundtable: Sandia National Laboratories, 37, Gerstner, T., 1999, Adaptive hierarchical ethods for landscape representation and analysis: Lecture Notes in Earth Sciences, 78, doi:1.17/bfb972 Haer, S., 1939, Terrain corrections for gravieter stations: Geophysics, 4, doi:1.119/ Haer, S., 1974, Topographic and terrain correction for airborne gravity: Geophysics, 39, doi:1.119/ Harrington, J. L., 2, Object-oriented database design: Morgan Kaufann Publishers, Inc. Kass, M., and Li, Y., 28, Practical aspects of terrain correction in airborne gravity gradioetry surveys: Exploration Geophysics, 39, doi:1.171/eg821 Li, Y., 21, 3D inversion of gravity gradioeter data: Expanded Abstracts, , 71st Annual International Meeting, Society of Exploration Geophysicists. Reid, A. B., 198, Aeroagnetic survey design: Geophysics, 45, doi:1.119/ Shara, P. V., 1966, Rapid coputation of agnetic anoalies and deagnetization effects caused by bodies of arbitrary shape: Pure and Applied Geophysics, 64, doi:1.17/bf Zhang, C., Mushayandebvu, M. F., Reid, A. B., Fairhead, J. D., and Odegard, M. E., 2, Euler deconvolution of gravity tensor gradient data: Geophysics, 65, doi:1.119/ Manuscript received 7 July 21; accepted 19 Deceber 21.

10 Rapid terrain corrections Exploration Geophysics 97 四分木メッシュ分割を用いた重力および重力勾配の高速地形補正 Kristofer Davis 1 M. Andy Kass 1 Yaoguo Li 1 1 要旨 : 351 (DEM) 31 キーワード : 重力, 重力勾配, データ処理, 四分木, 地形補正 Kristofer Davis 1 M. Andy Kass 1 Yaoguo Li EU(Eötvös unit) 1/351., DEM 31.

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Downloaded 18 Jul 2011 to Redistribution subject to SEG license or copyright; see Terms of Use at 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