Inversion theory:  
Practicalities: 2D DC resistivity as the example

Contents and objectives

In this section, fundamental concepts are implemented for a 2D DC resistivity data set. A synthetic situation is developed using software tools provided. Each step can be reproduced with tools and models included on the CD-ROM. Exercises and questions are embedded within the text to encourage experimentation with the methodology.

NOTE: detailed instructions about performing inversions are given in the WorkFlows of Chapter 7. The rest of this page is only an outline with a few details.

1. Introducing the scenario

Theory you have learned in earlier sections of this chapter will now be put into action. We will use a synthetic DC resistivity survey and the UBC-GIF's forward modeling & inversion program library called DCIP2D. An educational version of this program is provided on the CD-ROM, and a complete description and documentation are available via the "Software & manuals" chapter. The artificial 2D electrical conductivity structure of the Earth that is used to generate the data will be revealed later, but for now we will start with the data set and work through an inversion sequence as if we were doing a normal job.

There are several steps, as outlined in the table of contents above. The generic flow chart icons are included to help remind you of where in the inversion methodology we are currently working. There are occasional sidebars to provide clarifaction without interrupting the flow. Questions to direct your thinking are included via the questions icon . Click here to see all the questions on one page.

All examples were generated using facilities of DCIP2D, the UBC-GIF 2D DC/IP inversion program.

2. DC resistivity data, errors and prior information

First all four pre-requisites in the top box of the flow chart must be considered. They are 1. Field observations, 2. Error estimates, 3. Ability to forward model and 4. Prior knowledge.

The measurements include source current, resulting potentials, and the geometry of electrodes, but recorded data usually are apparent resistivities. Background information on DC resistivity surveys is included in Chapter 2 "Foundations", section e "Geophysical Surveys". However, plots of raw data are used here, and the conventional plotting procedure called pseudosections is not self explanatory. Therefore, a sidebar is provided here to briefly describe these  data plots. There is no other processing of data prior to inversion, except if voltages were not saved during the survey - then they must be derived by working the apparent resistivity formula backwards, assuming a source current of 1 Ampere. Also, of course, the input data must be formatted into a text file with the proper format. The input file used in this exercise can be seen here. In this file, columns are not labelled - they are (left to right) transmitter electrode positions along the line (2 columns), receiver electrode positions (2 columns), normalized potential measurement, and an estimate of the standard deviation of that measurement. This last column can be added automatically using the DCIP2D user interface, as explained in part 5 below.

Errors in the data arise for various reasons including inaccurate locations of electrodes and electrical noise (signals can be in the range of low milli-volts or on the order of microvolts). Also, if the real Earth is far from two dimensional near any electrodes, then "geologic errors" occur because the features causing data cannot be recovered perfectly with a strictly two-dimensional (2D) method.

Forward modelling of DC resistivity data based upon a 2D earth (discretized as shown below) is carried out using equations that are explained in section 3 (Principles) of "DC resistivity" in the "Geophysical Surveys" section of Chapter 2 "Foundations".

Prior knowledge includes field geology, "guesses" about physical property values, expected structures, etc. These will be used in the interpretation. In addition, the methodology makes use of implied prior knowledge as explained in earlier sections of this chapter. For example, we will be looking for smooth models that are close to a reference model that can be defined by the user. This reference model is often simply a uniform Earth with fixed electrical conductivity.

3. Discretizing the earth

2D models of the Earth are cross-sectional views of structure and materials directly under the survey line. A very important assumption is that there are no variations perpendicular to the cross-section. For inversions using the UBC-GIF methodology, the cross-section is discretized using rectangular cells.

Smaller cells are used to fill the zone where current has been injected. These determine the spacial resolution of the model. More cells may seem to be better, but with only limited data, the finest feasible resolution will be obtained using cells with widths equal to half the electrode spacing, and depths equal to a quarter of the electrode spacing.

The mesh used to define a discretized 2D model under the survey line. The survey extended from 0m to 200m. Zones outside this region are "Padding", explained in the text. Mouseover illustrates.

Larger cells are used around the main region of interest to allow for mathematically smooth transitions towards the edges of the domain. Without this buffer (or "padding") zone, it would be impossible to obtain sensible values for the region of interest. Larger cells are acceptible since this zone of the model will not be interpreted.

