Rapid interactive modeling of 3D magnetic anomalies

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Author's personal copy Computers & Geosciences 48 (2012) 308–315

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Rapid interactive modeling of 3D magnetic anomalies Fabio Caratori Tontini n GNS Science, 1 Fairway dr, Lower Hutt 5010, New Zealand

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 November 2011 Received in revised form 14 December 2011 Accepted 9 January 2012 Available online 18 January 2012

I implement a MATLABs function (for_3DFFT_mag) for calculating magnetic anomalies from a 3D distribution of magnetization, which can be loaded interactively through an user-friendly graphic interface. The forward calculation engine is based on a 3D Fast Fourier Transform computation, that gives accurate results in a very short computing time, making the use of this program particularly suitable for 3D interactive modeling of observed magnetic anomalies. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Magnetic anomalies Magnetization models Interactive modeling

1. Introduction Forward modeling techniques are widely used to interpret observed magnetic anomalies (Blakely, 1995). In this approach, a starting model based on pre-existing geological and geophysical knowledge of the magnetization distribution is interactively adjusted to reproduce the magnetic field observations. Although simple in principle, the calculation of magnetic anomalies generated by complex 3D models is a challenging problem. Commonly, the subsurface magnetization distribution is approximated by a set of prismatic cells with constant magnetization (Pignatelli et al., 2011). The model magnetic anomalies are then computed by summing the contribution of each cell at the observation locations. Despite this being a linear problem, i.e. the contribution of each cell to the given observed datum is a linear function of the cell’s magnetization, in the practical case we are often limited by the computing time needed to calculate the corresponding sensitivity matrix, as well as by the amount of computer memory needed to store it. For example, if we want to model a data-set made of N ¼ 100 100 observations by using a 3D grid made of M¼ 100 100 100 cells, we need to calculate and store a N M matrix, i.e. a matrix made of 1010 elements. If the matrix elements are described by double precision real numbers, the corresponding matrix size will be of the order 102102 GB. Calculating and storing a similar matrix may still be a problem even with modern desktop computer workstations, and the use of this matrix in an interactive forward modeling tool is a challenging computational task.

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Here I use a set of recently developed exact 3D forward modeling equations for the magnetic field (Caratori Tontini et al., 2009), based on a 3D Fast Fourier Transform (FFT) of the magnetization distribution. A 3D mesh is used to define the magnetization model, and the equations are used to calculate the field at the same nodes of the same 3D mesh. The field can then be calculated by a trilinear interpolation at any given location, provided this location is outside of magnetized material but is contained within the 3D mesh. Since the sensitivity matrix is not involved in the calculation, which consists of a simple 3D FFT and subsequent filter of the magnetization distribution, the method is very fast permitting a ‘‘real-time’’ computation of the magnetic anomaly each time the model is changed. The MATLABs implementation of the corresponding program has the great advantage of using all the pre-existing MATLABs graphical tools and functions which allows the method to be used in an interactive approach to interpret 3D magnetic anomalies. Furthermore, the software can be used as a sensitivity modeling tool, to evaluate expected magnetic anomalies from a given complex source 3D model before undertaking a magnetic survey, or to test the response of inversion programs for known source models.

2. Anomaly computation We briefly discuss here the forward modeling equations used to compute the magnetic field by the computer program proposed here. A complete description of the theoretical background and the numerical comparison with the exact calculations are discussed in Caratori Tontini et al. (2009). The 3D expression in the Fourier domain for the magnetic anomaly as a 3D function

Author's personal copy F. Caratori Tontini / Computers & Geosciences 48 (2012) 308–315

corresponding sensitivity matrix for 100 100 observation surface data could take several hours.

defined in the domain outside of the source volume is F ½DT ¼ m0

b k½M b k ½F 2

k

F ½M,

ð1Þ 3. Program execution

where F is the 3D Fourier transform operator; m0 ¼ 4p 107 Henry=m is the magnetic permeability of free space; b is a unit vector aligned with the ambient magnetic field; F b is a unit vector aligned with the source magnetization; M k is the 3D spatial wave-vector; M(x,y,z) is the 3D distribution of magnetization. From a computational perspective, Eq. (1) can be straightforwardly implemented by the following steps:

Interactively define a 3D grid of magnetization values M. Perform a discrete 3D FFT of the magnetization distribution M. Multiply the Fourier transform of the magnetization distribu

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b k½M b k=k2 . tion by the filter ½F Perform an inverse 3D FFT of the filtered magnetization to obtain the scalar value of the magnetic anomalies at the nodes of the 3D grid. Calculate the magnetic anomaly data on a given (not necessarily planar) observation surface by trilinear interpolation.

