ISMap and Emerge. In this document, we will combine the EMERGE multi-attribute transform algorithm with the geostatistical algorithms in ISMap.

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1 ISMap and Emerge In this document, we will combine the EMERGE multi-attribute transform algorithm with the geostatistical algorithms in ISMap. Before doing so, we will briefly review the theory of the EMERGE program. We will then discuss how the algorithm can be adapted to map analysis. CE7/R3 Last Updated: November 2005 Author: Brian Russell 1

2 Introduction to EMERGE EMERGE is a program that analyzes well log and seismic data by finding a relationship between the log and seismic data at the well locations. It uses this relationship to predict or estimate a volume of the log property at all locations of the seismic volume. EMERGE uses: A seismic volume (usually 3D). A series of wells which tie the volume. Each well contains target log, such as porosity, which is to be predicted. Each well also contains the information for converting from depth to time, usually in the form of a check-shot corrected sonic log. (Optional) One or more external attributes in the form of seismic 3D volumes. CE7/R3 Last Updated: November 2005 Author: Brian Russell 2

3 Linear Regression with Multiple Attributes In the EMERGE program, a target log is modeled as a linear combination of several attributes, using the following formula, which can be solved using a standard least-squares approach: L(t ) = w + w A (t ) + L + w where 0 L 1 1 = the log, w i M (t ), the attributes. CE7/R3 Last Updated: November 2005 Author: Brian Russell 3 A M = weights, and A i =

4 The Multi-Attribute Transform Applied to Map Data When we apply the multiattribute transform to map data, we compute M+1 weights such that the log value L(x,y) at a particular map value is a weighted sum of M attributes A i : L( x,y ) = w0 + w1a 1( x,y ) + L + w M A M ( x,y ) Again, the solution to this problem can be found by using a standard least-squares technique. A key problem is deciding which attributes to use, which is done using step-wise regression. Another important consideration is which of the attributes are statistically significant, which is done using cross-validation. The next slide shows this approach pictorially. CE7/R3 Last Updated: November 2005 Author: Brian Russell 4

5 The Multiattribute Map Transform Y X L = log value Attribute map 1 w 1 A 1 Attribute map 2 w 2 A 2 Attribute map M w m A m This figure shows the multiattribute map transform approach in schematic form. We need to compute the weights w i which, when multiplied by the attribute values, will produce the log value. CE7/R3 Last Updated: November 2005 Author: Brian Russell 5

6 Which attributes should we use? Typically, we can use any type of attribute, such as amplitude, phase, frequency, etc. In the following exercise, the following attribute slices are used: Inverted seismic amplitude, or impedance Seismic amplitude Amplitude envelope Instantaneous phase Instantaneous frequency Integrated seismic trace Trace length the total length of the trace over the window Each attribute (except trace length) is derived from the seismic volume by taking an RMS average over a user defined window, 10 ms for example, below a picked top. CE7/R3 Last Updated: November 2005 Author: Brian Russell 6

7 Finding the Best Attributes EMERGE and ISMap use a technique called step-wise regression to find the best set of attributes. The attributes, such as impedance, amplitude, phase, etc, are first arranged from lowest error/highest correlation with the log parameter to highest error/lowest correlation, as shown below. Notice that we have included nonlinear transforms of both the target log and the attributes. CE7/R3 Last Updated: November 2005 Author: Brian Russell 7

8 Step-Wise Regression 1. The step-wise regression technique uses the attribute with the lowest error as the first attribute. 2. We then find the combination of that attribute and one other attribute that produces the lowest error of every combination of two attributes. 3. We then find the combination of those two attributes and one other attribute that produces the lowest error of every combination of three attributes. 4. This is repeated for the desired number of attributes. For example, the table below shows the best combination of five attributes. CE7/R3 Last Updated: November 2005 Author: Brian Russell 8

9 Cross-Validation The list on the previous page shows that the best five attributes, with the errors shown on the right. The total error, using all the points on the map, will always go down as we increase the number of attributes. But the crossvalidation error, which is found by leaving each well value out in succession, and then predicting it from the other values, will only decrease for the first few attributes. This is shown graphically on the right, using the numbers from the previous table. From the validation error plot, we notice that only the first three attributes should be used in the final map. CE7/R3 Last Updated: November 2005 Author: Brian Russell 9

10 The Multiattribute Map The figure below shows an example of producing a multiattribute map, which is a weighted combination of the attributes shown earlier: CE7/R3 Last Updated: November 2005 Author: Brian Russell 10

11 Combining the EMERGE Algorithm with Geostatistics The final stage of the process is to combine the multilinear regression result with the well values using geostatistics. That is, the multiattribute transform will replace the inversion slice as the secondary variable. The first step is to re-compute the seismic to seismic variogram. We then re-compute the cokriging or KED result, as shown below: CE7/R3 Last Updated: November 2005 Author: Brian Russell 11