Inversion and Geodiversity: Searching Model Space for the Answers

Abstract

Geophysical inversion employs various methods to minimize the misfit between geophysical datasets and three-dimensional petrophysical distributions. Inversion techniques rely on many subjective inputs to provide a solution to a non-unique underdetermined problem, including the use of a priori model elements (i.e. a contiguous volume of the same litho-stratigraphic package), the a priori input model itself or inversion constraints. In some cases, inversion may produce a result that perfectly matches the observed geophysical data, but can still misrepresent the geological system. A workflow is presented here that offers objective methods to provide inputs to inversion: (1) simulations are performed to create a model suite that contains a range of geologically possible models; (2) stratigraphic variability is determined via uncertainty analysis to identify low certainty model regions and elements; (3) geodiversity analysis is then conducted to determine geometrical and geophysical extremes and commonalities within the model space; (4) geodiversity metrics are simultaneously analysed using principal component analysis to identify the contribution of different model elements toward overall model suite uncertainty; (5) principal component analysis also determines which models exhibit diverse or common geological and geophysical characteristics which (6) facilitate the selection of models as inputs to geophysical inversion. This workflow is applied to a three-dimensional model of the Ashanti Greenstone Belt, southwestern Ghana in West Africa in order to reduce the subjectivity incurred during decision making, explore the range of geologically possible models and provide geological constraints to the inversion process to produce geologically and geophysically robust suites of models. Results further suggest that three-dimensional uncertainty grids can optimize inversion processes and assist in finding geologically reasonable solutions.

Appendix: Description of Geodiversity Metrics

The following summary of geodiversity metrics is a compilation of the methods presented in Lindsay et al. (2013a, b).

1.1 A1. Geometrical Geodiversity Metrics

1.1.1 A1.1. Formation Depth and Volume

The shallowest extent of a stratigraphic unit is calculated by determining the depth of the shallowest voxel in the formation under study. The deepest extent is determined from the deepest voxel of a given formation (Fig. 19). Volumes are determined by multiplying the count of stratigraphic unit voxels with the voxel volume, or by simply displaying the voxel count (Fig. 19).

Fig. 19

Formation depth and volume calculations. The geological bodies are shown as a set of voxels rather than surfaces, according to the described method. Formation depth analysis detects the location of the deepest or shallowest voxel attributed to a particular geological unit. Volume analysis is performed with a voxel count of the geological unit. Reprinted from Tectonophysics, 594, Lindsay et al. (2013a), Copyright 2013 with permission from Elsevier

1.1.2 A1.2. Average Mean Curvature

Curvature is determined by finding the orientation of a normal section rotated around the surface normal $(\overrightarrow{N})$ at P, a given point on a folded surface (Lisle and Toimil 2007). The maximum magnitude of curvature along the intersection of the folded surface is principal curvature $k_1$. Curvature of the intersection of the surface and a normal section perpendicular to $k_1$ is principal curvature $k_2$ so that $k_1 > k_2$. Positive principal curvature indicates convex-upward and negative principal curvature indicates concave-upward. Mean curvature ($M$) is the arithmetic mean of $k_1$ and $k_2$

$$\begin{aligned} M=\frac{k_1 +k_2 }{2}. \end{aligned}$$

(1)

When $M >0$ an antiformal (convex) surface is observed, when $M <0$ a synformal (concave) surface is observed. $M = 0$ indicates the surface is a flat plane or a surface where $k_1 = -k_2$ known as a ‘perfect saddle’ (Lisle and Toimil 2007). Positive Gaussian curvature ($G)$

$$\begin{aligned} G=k_1 \cdot k_2 . \end{aligned}$$

(2)

indicates that both principal curvatures $k_{1}$ and $k_{2}$ have the same sign and the surface resembles a dome or a basin, if inverted. Negative Gaussian curvature indicates that principal curvatures have different signs and the surface resembles an antiformal or synformal saddle (Fig. 20) (Lisle and Toimil 2007; Mynatt et al. 2007). Please refer to Besl and Jain (1986), Lisle and Robinson (1995) and Lisle and Toimil (2007) for further details and case studies.