These steps are the key to explain the differences with the matrix approach in terms of execution times. As an example a 3D magnetization grid composed of 100 100 100 prismatic cells can be calculated in less than 1 s on a normal desktop PC. There is no virtual limitation to the number of cells that can be handled, apart from the users computer memory. The computation of a

3.1. Program layout The program interface is shown in Fig. 1. There is an input menu, for import/export of model and anomaly data files, and a help menu calling the program manual and additional information about the program. The three stacked panels on the left of the program window (Fig. 1) allow the user to generate the 3D mesh used to define the magnetization distribution, define the magnetic field and magnetization directions, and interactively build and visualize the 3D magnetization models. The right panel shows the map data, i.e. the observation surface, the topography surface, the observed magnetic anomaly and the calculated magnetic anomaly. These items will be described in detail in the following subsections. 3.2. Program menu The program requires a data input file to start the plotting and calculation commands (Fig. 2). The input file contains information about the observation location, the observed magnetic anomaly, and the topography elevation at the observations locations. The data input file format is an ASCII file with eight columns delimited by spaces (see attached example files), i.e. Observation Easting (m), Observation Northing (m), Observation Elevation (m), Magnetic Anomaly (nT), Error on Magnetic Anomaly (nT), Topography Elevation (m), Bottom Depth (m), Reference Model (A/m). The topography data are plotted as contour lines during the model building procedure

Fig. 1. Snapshot of the program main window.

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Fig. 2. File menu.

to allow the user to magnetize the cells inside the topography relief. The Bottom Depth and Reference Model channels are not relevant to the program calculations but are retained for compatibility with other existing inversion and modeling programs (Caratori Tontini et al., 2003, 2006, 2008). The Error on Magnetic Anomaly channel is an estimate of the error standard deviation on the magnetic measurements which is used to evaluate the least-squares norm of the differences between observations and forward calculations. The user can load a pre-existing input file by following the option Load input file in the File menu. If the user is interested in creating a synthetic data-set, the option Prepare scratch input file is available. In this case, a regular 2D data grid at constant altitude will be created by using an additional input window where the user can specify the data extent and the grid cell size. In this particular case, a flat topography will be generated at the same constant elevation of the data grid. The user can also load a pre-existing magnetization model by Load input 3D model. The 3D magnetization model file and the calculated anomaly data file (in the same format as the input data file) are exported by Write output 3D file and Write output anomaly file, respectively. The Exit option closes the program, clears all the variables and closes all the open figures. The Help menu is characterized by two entries, calling an .html file with a manual for the program and other information about the program release, respectively.

Fig. 3. Mesh parameters panel.

Fig. 4. Magnetic directions panel.

3.3. Mesh parameters panel The user can specify the mesh characteristics and number of voxels (volume pixels) by the input boxes in this panel (Fig. 3). By default, once an input data file is loaded, these values are initialized by estimating the 3D mesh size and extent from the data distribution, assuming the data are organized into a regular rectangular grid. In this case, a mesh is built with the same horizontal spacing and extent of the data-grid. The user can in any case redefine the mesh according to his/her own needs either to build a different 3D model, or to use irregularly sampled data. To avoid extrapolation errors however, the horizontal mesh extent should be made at least the same size as the horizontal data extent. However, if a 3D model is also loaded, the mesh is automatically generated according to the 3D model extent and thus it should not be changed by the user to avoid program errors. When the Accept input parameters button is pressed, the mesh is loaded into the program and the map plots appear in the main maps program panel. 3.4. Magnetic directions panel The input boxes in this panel are used to specify the geomagnetic field and magnetization directions (Fig. 4). These directions are fixed for the entire source. However, unlike existing programs based on a matrix approach, the user can change the magnetic directions any time during the program execution and changing the corresponding anomaly data and thus exploring the magnetic response of the

