A predictive GIS methodology for mapping potential mining induced rock falls

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1 University of Wollongong Thesis Collections University of Wollongong Thesis Collection University of Wollongong Year 006 A predictive GIS methodology for mapping potential mining induced rock falls Hani Zahiri University of Wollongong Zahiri, Hani, A predictive GIS methodology for mapping potential mining induced rock falls, M.Eng thesis, School of Civil, Mining and Environmental Engineering, University of Wollongong, This paper is posted at Research Online.

2 NOTE This online version of the thesis may have different page formatting and pagination from the paper copy held in the University of Wollongong Library. UNIVERSITY OF WOLLONGONG COYRIGHT WARNING You may print or download ONE copy of this document for the purpose of your own research or study. The University does not authorise you to copy, communicate or otherwise make available electronically to any other person any copyright material contained on this site. You are reminded of the following: Copyright owners are entitled to take legal action against persons who infringe their copyright. A reproduction of material that is protected by copyright may be a copyright infringement. A court may impose penalties and award damages in relation to offences and infringements relating to copyright material. Higher penalties may apply, and higher damages may be awarded, for offences and infringements involving the conversion of material into digital or electronic form.

3 Chapter 3 Weights-of-Evidence Method

4 3. Introduction Chapter 3 Weights-of-Evidence Method According to Sawatzky et al., (004) the Weights-of-Evidence (WofE) method is a quantitative method for combining evidences in support of a hypothesis. In the simple terms, the weights-of-evidence method constructs a probabilistic model in using known occurrences of a feature under consideration (eg. rock fall). The known occurrences of the feature are commonly referred to as the training points. The controlling or predictive factors, termed as evidential themes, are the measures of the datasets that cause the feature to occur (Kemp et al., 00 and Sawatzky et al., 004). The weights-of-evidence method combines different datasets as controlling factors of occurrence of a particular phenomenon by weighting each factor using a log-linear form of a ayesian based model (Sawatzky et al., 004 and Spiegelhalter, 986). The technique calculates the weights for each controlling or predictive factor based on its dominance in the training point theme () units within the area of a predictor () (onham-carter, 994). Equations 3- and 3- represent the general formulation of the weights in WofE method: + W = W = ln ln (3-) (3-) - This term is defined in Arcview as a synonym of map or layer

5 where = probability of object occurrence; ln = the natural log; = controlling (predictor) theme; = training point theme unit; W + = the level of positive correlation between the presence of the evidential (controlling) theme and the training points; W - = the level of negative correlation between the presence of the evidential (controlling) theme and the training points; The overall level of spatial association between the evidential theme and the training points is given by the parameter of Contrast, C, which is defined by Equation 3-3 (onham-carter et al., 989): C = W + - W - (3-3) The weights-of-evidence method was first applied in quantitative analysis for medical purposes in disease diagnosis. The symptoms were considered as evidences. ased on the defined hypothesis this patient has disease x, two weights were calculated for each symptoms, one for presence and one for absence of each symptom. The Contrast of weights measures the relationship between disease and the particular symptom. In the cases of other patients, the weights could then be used to calculate the probability of the occurrence of particular diseases based on the presence or absence of symptoms (Lusted, 968; Aspinall and Hill, 983; Spiegelhalter and Knill-Jones, 984). In geological application, the weights-of-evidence method was originally developed in a non-spatial mode (Reboh and Reiter, 983; McCammon, 989), using the ayesian approach to combine controlling evidences for mineral

6 prospects in an expert system. Campbell et al. (98) applied this method to combine evidences in map form and successfully predicted the extension of the Mount Tolman molybdenum deposit in Washington. Reddy et al. (99) applied the same approach for the prediction of base metal deposits in a greenstone belt. onham-carter used weights-of-evidence with the combination of spatial data analysis in the mapping of mineral potential (onham-carter and Thomas 973; onham-carter and Chung 983; onham-carter et al. 988, 989). onham- Carter et al. (989) and onham-carter (994) further described a ayesian based method through which the quantitative relationship between an indicator map and a mineral deposit pattern was analysed. After the development of Arc-WofE and Arc-SM extensions in 00, the weights-of-evidence method has been widely used in various fields of study using the GIS based approach. 3. The ayesian concept The ayes law provides the basis for the weights-of-evidence method. To illustrate this concept, the likelihood of rain for tomorrow example described by onham-carter (994) is used. Suppose we want to predict the likelihood of rain on the next day. According to historical data, it is known that on average 00 days of the year has rainfall. A reasonable estimate of the prior probability of rain in the next day might be the average proportion, 00/365. This initial or prior estimate can be modified if other relevant information is available. For example, an important factor that influences rainfall is the time of year. The prior 3

