aglo

MAG3D manual: 
Synthetic examples - Ver. 4.0
 

Introduction

We present two synthetic examples to illustrate Version 3.0 of MAG3D. Important functionalities of MAG3D are the ability to handle multicomponent borehole data and the wavelet compression for large scale problems. The two synthetic examples are constructed to show these features. The first model consists of two vertical prisms, and both surface and borehole data are simulated. We illustrate the inversion of individual data sets and joint inversion of surface and borehole data. The second example is intended to be large and is composed of several prisms buried at different locations and depths beneath a topographic surface. Aeromagnetic data are simulated for this example. The size of this example is too large for the direct approach to handle. We show that the wavelet compression allows the solution of such a large problem with little demand on computing resources.

Example 1

This figure displays the synthetic susceptibility model. It consists of two prisms buried in a non-susceptible background. The collar positions of three vertical boreholes are indicated on the plan-section. An inducing field in the direction I=65 and D=25 is assumed. The total field anomaly was calculated on the surface at an interval of 25 m along east-west lines spaced 100 m apart, resulting in a total of 175 data. Three-component data in easting, northing, and vertical directions are calculated in the boreholes. There are 16 observation locations spaced 25 m vertically in each hole, and the total number of data in three holes is 144.

Gaussian noise having a standard deviation of 2 nT has been added to all data and the resulting simulated observations are shown in the next figure where the surface data are shown as gray-scale contours and the three-component borehole data are plotted as functions of depth. For inversion, we use a m region and divide it into 24x24x16 cubic cells of 25 m on a side. This yields a total of 9216 cells. For all the inversions presented below, the distance weighting is used.

This figure shows the susceptibility model recovered from inverting the surface data alone. The true positions of the prisms are indicated by the white lines. This model clearly shows the presence of the shallow prism at the correct location but it does not give a clear indication of a separate, deeper prism. There is only a broad zone of low susceptibility extending from the high-susceptibility zone. The vertical extent of the anomaly is not well-defined.

models Click the button to see the susceptibility model recovered by inverting the three-component borehole data. The model shows two regions of high susceptibility at locations corresponding to the true prisms, and the recovered depths agree well with the true depths. However, the amplitude of the shallow prism is small and there is no clear separation from the deep prism. Despite this, the model provides a good result considering that there are only three widely separated boreholes and that the inversion has no explicit information regarding where to place the magnetic material.

Click to see the susceptibility model obtained from the joint inversion of the surface and three-component borehole data. This model combines the merits of the models from individual inversions. Both target prisms are well define in horizontal and vertical locations and their amplitudes are comparable to those of the true model. This model provides the best representation of the true model.

This is the total-field Anomaly Inversion. It shows the model recovered by inverting the surface and borehole total-fieldanomaly. The total field anomaly in the boreholes is first computed from the three-component data shown above. This simulates a more realistic situation since in many practical applications, only the total field anomaly can be extracted accurately from borehole measurements. The recovered model images both prisms. The success of this inversion demonstrates that single component borehole data can provide the information that is complementary to surface data when multicomponent data are not available.


Example 2

The true model is illustrated to the right. The figure showsthe relative sizes and positions of seven susceptible bodies buried in a non-susceptible background. The six smaller bodies are placed at shallow depths to simulate small-scale anomalies, and the large block is placed at a greater depth to generate a broad anomaly over which the small anomalies are superimposed.

The surface topography above this model is shown in the next figure. The elevation of the surface ranges mostly between 0 and 125 m, with a few points reaching 150 m.

synthmodel

dataSimulated aeromagnetic data calculated at a constant terrain clearance of 75 m are shown next. The data are located on a grid with a 50–m spacing in both directions. This simulates a normal data set that has been decimated - it is NOT recommended that "gridded" data are used for inversion. The inducing field is assumed to be in the direction I=65o and D=25o, and total field anomalies are calculated. The data have been contaminated by independent Gaussian noise having a zero mean and a standard deviation of 5 nT. The data show six anomalies due to the shallow blocks, but they provide no indication about the presence of the deep block.

To invert these data, a model region of 3.2 km by 3.2 km by 1.5 km is used. The top of the mesh is placed at the elevation of 125 m. The cell width is 50 m in both horizontal directions and the thickness varies from 25 m near the surface to 100 m at the bottom. After the surface topography is discretized onto the mesh, the resulting model contains a total of 110,000 cells. The corresponding sensitivity matrix requires more than 1.5 Gb to store, and that is beyond the memory limit of many workstations. When compressed using the Daubechies-4 wavelet at a reconstruction accuracy of 5%, a compression ratio of 76 is achieved. The compressed sensitivity matrix requires on 43.5 Mb of storage. The sensitivity calculation takes 245 minutes on a SUN Sparc20 workstation, or 175 minutes on a 233–MHz MMX Pentium PC.

3dmodelThe model objective function is specifie by choosing
S=0.0001, X=Y=Z=1, and a zero reference model.
The 3D weighting functions are all set to unity. The inversion is performed by setting the target misfit to the expected value of 3,600 and executing MAG3D with mode=1. The inversion uses 60 Mb of memory and lasts 146 minutes on the SUN Sparc20 workstation, or 110 minutes on a 233-MHz MMX Pentium PC having 64 Mb of memory (1995). Thus the entire procedure from the calculation and compression of the sensitivity to the inversion requires about 392 minutes on the SUN workstation, or 285 minute on the PC.

The last figure displays the susceptibility obtained from the inversion. The model is shown in six plan-sections. The different bodies in the true model are well- imaged. In particular, the large block at depth is cleary visible. The blank area in the section at Z=87.5 indicates the region above the topographic surface. The depth labelled in each section is referenced to the surface elevation of 125 m.

The large block at depth and the small block near the surface have a susceptibility of 0.08 SI unit, and the other blocks have a susceptibility of 0.05.

Click this button to see the true (synthetic) model.