Fig. 5. Magnetic slice model panel.

model to variations of the magnetic directions. This is of particular interest either to generate different synthetic models for varying magnetization directions, or to model magnetic anomalies when the magnetization directions are not well constrained by paleomagnetic measurements. 3.5. Magnetic slice model These panels control the model building and visualization and the corresponding anomaly calculation, i.e. the core of the program (Fig. 5). The model is built by using a set of horizontal cross-sections


Fig. 6. Interactive assignment of magnetization.

Fig. 7. The 3D model visualization tool.

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defined by the numbers of vertical voxels in the mesh parameters panel. I use a similar approach to the Polygon mode of Pignatelli et al. (2011). By default the model has zero magnetization everywhere, unless a 3D model has been loaded by using the File menu. The user selects the slice number to be magnetized by the corresponding input box in the panel. The user can copy the magnetization from a pre-existing slice to the current slice by typing the corresponding reference slice number in the appropriate input box. The magnetization value shall be introduced in the corresponding input box. This choice makes possible either to change the magnetization slice by slice, and also inside the chosen slice by drawing different polygons. By pressing the button Draw magnetization model the program opens a new window with a representation

Fig. 8. Data file parameters for the prism model.

of the slice magnetization model (Fig. 6). If the chosen slice cut through topography relief, the corresponding topography contour line is drawn by a white line as a reference for the user to magnetize the cells within the topography relief. The user can draw a polygon by sequentially pressing the mouse left button and the program will magnetize the cells covered by the polygons. Each slice can accept variable magnetization values by changing the magnetization of each polygon. For example, this is very important in the modeling of volcanic terrain with complex lithologies. The user can also demagnetize cells by this approach by setting the magnetization value to zero in the corresponding input box. When the slice magnetization model is built, the right mouse button pressure adds or substitutes the slice to the pre-existing 3D magnetization model and the control returns to the main program. The revised magnetic anomaly is then computed, making this procedure particularly useful to fit real data by a trial-and-error approach by comparing the revised theoretical model anomaly with the observations in the maps panel. The user can also recalculate the magnetic anomaly by pressing the button Anomaly evaluation, for example after the magnetic directions have been changed or after a different 3D model file is loaded. The program can also smooth the magnetic anomalies by a low-pass filter that can be interactively defined by the corresponding slider command. The filtered anomaly will be shown after the button Anomaly evaluation is pressed. The entire 3D model can also be explored in an interactive way. The button View 3D model opens a new window where the 3D model is shown by the intersection of three perpendicular cross-sections (Fig. 7). The user can explore this model by interactively moving through the vertical cross-sections by the keyboard arrows and changing the depth of the horizontal cross-section by the keyboard þ/. The user can also change the perspective of the 3D view by using the l/r keyboard (left/right), or the u/d keyboard (up/down). The shading can be changed by the s keyboard and the colorbar by the c keyboard. The user must press Esc to return to the main program and disconnect the visualization tool before any other operation is performed. When the button Statistics is activated, the map of the difference between observed and calculated anomaly is shown, together with the least-squares norm of the difference. This may be used to perform a quantitative match by a trial and error approach.

Fig. 9. Prism geometry.


4. Data and model examples I show two data and model examples to demonstrate the use of the program. In the first example, I will calculate a 3D model anomaly from scratch, while in the second scenario I will simulate a set of real observations with topography by the interactive generation of a 3D model. The users can reproduce these examples by using the files attached to the program distribution. 4.1. Prism example We test the response of the program to the magnetic anomaly generated by a simple prism. To this aim we use the command

