# Presentation: 115290

## 3D Geological Modeling: Solving as Classification Problem with Support Vector Machine

**3D Geological Modeling: Solving as Classification Problem with Support Vector Machine**

**A. Smirnoff, E. Boisvert, S. J.Paradis**

***Earth Sciences Sector***

***Groundwater***

## Objectives

- Find an algorithm for automating the 3D modeling procedure from sparse data
- Test the algorithm on available data
- Make conclusions about its applicability

## Possible Input Data

- Well data
- Surface geology maps
- Cross-section data
- Can be used alone or in combination

## Algorithms Currently in Use and Their Limitations

- Voronoi diagrams
- Potential fields
- Normally require too much information and/or additional procedures
- What if we only have a few sections to start with?

## 3D Reconstruction as a Classification Problem

## Available Classification Methods

**Bayesian classification**
- a priory knowledge of probabilities** **
**Nearest-Neighbor classifiers**
- extremely sensitive to parameter choice and scaling
**Decision trees**
- not flexible with many samples
**Neural networks**
- slow and difficult to use
**Support Vector Machine (SVM)**
- relatively new method
- becoming more and more popular

## SVM Algorithm

_**Input:**_ Take a set of training samples with known features and classes
_**Model:**_ Build a model (boundary) separating the training samples
_**Output:**_ Classify any new (unclassified) or test samples using the model

## Binary Reconstruction

**Binary Reconstruction**

## Input Data and Results

## Detailed Analysis (Class 1)

## Peeking into the SVM Black Box

- A simple case: two classes and two features (*e.g.,* length of petal and sepal in flowers)
- Training Set: known data vectors : *x**i*, where *i* = 1, ...., *l*

| Training Records (i) | Class Labels (yi) | Data Vector (xi) |  |
| --- | --- | --- | --- |
|  |  | Feature 1 | Feature 2 |
|  1 |  1 | 2 | 4 |
|  2 | -1 | 5 | 3 |
|  3 |  1 | 6 | 8 |
|  ... | ... | ... | ... |
|  l | -1 | 7 | 3 |

## Maximum Margin Separating Hyperplane (MMSH)

**Notes:**
Support vectors are points closest to the hyperplane. Those with distance 1/2 from MMSH.  Note, solution depends only on these data, the rest of data is not important.

## Hard Margin Classification-HMSH

## How to Maximize the Margin?

## Problem Formulation

## Soft Margin Classification - SMSH

**Notes:**
Again solution depends only on support vectors, the rest of the data is not important.
Note that C is one of parameters that needs to be adjusted.  It is the tradeoff between of the relative importance of maximizing the margin and fitting the training data.

## Non-Separable Data

## Non-Linear SVM

## Kernel Trick

## Kernel Functions

## How Did We Use SVM?

## C and   Grid Search

## Influence of C and 

**Influence of ****C and ******

## Multi-Class Classification

**Multi-Class Classification**

**X**

**Z**

**Y**

**1. Original**

**2. Training set**

**3. Output**

## Data Statistics and Results

**Data Statistics and Results**

| Class | To Classify | Training Set | % of Total | Success % |
| --- | --- | --- | --- | --- |
| 1. Organic  | 1162 | 48 | 0.01 | 18.76 |
| 2. Littoral | 3626 | 193 | 0.05 | 37.20 |
| 3. Clay | 12667 | 628 | 0.16 | 57.10 |
| 4. Esker | 19305 | 995 | 0.26 | 67.65 |
| 5. Till | 15118 | 747 | 0.19 | 45.72 |
| 6. Bedrock | 319905 | 14841 | 3.81 | 95.45 |

## Success per Class

**Success per Class**

## Area and Volume Comparison

**Area and Volume Comparison**

## Conclusions

- The SVM can successfully be used in single and multi-unit 3D geological reconstructions:
- Reasonable results are obtained with just a few training sections
- Parameters must be picked from the range: *C=2**-3**- 2**15**; **** = 2**4**- 2**9*
- Low C values - less details, more generalized model
- High C values - more details, less generalized model
- Additional Experiments Demonstrated:
- Number of units can vary (all units must be represented in training set)
- Sections can be arbitrarily located
- Other types of information (well data, surface geology maps) can be used

## References

- Abe, S., 2005.  Support Vector Machines for Pattern Classification. Springer-Verlag, London, 343 pp.
- Cristianini, N., Shawe-Taylor, J., 2000.  Support Vector Machines.  Cambridge University Press, 189 pp.
- Vapnik, V., 1995. The Nature of Statistical Learning Theory. Springer-Verlag, New York, 311 pp.