7 probability would be updated by multiplying by a factor that varies with the time of year. This can be expressed as: Rain Time _ of _ year = Rain* Time _ of _ year _ Factor (3-4) where Rain = prior probability, and Rain Time _ of _ year = posterior probability, which is the conditional probability of rain given the time of year, or prior probability multiplied by a factor. The Time-of-year factor would vary depending on the month of the year. Another factor that affects the likelihood of rain could be whether it has rained today or not. These two factors or sources of evidence for the proposition it will rain tomorrow can then be combined by the expression: Rain evidences = Rain* Time _ of _ year _ factor * Rain _ today _ factor (3-5) where Rain_today_factor is determined from historical data, and the effect of combined factors of evidence is the product of two factors. Several sources of data that provide predictors (evidence) regarding tomorrow s weather can be used to update prior probability. The prior probability can be successively updated with the addition of new predictor evidence, so that the posterior probability based on a piece of evidence can be used as the prior for using a new piece of evidence. Fundamental to the understanding of weights-of-evidence is the concept of prior and posterior probabilities. The prior probability defines the expected outcome of 4

8 an event in the absence of evidence, whereas the posterior probability is defined as: osterior probability = rior probability * factor for each predictor (evidential themes) (3-6) Let us further consider the problem of determining the probability of rock fall occurrence within a specified region T, made up of u unit cells (areas). In practice, the size of the unit areas depends on the quality of available data and spacing between known rock fall locations, termed as training points. As a rule, the method assumes that each training points occupies only one small unit area hence the probability of a point can be defined as probability per unit area. If N is the number of unit cells containing rock falls and NT is the total number of unit cells in an area of study (Figure 3.), the prior probability of the rock falls occurrence in the study area is N. N T Steep areas are usually prone to rock falls. The slope of an area is an important controlling factor in estimating rock fall probability in an area. Slope factor or evidence can be used to modify prior probability of rock fall occurrence. Figure 3. shows a hypothetical spatial relationship between an area T, training theme and predictor map (or evidential theme). 5

9 Figure 3. inary map showing the location of rock falls and Venn diagram summarising the spatial overlaps relationship between the map pattern and the rock fall pattern (After Harris et al., 000) The probability of rock fall occurrence, given the presence of evidence (eg. steep slope), can be expressed by Equation 3-7: = (3-7) To obtain an expression for the posterior probability of rock fall occurrence in terms of the prior probability and a multiplication slope factor we note that the conditional probability on the binary slope map,, given the presence of a rock fall is defined as: 6

10 = (3-8) In order to obtain an expression indicating the relationship between posterior probability and prior probability and an evidence factor, Equations 3-7 and 3-8 can be combined to solve for, showing the requested relationship: = (3-9) Hence the posterior probability of a rock fall, given the presence of the evidence equals the prior probability of rock fall multiplied by the factor. Similarly, the posterior probability of rock fall occurrence given the absence of evidence (flat areas) can be expressed by Equation 3-0: = (3-0) 3.. Odds and likelihood ratios According to onham-carter (994), the probability that an event will occur divided by the probability that it will not occur is defined as a ratio called odds. The weights-of-evidence method uses the natural logarithm of odds, known as the log odds or Logits. From Equation 3-0 the odds ratio can be expressed as: = (3-) 7

11 From the definition of conditional probability: = = (3-) Substituting this expression for into the numerator of the right side of Equation 3-, and rearranging terms yields the following: =.. (3-3) The odds of a rock fall O are equal to, or ; Substituting the definition of odds into Equation 3-3 reduces to: O = O (3-4) where O is the conditional (posterior) odds of given, O is the prior odds of and is known as Sufficiency Ratio (LS) (onham-carter, 994). Taking the natural logarithm of both sides of Equation 3-4 gives log e LS, the positive weight-of-evidence W + : Logit = Logit + W (3-5) + Similar algebraic expressions lead to the derivation of an odds expression for the conditional probability of given the absence of the evidence (see Equation 3-6). 8