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Prepare scratch input file in the File menu. We simulate a data file organized on a regular grid at an elevation of 1 m a.s.l. made of 51 51 data with 100 m cell-size, i.e. a data extent of 5000 5000 m2, by setting the corresponding parameters on the input window (Fig. 8). The file is saved (test_scratch.txt), and then imported by the command Load input file. When importing this file, the software recognizes the organization of the data into a regular grid, and introduces proper values for the horizontal extent of the mesh. The user can obviously change these values, but for this example we use the default values. We still have to define the vertical extent of the mesh, however. We want to calculate the magnetic response of a prism with a top depth of 1 km and a bottom depth of 2 km. To accomplish this we build a mesh with a vertical extent of 3000 m, ranging from 0 to 3000 m. We sample this mesh with 31 horizontal cross-sections spaced 100 m apart. We assume that the magnetization is dominated by the induced component, having an inclination of 601 and declination 201. We start building the model from slice 11, by drawing a prism with 5 A/m magnetization, located between 2000 m East and 3000 m East, 1500 m North and 3500 m North. We replicate these horizontal cross-section from slice 11 to slice 20. The full 3D model can be explored by the model visualization tool and a snapshot is shown in Fig. 9. The corresponding magnetic anomaly is shown in Fig. 10. No smoothing filter was applied. The corresponding 3D model is contained in the file test_scratch_model.txt.

4.2. Topography model

Fig. 10. Snapshot of the prism model magnetic anomaly.

In this case we simulate an airborne magnetic survey at a constant altitude of 1800 m conducted over a volcano edifice. The data file in this case is test_topography_uniform.txt. The 3D magnetization model is contained in the file test_topography_uniform_model.txt and it is shown in Fig. 11. It consists of a uniform 5 A/m magnetization distribution enclosed within the topographic relief.

Fig. 11. Topography model geometry.



Fig. 12. Snapshot of the topography model maps. The top left panel is the observation surface.

Fig. 12 shows a snapshot of the program with the maps of the observed magnetic anomaly, generated by using the numerical approach of Parker (1972), and the calculated magnetic anomaly by the 3D FFT method. The topography map is also shown. Despite a poor sampling of the mesh along the vertical direction, based on just 16 horizontal layers, the agreement between the calculated magnetic anomaly and the observations generated by the method of Parker (1972) is good. No smoothing filtering was applied.

Acknowledgments The paper benefited from the internal review by Grant Caldwell and Supri Soengkono. Two unknown reviewers provided important comments. This contribution was made possible through funding by the Royal Society of New Zealand by the Marsden Fund (Grant GNS1003).

Appendix A. Supplementary data 5. Conclusions We have shown a graphical interactive software program for generating and interpreting magnetic anomaly data by building complex 3D models. The calculations are performed by using a recently published fast forward Fourier transform engine which makes this program particularly useful for a 3D trial-and-error forward modeling approach. Further development will be to adapt this method into an inverse modeling tool. The program can be freely obtained by contacting the corresponding author.

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.cageo.2012.01.006.

References Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications. Cambridge Univ. Press, New York. Caratori Tontini, F., Cocchi, L., Carmisciano, C., 2006. Depth-to-the-bottom optimization for potential-field data inversion: the magnetic structure of the Latium volcanic region. Journal of Geophysical Research 111, B11104.


Caratori Tontini, F., Cocchi, L., Carmisciano, C., 2008. Potential-field inversion for a layer with uneven thickness: the Tyrrhenian Sea density model. Physics of the Earth and Planetary Interiors 166, 105–111. Caratori Tontini, F., Cocchi, L., Carmisciano, C., 2009. Rapid 3-D forward model of potential fields with application to the Palinuro Seamount magnetic anomaly (Southern Tyrrhenian Sea, Italy). Journal of Geophysical Research 114, B02103.

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Caratori Tontini, F., Faggioni, O., Beverini, N., Carmisciano, C., 2003. Gaussian envelope for 3D geomagnetic data inversion. Geophysics 68, 996–1007. Parker, R.L., 1972. The rapid calculation of potential anomalies. Geophysical Journal of the Royal Astronomical Society 31, 447–455. Pignatelli, A., Nicolosi, I., Carluccio, R., 2011. Graphical interactive generation of gravity and magnetic fields. Computers & Geosciences 37, 567–572.