Method of simultaneous imaging of different physical properties using joint inversion of multiple datasets

A system for simultaneous imaging of different physical properties of an examined medium from a simultaneous joint inversion of multiple datasets of physical field/signal measurements comprising:

• one or more sensors;
• and
• a computing system, the computing system comprising:
• a processor;
• and
• one or more physical non-transitory computer readable media having computer executable instructions stored thereon that when executed by the processor, cause the computing system to perform the following:
• measure with at least two sensors located in proximity to the examined medium at least one component of at least two physical fields and/or signals and record by a corresponding recording device observed geophysical data generated as a response from the examined medium;
• construct, based on the measured at least one component of the at least two different physical fields and/or signals, a Gramian space with a norm providing a measure of correlation between different model parameters and/or their transforms and/or their attributes that correspond to the at least two different physical fields and/or signals, the measure of correlation specifying a degree of similarity between a distribution of model parameters and/or their transforms and/or their attributes for a first one of the physical fields and/or signals and a distribution of model parameters and/or their transforms and/or their attributes for a second one of the physical fields and/or signals, wherein the Gramian space is a Hilbert space formed by integrable functions describing corresponding model parameters within a volume, V, of the examined medium with an inner product operation defined by a corresponding Gram matrix;
• determine in the Gramian space a nonnegative Gramian functional equal to the determinant of the Gram matrix of at least two model parameters and/or their attributes;
• determine smoothing or focusing stabilizing functionals for producing smooth inverse images or images with sharp boundaries;
• determine predicted geophysical data based in part on the at least two model parameters and/or their attributes;
• construct a parametric functional defined as a linear combination of misfit functionals for at least two observed geophysical data and predicted geophysical data, smoothing or focusing stabilizing functionals, and the Gramian functional, wherein the Gramian functional is a measure of the distance between the distributions of the different model parameters and/or their transforms and/or their attributes in the Gramian space, wherein a shorter distance between the distributions of the different model parameters and/or their transforms and/or their attributes in the Gramian space is more indicative of the different model parameters and/or their transforms and/or their attributes corresponding to a single physical object in the examined medium;
• minimize the parametric functional with the Gramian functional, in order to enforce the correlations between the model parameters and/or their transforms and/or their attributes to thereby determine the different model parameters and/or their transforms and/or their attributes that have shorter distances between them;
• determine multi-modal model parameters, which correspond to the minimum of the parametric functional, by solving a minimization problem for the parametric functional using linear and/or nonlinear optimization methods;
• output the multi-modal parameters;
• evaluate the multi-modal parameters against termination criteria;
• if the termination criteria are not satisfied, the output multi-modal parameters are updated model parameters used in determining new multi-modal parameters;
• and
• if the termination criteria are satisfied, cause a drill to modify a path through the examined medium based on the outputted multi-modal parameters.

A system comprising:

• a drilling apparatus;
• a plurality of sensors;
• at least one processor;
• and
• a non-transitory computer readable medium storing instructions thereon that, when executed by the at least one processor, cause a computing system to:
• receive sensor data comprising a plurality of physical signals generated by an examined medium and recorded by the plurality of sensors positioned in proximity to the examined medium, the plurality of physical signals comprising a first physical signal and a second physical signal;
• generate, based on the received sensor data, a Gramian space indicating a measure of correlation between model parameters and associated attributes that correspond to the plurality of physical signals, wherein the measure of correlation indicates a degree of similarity between a first distribution of model parameters and the attributes for the first physical signal and a second distribution of model parameters and the attributes for the second physical signal;
• determine a non-negative Gramian functional based on a determinant of the Gramian space, the Gramian functional comprising a measure of distance between the distribution of model parameters and the second distribution of model parameters, the measure of distance comprising a measure of similarity in physical properties of the examined medium at different locations between physical signals detected by the plurality of sensors at the different locations;
• determine a stabilizing functional for producing smooth inverse images or images with sharp boundaries;
• determine predicted geophysical data based on the model parameters and associated attributes;
• generate a parametric functional including a linear combination of misfit functionals for two or more of the determined non-negative Gramian functional, stabilized function, and predicted geophysical data;
• minimize the parametric functional with the Gramian functional to enforce correlations between the model parameters and associated attributes;
• determine multi-modal model parameters corresponding to a minimization of the parametric functional;
• and
• cause the drilling apparatus to drill a hole along a path through the examined medium based on the determined multi-modal model parameters corresponding to the minimization of the parametric functional.