Kriging Estimation Using Precision Matrices and Sub-Space Segmentation
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Solution Overview
Problem
Kriging methods in oil exploration require significant computational power, especially when dealing with large datasets, leading to inefficiencies and sensitivity to spatial distribution of observation data, resulting in neighborhood artifacts and high computation times.
Innovation Solution
A computerized method that estimates quantities by using precision matrices in sub-spaces, where the precision matrix is the inverse of the covariance matrix, allowing for efficient estimation by determining neighboring sub-spaces and utilizing Schur complements to update matrices, reducing computational load and minimizing neighborhood changes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If kriging is used to estimate quantities at locations without measurements, then the precision of quantity estimation is improved, but the computation time increases significantly
Solution Approach 1:
The patent divides the space into multiple sub-spaces and processes each sub-space separately using local covariance matrices instead of inverting one large global covariance matrix. This segmentation reduces the computational complexity from O(n³) for a full matrix to multiple smaller O(k³) operations where k << n, significantly reducing computation time while maintaining estimation precision.
Solution Approach 2:
The patent pre-computes and stores the precision matrix (inverse of covariance matrix) for each sub-space before actual estimation queries. When estimation is needed, the pre-computed precision matrices are directly applied rather than computing from scratch, eliminating redundant calculations and reducing real-time computation time.
2Measurement precision
If the number of observation data points is increased to improve estimation accuracy, then the precision of quantity estimation is improved, but the computational power required increases
Solution Approach 1:
By partitioning the data into sub-spaces, the patent allows using many observation points overall while only inverting small local covariance matrices. Each sub-space uses its own set of observation data points, and the results are combined to achieve global estimation accuracy without requiring the computational power to invert a massive global matrix.
Solution Approach 2:
The patent applies local reasoning by using observation data points nearest to each point being estimated, creating local covariance matrices tailored to each region. This local quality approach ensures high estimation accuracy using only relevant local data points rather than forcing all data points into a single global model, reducing the computational burden.
3Loss of time
If local reasoning with nearest neighbors is used to reduce computation time, then the computation time is reduced, but neighborhood artifacts increase
Solution Approach 1:
The patent carefully designs sub-space partitions that are large enough to contain sufficient neighboring points for stable local covariance estimation, avoiding the artifacts of overly small neighborhoods. By segmenting into appropriately sized sub-spaces rather than using arbitrary small neighbor sets, the method maintains estimation stability while still reducing computation time compared to global methods.
Data Source
AI summary
The invention relates to a method for estimating a set of variables associated with sites of an area, the method comprising the following steps: a) a set of observation data is supplied, b) a variable associated with a site of a first sub-area is estimated by krigeing, using the precision matrix of the first sub-area, c) a second adjacent sub-area is determined, d) a variable associated with a site of the second sub-area is estimated by krigeing, on the basis of the first sub-space, the second sub-space, and the precision matrix determined for the first sub-space.