GRAV3D manual: 
Synthetic examples

4.1 Introduction

This page includes two synthetic examples to illustrate the GRAV3D Library. The first model consists of a dipping slab of positive density contrast buried in a uniform background. This is a relatively small model and we use it to illustrate both the basic operation of the library as well as the incorporation of variable bound constraints. The second model consists of several blocks of different configurations buried in a uniform background. This model has a relatively large number of observations and the number of cells in the inversion is large. It is used to illustrate the utility of wavelet compression in speeding up the inversion of large data sets. For both examples, we calculate the synthetic data from the true model and then add uncorrelated Gaussian noise to simulate noisy observations.

4.2 Example 1

Figure 3a below shows one cross-section and two plan-sections of the true model. This model consists of a dipping dyke in a uniform background. The dyke has a westerly dip of 45^o and width of 200 by 300 m in the easting and northing directions. It extends from 50 m to 400 m in depth. The density contrast is 1.0 g/cm³.

The observations are simulated above the surface at an elevation of 1 m, and over a 21 by 21 grid with a grid interval of 50 m in both directions. The standard deviation of noise added to the data is 2% plus 0.01 mGal. The noisy data are displayed in Figure 1. It shows a peak that around (600, 500) m and it decays steeply on the east side but slower on the west side due to the dipping structure of the dyke. The small-scale variations reflect the additive noise in the data.

Figure 1: Measured gravity anomaly (mGal), with noise.

Figure 2: Predicted data from Result 1.

Result 1: positive density contrasts

Since the simulated data are entirely positive, it is reasonable to invert for a model consisting of entirely positive density contrast. We accomplish this by imposing a lower bound of 0.0 for positivity and a large upper bound of 4.0. The model objective function has the length scales set to 100 m in all three directions, and the simple depth weighting with default parameters is used. The predicted data from this inversion is shown in Figure 2 above; it is a smooth representation of the noisy data.

The recovered model is shown in one cross-section and two plan-sections in Figure 3b. The model is characterized by a broad density high at the location corresponding to the true dipping dyke and there is clear indication of a dipping structure. The recovered density has a minimum of 0 as constrained by the lower bound of 0. The maximum value is slightly great than 1.0 g/cm³, which is in fact very close to the true maximum density value in this case. Overall, we have a reasonably good inversion that delineates the essential structure of the true model.

Result 2: tighter upper bound constraint

The second inversion result was obtained using a slightly tighter upper bound to illustrate the use of simple upper bound. This is useful when a reliable estimate of maximum density contrast is available. Imposition of such a bound can often improve the solution. For this inversion, we have set the lower and upper bounds to be 0.0 and 0.8. The maximum value of density contrast in the true model is 1.0. The result is shown in Figure 3c. This model is not very different from the one recovered in the preceding inversion, but the anomalous density appears to be slightly wider. This is to be expected since we now have a smaller density contrast and the required anomalous body should have a large dimension to reproduce the same observed anomalies.

Result 3: variable bounds

One of the new features in the GRAV3D Version 2.0 is the ability to impose variable bounds on the density contrast to be recovered in the inversion. This provides users with one more tool for inputting geological information to improve the inversion. For instance, we may expect a lower density contrast in the region of an orebody than that in another region. Similarly, one region of the subsurface may have a negative contrast while the rest has a positive contrast. In special cases, imposing a lower and upper bounds that are very close to each other will effective fix the recovered density contrast to a known value during the inversion.

In this example, we illustrate both the variable bounds as well as the use of tight bounds to fix the model values. We invert the data introduced above by imposing a lower bound of zero constant throughout the model, and a variable upper bounds shown in Figure 3d. We assume that the top surface of the dipping dyke is known, but we do not know the lateral extent of the dyke nor its depth extent. Therefore, we impose an upper bound of 0.01 above the dyke and an upper bound of 1.0 below that surface. This effectively constrains the density contrast above in the upper region of the entire model to being very close to zero while allowing the density contrast to vary between 0 and be as high as 1.0 g/cm³ as required by the data.

The recovered model is shown in Figure 3e. The top surface of the dipping dyke is well imaged as expected because of the imposed bound constraints. However, constraints on the top surface of the dyke has greatly helped image the bottom surface of the dyke. Furthermore, the lateral extent of the dyke is well imaged although we have not constrained it at all. This demonstrates the advantages of imposing specific geologic information through variable bounds that help better define the targets.

It should be noted that it is possible to "create" nearly any model you like using bounding constraints. They should not be employed unless geological information is very reliable.

4.3 Example 2

The figure to the right displays a volume rendered image of the large test model. Move your mouse over to see the recovered model. It consists of five blocks of different density contrasts in a uniform background. There is one large dipping dyke to the left that extends to a large depth. Four smaller blocks of various shapes are located at shallower depths to the right. Three cross-sections of this "true" model are shown via mouseover in last figure below. This figure indicates the vertical lateral locations the five blocks. The model occupies a volume of 2.5 km by 2.5 km by 1.25 km.

The figure below left shows the gravity data simulated on the surface over a 51 by 51 grid of 50-m grid spacing. A total of 2601 data is generated. Noisy observations were simulated by adding Gaussian random noise. The data plot shows four of the five anomalies and the effect of the added noise.

Gravity data simulated over the large model.

Predicted data based on the recovered model.

To invert this data set, we discretize the model region using 50-m cubes. This results in 62500 cells and the corresponding sensitivity matrix requires over 600 Mb to store. We use this example to illustrate the wavelet compression of the GRAV3D version 2.0. Using Daubechies-4 wavelet and a reconstruction accuracy of 5%, a compression ratio of 30 was achieved. The resulting matrix is stored in 42 Mb. With the compressed sensitivity matrix, the inversion was carried out readily without much demand on computer memory or CPU time.

The predicted data from the inversion are shown in the figure above right, which is a smooth version of the observation. The recovered density contrast model is shown via mouseover in the first Example 2 figure above. The cutoff value is 0.17 g/cm³. All five anomalous blocks are imaged.

The recovered model is shown in three crosssections in the figure to the right. Move your mouse over to see the true model. The amplitudes of the recovered anomalous blocks are lower than the true value. The depth extent of the large dipping block is also smaller. This is expected since the area of the data is limited. Over all, this is a good representation of the true model, and the inversion utilizing the wavelet compression has allowed the inversion to be carried out with very moderate demand on computational resources.

NOTE: All figures are shown on one page here (suitable for printing).