Fig. 20

Short-distance neighbour calculation constrained to locating the six closest neighbours in a regular grid made of cells with equidistant axes. The sample location assigned with a stratigraphic ID of ‘1’ is surrounded by cells assigned with IDs of ‘1’, ‘2’ and ‘3’ resulting in a neighbourhood relationship metric result of ‘3’. Modified from Lindsay et al. (2013a)

1.1.3 A1.3. Neighbourhood Relationships and Contact Surface Area

Neighbourhood relationships are calculated with a $k$-nearest neighbour algorithm ($k$-NN). The locations of k nearest neighbours around a point of reference are found, and from that their Euclidean distance can be calculated (Bremner et al. 2005; Friedman et al. 1977). The Euclidean distance is used to constrain which voxels are counted as neighbours and the shortest distances measured along eastings, northings and depth axes resulting in a six-neighbour relationship (Fig. 20). The surface area of the contact between stratigraphic units is identified together with the proportion of overall contact relationships within the three-dimensional volume.

1.2 A2. Geophysical Geodiversity Metrics

1.2.1 A2.1. Root Mean Square

A root-mean-squared value, or ‘rms’, is

$$\begin{aligned} x_\mathrm{rms} =\sqrt{\frac{1}{n}( {x_1^2 +x_2^2 +\cdots +x_n^2 }),} \end{aligned}$$

(3)

where $x_{1}$, $x_{2} \ldots x_{n}$ equals the difference between one value and another in a grid.

1.2.2 A2.2. Standard deviation

The standard deviation ($s$) of an image is taken as a measure of value spread over a grid

$$\begin{aligned} s=\left( {\frac{1}{n-1}\mathop \sum \limits _{i=1}^n ( {x_i -\bar{x}})^2}\right) ^{\frac{1}{2}}. \end{aligned}$$

(4)

1.2.3 A2.3. Entropy

Information entropy ($E)$ is used to measure the average bits per pixel over an entire image, representing its global information content (Batty 1974; O’Gorman et al. 2008; Shannon 1948)

$$\begin{aligned} E=\mathop \sum \limits _i^N p_i \log p_i , \end{aligned}$$

(5)

where $E$ is the sum of all products of $p$ (probability) of each possible outcome ($i)$ out of $N$ total possible outcomes. $E = 0$ indicates that the image is dominated by large regions of the same value. If in a 1-bit system of two integer values 0 and 1, $E_\mathrm{max} = 1,$ the image is made of equal proportions of possible values in this case. $E = 1$ reflects that it is equally likely to find a ‘0’ or a ‘1’ in a given image.

1.2.4 A2.4. Two-Dimensional Correlation Coefficient

Two-dimensional correlation coefficients are typically calculated in geophysical and engineering applications to track changes in two- and three-dimensional objects. The two-dimensional correlation coefficient $r$ is calculated using

$$\begin{aligned} r=\frac{\mathop \sum \nolimits _m \mathop \sum \nolimits _n ( {A_{mn} -\bar{A} })( {B_{mn} -\bar{B} })}{\sqrt{\mathop \sum \nolimits _m \mathop \sum \nolimits _n ( {( {A_{mn} -\bar{A} })^2})( {\mathop \sum \nolimits _m \mathop \sum \nolimits _n ( {B_{mn} -\bar{B}})^2})} }, \end{aligned}$$

(6)

where $\bar{A}$ is the global mean of set one and $\bar{B}$ is the global mean of set two. The purpose of this technique is to identify when patterns in two different sets resemble one another.

1.2.5 A2.3. Hausdorff Distance

The Hausdorff distance measures how far points in two different subsets are from each other. The distance can then be used to understand the level of resemblance two superimposed objects have to each other (Huttenlocher et al. 1993; Olson and Huttenlocher 1997; Rucklidge 1997; Sim et al. 1999; Wang and Suter 2007). The Hausdorff distance, $d_\mathrm{H}$, between two sets is

$$\begin{aligned} d_\mathrm{H} ( {X,Y})=\max \left\{ {{\begin{array}{l} \mathrm{sup} \\ {x\in X} \\ \end{array} }{\begin{array}{l} \mathrm{inf} \\ {y\in Y} \\ \end{array} }d( {x,y}),\quad {\begin{array}{l} \mathrm{sup} \\ {y\in Y} \\ \end{array} }{\begin{array}{l} \mathrm{inf} \\ {x\in X} \\ \end{array} }d( {x,y})} \right\} , \end{aligned}$$