Topography must be approximated using rectangular cells. When a "default" mesh is used with UBC-GIF codes, the program builds a discretized 2D Earth with cells in the zone under the electrodes that are 0.5S in width and 0.25S in depth (where S is minimum electrode spacing). Outside this region, cells are added with increasing size towards the periphery.

4a. Getting started with DCIP2D - the data

The first task is to inspect the data set itself in order to gain first impressions and to ensure that sensible errors are applied. (Find the educational version of the DCIP2D program library in Chapter 10, "Software & manuals".)

A

For some hints on familiarization with the data, click the questions button to the right.

Above: Data pseudosection for our synthetic data set. <== Left: The graphical user interface for DCIP2D.

B

When first examining a DC resistivity data several questions should be considered. The two main objectives are to gain some appreciation for what is likely to be in the ground prior to carrying out the inversion. Click the question button for some suggestions of applicable things to think about.

4b. Misfit criterion: specifying errors

We need to judge the reliability of the data set, and to specify errors for the data accordingly. This is necessary so that a suitable degree of misfit can be applied in the inversion. Making this judgement requires some knowlege about the acquisition of data in the field, and for the likelihood that there were problems or external sources of noise.

True statistical estimates of error from the field can only be obtained if many versions of the data set were obtain. Since this is almost never done, it is common to assume there are random Gaussian errors, and standard deviations for each datum can be applied using both a percentage of the datum and a minimum value, or offest. The DCIP2D user interface allows you to apply these noise estimates. For the data set used in our synthetic example, the added noise is a random Gaussian value based upon 5% of each datum plus a 0.001 Volt minimum.

Pseudosection display in DCIP2D, with the properties dialogue for a single datum.

C

If you have some field experience, some questions you might like to consider regarding errors are in Question Set C. Errors can also be adjusted for individual data points if you suspect any datum is particularly noisy. For example, it is not uncommon for all data values recorded at one electrode location to have additional noise, due for example to a poor electrical contact, a nearby metallic fence, or other reasons. Specifics for every datum can be examinied in the GUI by clicking on any data point (see the previous figure).

The standard deviations we specify for each data value are the in the measure of misfit, which was explained in the previous "Norms and Misfit" section. The misfit measure equation is reproduced to the right. The inversion will have to find a model which can cause predicted data that match observations to within a specified degree, using this equation to carry out the misfit test. The user must specify what value of this function will be acceptible. More discussion on this is given in the "Adjusting misfit" section below.

5. Specifying the desired optimal model

As explained in the previous "Norms and Misfit" section, the so-called model objective function is used to define the type of "optimum" model the inversion algorithm is looking for. This function is a way of quantifying desirable features of a physical property model. The inversion chooses an optimal model by searching for a model which will minimize this function subject to the constraint that the chosen model can generate predicted data that satisfy the misfit criteria. The model objective function is

,

and it is defined using two components:

1. The algorithm will try to find a model that is as close as possible to a reference model defined either as a half space (by default a halfspace with a resistivity equal to a weighted average of measured apparent resistivities), or as some model with more complexity. The latter option can be invoked by the user if there is enough prior knowlege.
2. The model will be as smooth as possible in the X and Z directions.

In fact, the significance of each component is controlled using the "Alpha" coefficients s, x, and z in the equation above. That means that the user can request a model that emphasizes component 1 or component 2.

Default values of these coefficients are determined by the program based upon the length scales of the survey and mesh. The inversion's task is to find a model that minimizes this relation; the result will be an optimal model.

For DCIP2D, the default specifications for the model norm have been found to work well as a first attempt, but experimentation and adjustment of the parameters defining the desired model type is expected during the course of inversion processing. This will be discussed in the "Alternative models" section below.

6a. First inversion

The first inversion should be run only after learning as much as possible from the raw data, including how to set errors properly. The DCIP2D user interface has defaults for all parameters except the input file name.

If our synthetic data set has errors assigned using 5% + 0.001V minimum, then running the first inversion with all default parameters will produce a reasonable initial solution. However, with synthetic data that have been contaminated with small amounts of Gaussian random noise, we can probably do better using a target misfit, , that is specified at less than N, the number of data. In our case, a first inversion with Chifact = 0.5 works well, and we will discuss the outcome of this inversion. A sidebar shows  predicted data and models as the inversion progresses.

6b. Evaluating results

The GUI after completing the inversion is shown in the next figure. Point your mouse to either model , pre, or log buttons to display an image of the corresponding GUI window for the inversion that was just completed.

