Reliability measures for statistical prediction of geophysical and geological parameters in geophysical prospecting

• US 6,442,487 B2
• Filed: 12/04/2000
• Issued: 08/27/2002
• Est. Priority Date: 12/06/1999
• Status: Expired due to Term

• Current Assignee: Exxonmobil Upstream Research Company (Exxon Mobil Corporation)
• Original Assignee: Exxonmobil Upstream Research Company (Exxon Mobil Corporation)
• Inventors: Kim, Chul-Sung
• Primary Examiner(s): McElheny, Jr., Donald E.
• Application Number:
  US09/729,576
• Publication Number:
  US 20010044698A1
• Time in Patent Office: 631 Days
• Field of Search: 702/6, 702/11, 702/12, 702/13, 702/14, 703/10, 703/5, 703/2, 367/73
• US Class Current: 702/6
• CPC Class Codes: G01V 1/32 Transforming one recording into another or one representation into another
• CPC Class Codes: G01V 1/50 Analysing data

Abstract

A method for assessing reliability of a prediction model constructed from N training data attribute vectors and N associated observed values of a specified parameter. Each training data attribute vector includes seismic attributes obtained from seismic data traces located at or near a well and each associated observed value is obtained from well log or core data from the well. A predicted value of the specified parameter is determined for each of the N training data attribute vectors, from the training data attribute vectors and the prediction model. A residual is determined for each of the N training data attribute vectors, as the difference between the associated observed value and the predicted value of the specified parameter for the training data attribute vector. An attribute vector for the designated location is determined.

Claims

1. A method for assessing reliability of a predicted value of a specified geological or geophysical parameter at a designated location, the predicted value obtained from a prediction model constructed from N training data attribute vectors and N associated observed values of the specified parameter;

each training data attribute vector including one or more seismic attributes obtained from seismic data traces located at or near a well, each associated observed value of the specified parameter obtained from well log or core data from the well, the method comprising the steps of;

determining a predicted value of the specified parameter for each of the N training data attribute vectors, from the training data attribute vectors and the prediction model;

determining a residual for each of the N training data attribute vectors, the residual being the difference between the associated observed value of the specified parameter for the training data attribute vector and the predicted value of the specified parameter for the training data attribute vector;

determining an attribute vector for the designated location;

determining the predicted value of the specified parameter at the designated location, from the attribute vector for the designated location and the prediction model;

determining N basic probability distributions from the N training data attribute vectors, the N associated observed values, the N residuals, and the predicted value of the specified parameter at the designated location;

determining N basic probability assignments for each of three hypotheses that the predicted value of the specified parameter at the designated location is reliable, unreliable, and unpredictable, respectably, from the N basic probability distributions; and

determining a reliability value, an unreliability value, and an unpredictability value for the predicted value of the specified parameter at the designated location as combinations of the N basic probability assignments for each of the hypotheses that the predicted value of the specified parameter at the designated location is reliable, unreliable, and unpredictable, respectively.

2. The method of claim 1, wherein the basic probability distributions m_i(z) over the specified parameter z are defined for i=1, . . . , N, by

3. The method of claim 2, wherein the normalized distance d_i is defined for;

=1, . . . , N, by $d_i=\sqrt{\underset{j=1}{\overset{J}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\left(x_j^n-x_{i,j}^n\right)}^2/J},$

wherein x_jⁿ and x_i,jⁿ are normalized j^th elements of the attribute vector x for the designated location and the N training data attribute vectors x_i, respectively, and J is the dimension of the attribute vector x.

4. The method of claim 3, wherein the normalized j^th elements of the attribute vector x, x_jⁿ and x_i,jⁿ, are defined by

5. The method of claim 1, in which the basic probability assignments m_i(H_k) are defined for i=1,. . . , N, and k=0, 1, and θ

, by $m_i<br><br>\left(H_0\right)={\int <br><br>}_{\stackrel{⋒<br><br>}{Z}-\mathrm{\gamma <br><br>}}^{\stackrel{⋒<br><br>}{Z}+\mathrm{\gamma <br><br>}}<br><br>m_i<br><br>\left(z\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>z,<br><br>m_i<br><br>\left(H_1\right)={\int <br><br>}_{-\mathrm{\infty <br><br>}}^{\stackrel{⋒<br><br>}{Z}-\mathrm{\gamma <br><br>}}<br><br>m_i<br><br>\left(z\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>z+{\int <br><br>}_{\stackrel{⋒<br><br>}{Z}+\mathrm{\gamma <br><br>}}^{+\mathrm{\infty <br><br>}}<br><br>m_i<br><br>\left(z\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>z,\text{\hspace{1em}<br><br>}<br><br>\mathrm{and}$ 

m_i(H_θ

)=1−

m_i(H₀)−

m_i(H₁),

wherein H₀, H₁, and H_θ

are the hypotheses that the predicted value of the specified parameter at the designated location is reliable, unreliable, and unpredictable, respectively, m_i(z) are the N basic probability distributions, {circumflex over (z)} is the predicted value of the specified parameter at the designated location, and γ

is a prediction error level.

6. The method of claim 5, wherein the reliability value Bel (H₀), the unreliability value Bel (H₁), and the unpredictability value Bel (H_θ

) are calculated by

7. The method of claim 6, wherein

8. The method of claim 5, further comprising the steps of:

calculating a sample standard deviation σ

_t for the N residuals; and

setting the prediction error level γ

equal to the sample standard deviation σ

_t.

9. The method of claim 5, further comprising the steps of:

selecting an error tolerance limit; and

setting the prediction error level γ

equal to the error tolerance limit.

10. The method of claim 2, further comprising the steps of:

defining a random variable z^t_i having a normal distribution with a mean z_i and the standard deviation σ

, for i=1, . . . , N;

calculating a modified γ

-level reliability Bel_i′

(H₀), for i=1,. . . , N, based on a restricted set of the training data triplets (x_j, z_j, r_j) with j=1, . . . , N and j≠

i;

calculating a probability P(|z_i^t−

{circumflex over (z)}_i|<

γ

|z_i), for i=1, . . . , N;

calculating an average correlation between the modified γ

-level reliabilities Bel_i′

(H₀) and the probabilities P(|z_i^t−

{circumflex over (z)}_i|<

γ

|z_i); and

determining the values of the standard deviation σ and

the maximum prediction distance d_max, which maximize the average correlation by an optimization algorithm.

11. The method of claim 10, wherein the modified γ

-level reliability Bel_i′

(H₀) is defined by

12. The method of claim 10, wherein the average correlation ρ

is defined by $\mathrm{\rho <br><br>}<br><br>\left(\mathrm{\sigma <br><br>},d_{\max }\right)=\frac{1}{N}<br><br>\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>P<br><br>\text{\hspace{1em}<br><br>}<br><br>(<br><br>Z_i^t-{\stackrel{⋒<br><br>}{Z}}_i<br><br><<br><br>\mathrm{\gamma <br><br>}<br><br><br><br>Z_i)<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{Bel}}_i^{\mathrm{\prime <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(H_0\right).$

13. The method of claim 10, wherein the optimization algorithm is an exhaustive search.