{"id":63,"date":"2022-07-14T00:09:15","date_gmt":"2022-07-14T00:09:15","guid":{"rendered":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/chapter\/checks-on-inversion-results\/"},"modified":"2022-07-23T03:19:38","modified_gmt":"2022-07-23T03:19:38","slug":"checks-on-inversion-results","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/chapter\/checks-on-inversion-results\/","title":{"raw":"4.6 Checks on Inversion Results","rendered":"4.6 Checks on Inversion Results"},"content":{"raw":"
\r\n

Tomographic inversion results are strongly affected by selected inversion parameters and regularization criteria, especially in the presence of large measurement errors. It is instructive, therefore, to run multiple inversions to gain insight into the effects of different software settings, which is the philosophy of the depth of investigation (DOI) analysis seen earlier. Rarely are default inversion settings appropriate and the inversion should be guided by prior information. Prior information that may be useful includes past geophysical results, (hydro)geologic maps, and drillers\u2019 logs. If inverted electrical conductivity cross sections are inconsistent with such prior information, this could indicate that settings are suboptimal or that assumptions (e.g., 2-D heterogeneity) are violated. Table\u00a0<\/strong>1<\/strong> lists some common problems and their associated symptoms and solutions. We emphasize that Table\u00a0<\/strong>1<\/strong> is by no means exhaustive in terms of the symptoms, problems, and solutions and relationships between them. Rather, this is meant as a starting point for practitioners to begin thinking about the roles of various inverse settings.<\/p>\r\n

Table\u00a0<\/strong>1<\/strong>\u00a0<\/strong>-<\/strong>\u00a0Common problems with inversion settings, and the associated symptoms and solutions.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Symptom<\/strong><\/td>\r\nPossible problems<\/strong><\/td>\r\nSolution<\/strong><\/td>\r\n<\/tr>\r\n
\r\n

Minimum\/maximum estimated electrical conductivity too low\/high compared to expected values<\/p>\r\n<\/td>\r\n

\r\n

The inversion may be overfitting the data<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Stop inversion at an earlier iteration or increase the assumed measurement error<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Non-random outlier data may be present<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Check the dataset for outliers and edit<\/li>\r\n \t
  • Try the L1 norm for data misfit<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Tomogram is speckly or looks like a checkerboard<\/p>\r\n<\/td>\r\n

\r\n

The inversion may be overfitting the data<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Stop inversion at an earlier iteration or increase the assumed measurement error in an Occam\u2019s inversion<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Non-random outlier data may be present<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Check the dataset for outliers and edit<\/li>\r\n \t
  • Try the L1 norm for data misfit<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

The inversion cannot match the data to within the reciprocal error<\/p>\r\n<\/td>\r\n

\r\n

The optimization algorithm may be caught in a local minimum<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Change optimization tolerances<\/li>\r\n \t
  • Increase number of iterations<\/li>\r\n \t
  • Update the Jacobian more frequently<\/li>\r\n \t
  • Change the starting model<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

The finite-difference or finite-element grid may be too coarse<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Refine the grid or mesh<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

The inversion grid may be too coarse<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Increase the number of inversion parameters<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Non-random outlier data may be present<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Check the dataset for outliers and edit<\/li>\r\n \t
  • Try the L1 norm for data misfit<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

The tomogram does not look like expected geology<\/p>\r\n<\/td>\r\n

\r\n

Regularization criteria may be smoothing\/blunting the tomogram too much<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Try robust model misfit instead of L2 model misfit<\/li>\r\n \t
  • Use anisotropic regularization<\/li>\r\n \t
  • Try different regularization criteria<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Electrical conductivity may not correlate with lithology<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Another geophysical technique may be needed<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Resolution may be above the scale of the pertinent heterogeneity.<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Another geophysical technique may be needed or array type and\/or electrode spacing used may need to be reconsidered<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Two (or more) tomograms that share a borehole appear inconsistent at the borehole<\/p>\r\n<\/td>\r\n

\r\n

Electrical anisotropy<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Use an inversion package with a forward model that allows for electrical conductivity anisotropy<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Outlier data are present in at least one dataset<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Check the dataset for outliers and edit<\/li>\r\n \t
  • Try the L1 norm for data misfit<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Parsing data into individual tomograms when they should be considered together<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Invert all data at once, rather than breaking it into pieces<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n
\r\n

Tomograms show vertical streaking, or high or low electrical conductivity patches only at boreholes<\/p>\r\n<\/td>\r\n

\r\n

Resolution may vary greatly from the sides to the middle of the tomogram<\/p>\r\n<\/td>\r\n

\r\n
    \r\n \t
  • Create synthetic or hypothetical forward models of the experiment to evaluate resolution and likely artifacts<\/li>\r\n \t
  • Examine plots of the inversion\u2019s sensitivity or resolution matrix<\/li>\r\n \t
  • Explicitly model the boreholes<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"
    \n