(7)

where $X$ and $Y$ are two non-empty subsets of a metric space ($M, d)$, ‘sup’ is the supremum, ‘inf ’ is the infimum and $x$ and $y$ are points within sets $X$ and $Y$, respectively. The supremum defines the upper bounds of subset $Y$ within set $X$, whereas the infimum defines the lower bounds of subset $Y$ within set $X$. First, determine the shortest distance from point $x_{1}$ to any point in $Y,$ then the shortest distance from point $x_{2}$ to any point in $Y,$ point $x_{n}$ to any point in $Y.$ The Hausdorff distance is the largest distance of those measured from $x_{1}, x_{2 },{\ldots },x_{n}$.

1.3 A3. Two-Stage Principal Component Analysis Procedure

The following summary of PCA is a compilation of the methods presented in Lindsay et al. (2013a, b). Common sources of model suite variability can be identified by combining metric variability into principal component analyses. Thus, PCA has been chosen to understand the multidimensional problem presented in this manuscript. PCA is a multivariate exploratory data technique that allows complex data interactions to be displayed (Jolliffe 2002). PCA transforms the data orthogonally and highlights relevance by re-organization of the data with respect to the attribute under analysis. Central to PCA is conversion of the potentially correlated original variables, i.e. the geodiversity metrics, into uncorrelated principal components. Data conversion is performed so the first principal component displays the greatest variance, with each subsequent component displaying progressively lower degrees of variance. Each component contains a contribution of variability from all the metrics submitted to the PCA, while each following component contains the next highest possible degree of remaining variance, with the proviso that it is uncorrelated to the preceding components (Jolliffe 2002).

The MATLAB princomp function (http://www.mathworks.com.au/help/toolbox/stats/princomp.html) was used to calculate the coefficients, or loadings, of metric linear combinations. Principal components are calculated in the following manner:

• 1. Statistical calculations such as the mean, subtraction of deviations from the mean and covariance matrix calculation.

• 2. Sort the eigenvalues and eigenvectors of the covariance matrix in descending order.

• 3. Determine eigenvector contribution to eigenvalues.

• 4. Basis vector determination.

• 5. Project $z$-score-converted original dataset onto the basis vectors.

Loading vectors show the amount of variability any metric has toward a principal component (see Fig. 21). The length of a vector and the angle it has with the principal component axis indicate the strength of the relationship. The longer the vector, the more variability contained within the metric it represents. The smaller the incidence angle of the loading vector to the principal component axis, the more related the vector is to that principal component. Stage one analyses individual stratigraphic units within a metric, to find those which best describe variability within each metric. The first stage is critical to identifying: (1) redundant metrics, and filtering them from analysis and; (2) retaining those metrics that do adequately describe variability in the model suite for analysis in stage two. The filtering of redundant metrics is only performed for those metrics that analyse each stratigraphic unit (depth, volume and short-distance neighbourhood relationship). Metrics that represent the model with a single value, such as the geophysical metrics (refer Sect. A2), are input directly into stage two.

Fig. 21

Two-stage PCA method used in this contribution shown as a flowchart. Stage one geodiversity analyses of the volume and depth of formations ‘A’, ‘B’, ‘C’ and ‘D’ are above the solid black line. Note formations are subscripted ‘V’(volume) or ‘D’(depth) to indicate the relevant geodiversity metric. Stage two results are below the black line and are where all the metrics (volume and depth, in this example) are simultaneously examined with PCA. Note that any metrics, not just depth and volume, can be included in stage two. The volume of formations B and C (circled) and the depth of formations A and D are shown in stage one to contribute most to model suite variability. The most influential metric for model suite variability, the depth of formation A, is revealed in stage two. Reprinted from Tectonophysics, 594, Lindsay et al. (2013a), Copyright 2013 with permission from Elsevier

Stage two repeats the PCA process, with all the metrics described in Sects. A2 and A3. A simplified example of the two-stage PCA process is shown in Fig. 21. Stratigraphic units B and C contribute most to the first principal component and second component variability respectively for the volume metric. Units A and D contribute most variability to the first and second principal components respectively for the depth geodiversity metric. Units B and C represent the volume metrics, and units A and D represent the depth metrics. Figure 21 shows that in stage two, where the volume and depth metrics are combined, depth of unit A has been determined to be the most influential in terms of model suite variability.