DCIP3D examples

DCIP3D manual:
Example 1

Introduction

We present two synthetic examples to illustrate DCIP3D Version 1.0. The main features of the program library are (i) the ability to handle data acquired both on the topographic surface and in boreholes, (ii) wavelet acceleration of the 3D inversion, and (iii) different methods for choosing the optimum regularization parameter. The first example serves to test the basic features and numerical aspects of the programs, while the second example demonstrates their performance in the presence of surface topography. The inversions are then carried out using different parameter settings. In both cases, the data to be inverted were obtained by adding independent Gaussian random noise of standard deviation 5% to the forward modelled data for the true model.

Modelling and inversion with electrodes on mesh nodes:
The bulk of this page uses black and white figures to describe the synthetic model, resistivity and IP data (surface and borehole) generated with electrodes located to coincide with mesh nodes, and resulting 3D models.

Modelling and inversion with electrodes that do NOT coincide with mesh nodes:

Colour images near the end of this page show resistivity AND chargeability models recovered from inversion of synthetic data generated over the same "5 - prism" model with electrode positions that do NOT coincide with mesh nodes. This innovation occured after DCIP3D version 1.0 was released.
There are also results of synthetic modelling (and inversion) involving a conductive block buried under a topographic pyramid, including several lines of surface pole-dipole data, and one line of data with sources inside a borehole and potentials measured at the surface.

Example 1: Five–prism Model

The first model is comprised of five anomalous rectangular prisms embedded in a uniform halfspace of 1 mS/m. There are three surface prisms simulating nearsurface distortions, and two buried prisms simulating deeper targets. DC resistivity and IP data from both surface and crosshole experiments have been computed. For the forward modelling and inversion, we have used a mesh that is made up of cells that are 50m x 50m x 25m in the central region. The mesh has been extended outwards and downwards with cells of increasing sizes. The total number of cells is 15,548. Majority of inversions are performed by using the wavelet compression of sensitivity matrices. We have used Daubechies-4 wavelet (daub2) and required the reconstruction accuracy to be 0.02. One set of inversions is carried out without the compression to demonstrate the consistency with the results obtained with compression.

Figure 1 shows a 3D view of the true model. The conductivity and chargeability of each region are listed in the table below. For the surface experiment, eleven east-west lines were used spaced 100 m apart. Data are simulated for a pole-dipole array with a=50 m and n=1 to 6. A cross-hole experiment is also simulated for four boreholes placed around the region of anomalous conductivity and cross-hole poledipole data generated.

Prism	Conductivity (mS/m)	Chargeability (%)
S1	10	5
S2	5	5
S3	0.5	5
B1	0.5	15
B2	10	15

NOTE: the outcome of experiments is given in figures below, and explanations are in the figure captions. Click radio buttons to see the specific image. Figures are not numbered in order for reasons having to do with the history of this example document.

Figures of the resistivity data

Figure 3 “Five-prism: Apparent Conductivity” shows the plan-maps of the surface DC data in the form of apparent conductivity in mS/m. Each map displays the data of the same n-spacing. The original potential data have been contaminated with uncorrelated Gaussian noise. The data clearly show the presence of three surface prisms but little can be seen that indicates the deeper prisms. There are a total of 1089 observations.

Figure 6 “Five-prism: Predicted Apparent Conductivity: Mode=1” shows the predicted data from the inversion using mode=1 of DCINV3D (Figure 5). This can be compared with the observed data shown in Figure 3. The inversion result reproduces these data well. This example serves to show the fidelity of the predicted data in the inversion. The subsequent inversions all reproduce the observation to a similar degree and so will not be shown individually.

Figure 10 “Five-prism: Predicted Apparent Conductivity: AIM” shows the predicted data from the DCAIM3D inversion. By comparing this plot with the data in Figure 3, it can be seen that the inverted model does not reproduce the data as well as in the inversion done with DCINV3D (Figure 6). The data misfit is equal to 4546, which is greater than the expected misfit of 1089.

Figures of the chargeability data

Figures of the conductivity model

Figure 2 “Five-prism: Conductivity Model” displays the distribution of the true conductivity, shown in Figure 1, in one cross-section and two plan-sections. The cross-section shows the vertical position of the four major prisms while the two plan-sections display the horizontal locations of the five prisms.

Figure 5 “Five-prism: Surface DC Inversion: Mode=1” shows the recovered conductivity obtained from inverting the surface DC resistivity data in Figure 3. Outlines of the true prisms are also overlaid. This will be the basic format for displaying the recovered conductivity and chargeability models in this series of tests. The inversion uses mode=1, in which the correct target misfit is specified and a minimum-structure model that reproduces the data to that target value is constructed. The recovered model shows all three surface prisms and the buried conductive prism well, but only shows some indication of the buried resistive prism.