    Tomographic inversion results are strongly affected by selected inversion parameters and regularization criteria, especially in the presence of large measurement errors. It is instructive, therefore, to run multiple inversions to gain insight into the effects of different software settings, which is the philosophy of the depth of investigation (DOI) analysis seen earlier. Rarely are default inversion settings appropriate and the inversion should be guided by prior information. Prior information that may be useful includes past geophysical results, (hydro)geologic maps, and drillers\u2019 logs. If inverted electrical conductivity cross sections are inconsistent with such prior information, this could indicate that settings are suboptimal or that assumptions (e.g., 2-D heterogeneity) are violated. Table\u00a0<\/strong>1<\/strong> lists some common problems and their associated symptoms and solutions. We emphasize that Table\u00a0<\/strong>1<\/strong> is by no means exhaustive in terms of the symptoms, problems, and solutions and relationships between them. Rather, this is meant as a starting point for practitioners to begin thinking about the roles of various inverse settings.<\/p>\n

    Table\u00a0<\/strong>1<\/strong>\u00a0<\/strong>–<\/strong>\u00a0Common problems with inversion settings, and the associated symptoms and solutions.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    Symptom<\/strong><\/td>\nPossible problems<\/strong><\/td>\nSolution<\/strong><\/td>\n<\/tr>\n
    \n

    Minimum\/maximum estimated electrical conductivity too low\/high compared to expected values<\/p>\n<\/td>\n

    		\n

    The inversion may be overfitting the data<\/p>\n<\/td>\n

    		\n
      \n
    • Stop inversion at an earlier iteration or increase the assumed measurement error<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Non-random outlier data may be present<\/p>\n<\/td>\n

    		\n
      \n
    • Check the dataset for outliers and edit<\/li>\n
    • Try the L1 norm for data misfit<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Tomogram is speckly or looks like a checkerboard<\/p>\n<\/td>\n

    		\n

    The inversion may be overfitting the data<\/p>\n<\/td>\n

    		\n
      \n
    • Stop inversion at an earlier iteration or increase the assumed measurement error in an Occam\u2019s inversion<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Non-random outlier data may be present<\/p>\n<\/td>\n

    		\n
      \n
    • Check the dataset for outliers and edit<\/li>\n
    • Try the L1 norm for data misfit<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    The inversion cannot match the data to within the reciprocal error<\/p>\n<\/td>\n

    		\n

    The optimization algorithm may be caught in a local minimum<\/p>\n<\/td>\n

    		\n
      \n
    • Change optimization tolerances<\/li>\n
    • Increase number of iterations<\/li>\n
    • Update the Jacobian more frequently<\/li>\n
    • Change the starting model<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    The finite-difference or finite-element grid may be too coarse<\/p>\n<\/td>\n

    		\n
      \n
    • Refine the grid or mesh<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    The inversion grid may be too coarse<\/p>\n<\/td>\n

    		\n
      \n
    • Increase the number of inversion parameters<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Non-random outlier data may be present<\/p>\n<\/td>\n

    		\n
      \n
    • Check the dataset for outliers and edit<\/li>\n
    • Try the L1 norm for data misfit<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    The tomogram does not look like expected geology<\/p>\n<\/td>\n

    		\n

    Regularization criteria may be smoothing\/blunting the tomogram too much<\/p>\n<\/td>\n

    		\n
      \n
    • Try robust model misfit instead of L2 model misfit<\/li>\n
    • Use anisotropic regularization<\/li>\n
    • Try different regularization criteria<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Electrical conductivity may not correlate with lithology<\/p>\n<\/td>\n

    		\n
      \n
    • Another geophysical technique may be needed<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Resolution may be above the scale of the pertinent heterogeneity.<\/p>\n<\/td>\n

    		\n
      \n
    • Another geophysical technique may be needed or array type and\/or electrode spacing used may need to be reconsidered<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Two (or more) tomograms that share a borehole appear inconsistent at the borehole<\/p>\n<\/td>\n

    		\n

    Electrical anisotropy<\/p>\n<\/td>\n

    		\n
      \n
    • Use an inversion package with a forward model that allows for electrical conductivity anisotropy<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Outlier data are present in at least one dataset<\/p>\n<\/td>\n

    		\n
      \n
    • Check the dataset for outliers and edit<\/li>\n
    • Try the L1 norm for data misfit<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Parsing data into individual tomograms when they should be considered together<\/p>\n<\/td>\n

    		\n
      \n
    • Invert all data at once, rather than breaking it into pieces<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    \n

    Tomograms show vertical streaking, or high or low electrical conductivity patches only at boreholes<\/p>\n<\/td>\n

    		\n

    Resolution may vary greatly from the sides to the middle of the tomogram<\/p>\n<\/td>\n

    		\n
      \n
    • Create synthetic or hypothetical forward models of the experiment to evaluate resolution and likely artifacts<\/li>\n
    • Examine plots of the inversion\u2019s sensitivity or resolution matrix<\/li>\n
    • Explicitly model the boreholes<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n","protected":false},"author":1,"menu_order":20,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-63","chapter","type-chapter","status-publish","hentry"],"part":128,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":5,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions"}],"predecessor-version":[{"id":411,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/63\/revisions\/411"}],"part":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/128"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/63\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/media?parent=63"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapter-type?post=63"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/contributor?post=63"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/license?post=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}