See text for notes on images obtained by pointing to either of the buttons on the image to the right. Reset to blank

How should you examine these results to determine if the inversion was successful at returning a reasonable model? There are five aspects to observe and consider:

1. A log file is produced containing information about how the inversion progressed. For a quick assessment of whether your inversion run proceeded normally, point to the button and look at the last few lines of the resulting text file. Here are some questions to consider when using the log file. Arrows in the figure correspond to colours in this list:
   • Did the inversion end with “convergence?”
   • What value of target misfit was specified?
   • Was this target misfit achieved?
   • How many iterations were performed?
      
2. Next, predicted data should be compared to observations. Pointing to the button displays pseudosections of the observations and the predicted data. They should look nearly identical. To see variations more closely, click the "diff" button in the data viewing window. This changes the second pseudosection to a "misfit map," which shows the differences between the two data sets. This is what is shown above. The misfit map should look random, with maximum values of some small percentage of the measured data (based upon noise specifications).
    
3. Now look at the resulting model. The button brings up the model viewing window with the complete model displayed, including all the padding cells. Select "Padding cells" in the "Options" menu in this window to specify how many padding cells to drop from the display. The image above has 3 cells dropped from either side, and 4 cells dropped from the bottom. You can also adjust the minimum / maximum values for the colour scale - necessary for comparing various models.
    
4. The progress of the inversion during it's iterations should also be checked. In the model viewing window, the algorithm's progress can be displayed graphically by selecting the "Curves" toolbar button in the "View" menu. The resulting graph shows how the values of misfit and model norm varied at each iteration. The algorithm is programmed to add structure gradually in order to find a model that explains the data - i.e. it works on reducing the misfit value (blue curve) until the target misfit is reached. Then it must try to minimize the model norm without changing misfit. Thus, you should see a slight drop in the model norm value (red curve) until no more adjustments can be made to improve the situation.
    
5. Geologically reasonable? It is important to decide whether the resulting model is geologically reasonable. This final consideration is more subjective. A simple example is shown here, in which data produced by calculating data over the "true" 2D model (top right) are inverted twice to produce two inversion results which are both inadequate. The image labelled "underfit" is a model recovered when the target misfit was too large. The program has stopped looking for details when predictions look only somewhat like observations. The image labelled "overfit" is a model recovered when the program has tried too hard to find details that explain every nuance in the observations. The corresponding situations were discussed using the UBC-GIF Linear Inversion Applet in this chapter, Section d., "Optimization".

7. Second inversion

Your first inversion rarely produces an optimal result. More inversions must be run to obtain alternate models. Alternate models are obtained by changing either the model norm or the value of misfit. Changing the model norm can be done in two ways: a "reference model" can be adjusted, or the degree of flatness and smallness can be adjusted. See the last part of "Feasible model norms" in the "Norms and misfit" section of this chapter.

First you should identify what values the program set for its default run (see question set D above), and then you should adjust one of these to obtain a second result.

D

Here we will start by specifying a different reference model. One with a value similar to the first model's lowest conductivities is a good choice.

E

Upon completion there will be a second model can be compared with the first, default model. These are both shown in the figure below. Typical questions to consider are given in question set E.

Resistivity model using default reference model.

Resistivity model using reference model of 1000 Ohm-m.

8. Depth of investigation (special for 2D DC/IP inversions)

Models produced by inversion of DC resistivity data appear to fade with depth, but how can we gain a more quantitative understanding about which regions of the model are reliably constrained by our measurements? If there are at least two reasonable models obtained using different reference models, the two models can be compared to identify which regions of the model most significantly affect the measurements. Results of doing this are explained next.

F

Using the DCIP2D GUI, the method is applied within the model viewing window, using "Depth of investigation" option in the "Options" menu. There must be a second model that was recovered using the same mesh as the one being observed. Results for our synthetic model can be seen as soon as the two inversions described above have been done. Any two inversions can be used so long as the regions constrained by the norm are different. Four versions are shown here, and typical questions that should be considered with such results are given in question set F.

Default model (chifact = 0.5), depth of investigation (DOI) via a model that used mref=0.001S/m. Same model, DOI via a model that used mref=0.02S/m. Model using mref=0.001S/m, DOI via a model that used mref=0.02S/m. Model using mref=0.02S/m, DOI via a model that used mref=0.001S/m. Evidently, all versions will be interpreted similarly. The left buried conductor appears to be small with more resistive material beneath approximately 40m depth. The stronger buried conductor to the right, however, does not have a bottom that has been imaged by this survey. DC current tends to accumulate in conducting regions so investigation depths in conductive ground tend to be shallower.