12 O = O (3-6) The term is called the Necessity Ratio (LN) (onham-carter, 994). In weights-of-evidence terminology, W - is the natural logarithm of LN, or log e LN that is expressed as: Logit = Logit + W (3-7) oth LS and LN are also called likelihood ratio and can be calculated from the available field data (see Figure 3.) Figure 3. iagrammatic representation of the weights calculation in weights-ofevidence method (After Harris et al., 000) 3.. Combining atasets The conditional probability of a rock fall occurring, given the presence of two predictive map patterns, (eg. slope) and (eg. cliff height), can be expressed as: 9

13 30 = (3-8) which can be expressed as + = = (3-9) Equation 3-9 is the ayes Rule for two exclusive hypotheses, and, with = +. If and are conditionally independent then: = (3-0) and Equation 3-9 reduces to:. = (3-) Note that Equation 3- is similar to Equation 3-9, except that the multiplying factors for the two themes are used to update the prior probability to give the posterior probability. Using the odds formulation, the conditional or posterior odds can be expressed for two themes and as: * * LS LS O O = (3-)

14 or the log-linear weights-of-evidence form: Logit W (3-3) + + = Logit + W + where the subscripts and refer to the likelihood ratio or weights determined independently for evidential themes and respectively. The weights for each of the two themes (predictors) are calculated in exactly the same way as the weights for a single theme. As pointed out by onham-carter (994), there are now four different ways of combining two binary map patterns. In addition to Equation 3-3, three other ways could be formulated as below: + = Logit + W + Logit W (3-4) + = Logit + W + Logit W (3-5) = Logit + W + Logit W (3-6) With 3 datasets as evidence, there are 3, or 8, possible combinations. In general with n map there are n possible combinations. The general expression for combining i=,, 3 n maps is: for the likelihood ratio, and for the weights. n 3... n = O * LS i i= O (3-7) n n = logit + W i i= Logit (3-8) 3

15 Equations 3-7 and 3-8 are the computing formulae for combining a set of evidential themes with the ayes model which is implemented in GIS environment. In general, if the i-th evidence is absent instead of presence, the LS becomes LN and W + becomes W -. Where data is missing for a particular theme in some locations, the likelihood ratio is set to, or the weight is set to Implementation of weights-of-evidence method Applying weights-of-evidence method to map rock fall potential will entail the following steps: Choose a series of maps that are useful for providing evidential themes for rock falls occurrence. These may include maps of cliff heights and slope. For each map, determine the optimum reclassification scheme to convert the themes to binary or ternary format. This maximises the spatial association between the map and the occurrence points. Calculate the weights for each predictive map or evidential theme. Check for conditional independence between the evidential datasets. In some cases, maps may have to be combined or rejected to avoid conditional independency. Combine the evidential themes using calculated weights. Generate result (response) theme showing posterior probabilities. Figure 3.3 is a schematic of the main steps of the GIS based WofE method implementation. The implementation steps are also summarised in Figure

Selecting evidential themes and providing

16 Selecting evidential themes and providing GIS-based inventory Weighting evidential themes with respect to the spatial association with training points Training point theme Weighting Table Combining weighted evidential themes and generate response theme Response theme osterior robability Figure 3.3 Schematic GIS based weights-of-evidence method procedures 33

17 roblem definition Selection of controlling factors ata editing, manipulation and formating in GIS framework Setting of initial parameters (efine area of study, unit cell and training points) rimary weighting evidential themes Theme reclassification Secondary weighting evidential themes Check conditional independence Generate response themes and Interpretation Figure 3.4 General procedure for weights-of-evidence implementation 34

18 Among the stages, in first stages the geometry of the problem, goals, available models, available data, limitations and creation a prevision of the results, are the main issues. Selecting reasonable controlling factors to contribute to the model as evidential themes is the next step. This is followed by compilation of GIS based data inventory including data transferring, editing and manipulation. uring the primary weighting step, weights are calculated for each evidential theme with respect to the spatial association between the training points and the evidential themes. Themes are then reclassified based on primary weighting results in order to convert from multi-class themes to binary or ternary pattern Weighted themes are then combined together using calculated weights for each one. The response theme shows the probability of rock fall occurrence within the area of study. 35

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