Gramian Space Joint Inversion for Subsurface Imaging
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Solution Overview
Problem
Existing methods for joint inversion of geophysical data are inadequate for capturing geological complexity, as they often require prior knowledge of specific analytical or statistical relationships between different physical properties and do not account for structural correlations, limiting their ability to accurately image subsurface structures.
Innovation Solution
The method involves constructing a Gramian space of model parameters and their transforms, using a Gram matrix to determine nonnegative functionals that characterize linear dependency and structural similarity, allowing for the minimization of a parametric functional to enforce correlations between model parameters without requiring specific prior knowledge, and using smoothing or focusing stabilizing functionals to produce inverse images with sharp boundaries.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If analytic, empirical or statistical correlations between different physical properties are used for joint inversion, then the inversion can be performed with known relationships, but the method fails when specific forms of relationships are unknown or when structural correlations exist without analytic relationships
Solution Approach 1:
The patent introduces a Gramian matrix as an intermediary mathematical construct that operates on the model parameters themselves rather than requiring knowledge of their physical relationships. The Gramian matrix G with elements G_ij = <m_i, m_j> serves as a mediator that enforces structural similarity between different physical properties through their inner products, allowing joint inversion to proceed without needing to know the specific analytic or statistical relationships between properties like resistivity and seismic velocity
Solution Approach 2:
The patent transforms the joint inversion problem by changing the parameter space from individual physical properties to a Gramian space of inner products between model parameters. By working with the Gramian matrix and its eigenvalues rather than directly with the physical property relationships, the method enables inversion without requiring prior knowledge of the specific functional relationships between different geophysical properties
2Ease of manufacture
If traditional joint inversion methods are used, then computational simplicity is maintained, but the ability to capture geological complexity and structural correlations is insufficient
Solution Approach 1:
The patent adds a new dimension to the joint inversion problem by introducing the Gramian space of model parameters as an additional mathematical dimension. Instead of inverting only in the space of physical properties, the method operates in the expanded space that includes Gramian inner products, enabling the capture of structural correlations and geological complexity while maintaining computational tractability through the mathematical properties of the Gramian matrix
3Measurement precision
If multiple geophysical datasets are jointly inverted, then comprehensive subsurface imaging is achieved, but the requirement for prior knowledge of relationships between properties increases complexity
Solution Approach 1:
The patent extracts the essential structural information from multiple geophysical datasets by focusing on the Gramian inner products between model parameters rather than requiring extraction and specification of complex physical relationships. This extraction approach isolates the structural correlations that are common to all datasets while removing the need to model specific physical property relationships, thereby maintaining imaging accuracy while reducing method complexity
Data Source
AI summary
A method for the simultaneous imaging of different physical properties of an examined medium from the simultaneous joint inversion of multiple datasets of physical field measurements is described. The method introduces Gramian spaces of model parameters and/or their transforms, and Gramian constraints computed as the determinants of the corresponding Gram matrices of the model parameters and/or their transforms. Gramian constraints are introduced as additional regularization terms, and their minimization enforces the correlation between different model parameters and/or their transforms. The method does not require a priori knowledge about specific analytical or empirical or statistical correlations between the different model parameters and/or their attributes, nor does the method require a priori knowledge about specific geometric correlations between different model parameters and/or their attributes. The method is a generalized in that it can be applied to the simultaneous joint inversion of any number and combination of physical field measurements.