Figure 7 “Five-prism: Surface DC Inversion: Mode=3” is the recovered conductivity model from the inversion of surface data using mode=3 of DCINV3D. The correct standard deviations of the errors are supplied to weight the data, but the inversion algorithm uses the L-curve criterion to determine an optimum regularization parameter and the corresponding data misfit. The L-curve estimate produced a final data misfit of 824, which is slightly lower than the expected misfit of 1089. The model is similar to that obtained from mode=1 (Figure 5).

Figure 8 “Five-prism: Surface DC Inversion: Mode=2” shows the recovered conductivity model obtained by inverting the surface data using mode=2 of DCINV3D in which the regularization parameter

is set to a constant value of 0.0885. This value of

was chosen because it was the last value estimated using the L-curve in the previous example. The final
data misfit was 826.

Figure 9 “Five-prism: Surface DC Inversion: AIM” shows the recovered conductivity model obtained, using DCAIM3D, after the third iteration. This model is not expected to reproduce the observations to the expected misfit. Consequently, it is not as good as the previous ones that were obtained using DCINV3D. However, it took approximately one quarter of the time to produce compared to mode-3 of DCINV3D.

Figure 19 shows the DC results obtained from the joint inversions of both surface and cross-hole data. Note that the buried blocks are more visible than when only the surface data was inverted (Figures 5 and 14). This is because the borehole data are more sensitive to the vertical location of the anomalies while the surface data can better define the horizontal locations. The two data sets contain complementary information and the joint inversion improves the resolution of the recovered model.

Figures of the chargeability model

Figure 11 “Five-prism: Surface IP Inversion: Mode=1” shows the recovered chargeability model obtained from the inversion of surface IP data using mode=1 of IPINV3D. For this inversion, we have calculated the sensitivity using the recovered conductivity model in Figure 5. The surface blocks are clearly visible in the constructed chargeability model, and the buried chargeable material at depth is imaged but there is no indication of two separate prisms.

Figure 13 “Five-prism: Surface IP Inversion: Mode=3” shows the recovered chargeability model obtained by inverting the surface data using mode=3 of IPINV3D, where the regularization parameter is calculated using GCV. This model appears to be smoother and attains a smaller amplitude because the estimated regularization parameter is slightly greater than the optimal value. This result is quite satisfactory and demonstrates the utility of GCV estimate in practical applications where the data noise is often poorly known.

Figure 14 “Five-prism: Surface IP Inversion: Mode=3, DC:AIM” shows the recovered chargeability model obtained by inverting the surface data using the sensitivities calculated from the conductivity model recovered in DCAIM3D inversion (Figure 9). Since the conductivity is a poorer approximation, the GCV option (mode=3) is used to estimate the regularization parameter. Since this result is similar to the previous inversion (Figure 13) where the sensitivities were calculated using the conductivity from DCINV3D, it illustrates that a full DCINV3D inversion is not always necessary to get a good IP result.

Figure 15 “Five-prism: Surface DC Inversion: Mode=1, No Compression” shows the recovered conductivity and
Figure 16 “Five-prism: Surface IP Inversion: Mode=1, No Compression” shows the recovered chargeability models obtained from inversions in which the wavelet compression of sensitivity matrix is not used. This set of figures are presented to show the savings in the CPU time that can be produced by using wavelet compression. These figures show that inversion results are similar to those in Figures 5 and 11. However, to obtained these similar results, the inversions without compression took much longer to complete. It took 33% longer to produce Figure 15 then Figure 5; and Figure 16 took 10 times longer than Figure 11.

Figure 20 shows the IP results obtained from the joint inversions of both surface and cross-hole data. Note that the buried blocks are more visible than when only the surface data was inverted (Figures 5 and 14). This is because the borehole data are more sensitive to the vertical location of the anomalies while the surface data can better define the horizontal locations. The two data sets contain complementary information and the joint inversion improves the resolution of the recovered model.

Figures of cross-borehole data

Example 1: electrodes NOT on mesh nodes

Results of inverting data generated using electrode positions that were not constrained by mesh nodes are given below. Side-by-side comparisons show the difference between working this way and forcing electrodes to be on mesh nodes.

Results using a mesh which does NOT coincide with conductive regions; data values are NOT on nodes.	Results using a mesh which coincides with conductive regions; data values are on nodes.
Slice through resistivity model under 480N.	Slice through resistivity model under 475N.
Slice through chargeability model under 480N.	Slice through chargeability model under 475N.
Slice through resistivity model at 30m depth.	Slice through resistivity model at 30m depth.
Slice through chargeability model at 30m depth.	Slice through chargeability model at 30m depth.
Slice through resistivity model at 150m depth.	Slice through resistivity model at 150m depth.
Slice through chargeability model at 150m depth.	Slice through chargeability model at 150m depth.