9. Alternative models

When geophysical models created by this inversion methodology are used to make geological interpretations, it is a crucial that more than one model should be used. The reason is that with any single model, many questions can be posed. Only by comparing several results can you begin to gain reliable answers to these questions. For example:

• Which of the features in the model actualy emulate the earth?
• To which regions of the earth are data most sensitive?
• How deep are we seeing?
• Are anomalous regions in the model open or closed at depth?
• Which features are in fact due to the particular objective function chosen?

There are three main approaches to finding alternate models, and these are outlined next.

9a. Adjusting the reference model

Our second inversion done above involved specifying an alternative reference model. This is a powerful tool for exploring which regions of the model are reliably defined by the data. This point is emphasized in the interactive figure below, which shows five models recovered for a new, synthetic model, which is somewhat more complex than the small example we have been using. All recovered models (2. through 6.) are equally valid because they can each reproduce the data to within the specified misfit, but they are different because of the specific reference model used for each.

1. True model 2. mref = 40 Ohm-m 3. mref = 400 Ohm-m 4. mref = 4,000 Ohm-m 5. mref = gradient from 400-400,000 Ohm-m 6. mref = gradient from 400-0.04 Ohm-m

9b. Adjusting the model norm

Recall that the "ruler" or model norm used to distinguish between different models involves two parts. For 2D problems, this mathematical ruler looks as follows. It has one part which measures how close the resulting model (m) is to a reference (m₀), and a second part that measures how smooth the model is. In fact, the smoothness in the horizontal and vertical directions are measured separately so the second part of our norm contains two components.

The UBC-GIF inversion routines make the reference model, m₀, and the three 's available for adjustment. What is the effect of adjusting these coefficients? Recall that this function will be minimized. If one of the 's is very small, then the corresponding term will contribute little to this minimization. So, reducing the size of s will result in smoothness in the X- and Z-directions in favour of closeness to the reference. Reducing the size of _X or _Z will result in models that are preferentially smooth in the X- or Z-directions respectively. Images below illustrate the effect of different choices for values of the three 's .

Click buttons to see models resulting when the three parameters are modified as follows: (s, x, z) = (.001,    1,    1) (s, x, z) = (   1, .001, .001) (s, x, z) = (.001,    1, .001) (s, x, z) = (.001, .001,    1) Are all models equally capable of reproducing the data? Yes, predicted data for all results are very similar. You can try this yourself, and consider the questions in question set G.

G

What about the actual values to specify for the parameters? This will be outlined in the "Tips for Success" section of this chapter.

9c. Adjusting misfit

Finally, what is the effect of adjusting the target misfit value to numbers other than N, the number of data? An initial comment was made above, but you can explore the consequences of adjusting the misfit by changing Chifact.

In fact, it is instructive to generate models for a range of values, and use the resulting values of and to generate the Tikhonov Curve (first introduced in the "Optimization" section of this chapter).

H

The results of inverting our example data with a range of values for Chifact are summarized in the following table and corresponding Tikhonov Curve. Some questions to consider are in question set H.

Table of values resulting from inversion using five values of Chifact.
Click to see models recovered using each value of Chifact. These results can be reproduced yourself. Chifact =0.17 Chifact =0.2 Chifact = 0.5 (Model recovered earlier) Chifact =1 Chifact =3 Chifact =10

10. Conclusions

After working through this page, some relevant closing remarks are:

• Mathematical non-uniqueness can be quite severe. We have seen many models that can be interpreted, and all are equally capable of generating the measured data set. We know that no single answer is possible because we can never sample the earth well enough (i.e. collect enough data) to render the problem over-determined, unless the earth is modeled with only a few parameters (i.e. cells).
• Some models are not geologically reasonable. This is subjective, but over-fit and under-fit results should be detectible by assessing the smoothness or roughness of the result.
• Finding alternate models is facilitated by altering either
  • the reference model, or
  • the alphas, or
  • the target misfit value.
• Interpreting geologic or geotechnical properties should never proceed without several models to work from. It is crucial that you be able to defend a model in terms of (i) zones that are reliable because their features depend upon measurements, and (ii) zones that should not be used in the interpretation because they depend upon the methodology rather than on field measurements.