ResxDInv course, Peru 7. April 2021

• Follow-up on Tuesday’s course• Recap: Short live demo• Pseudosections• Inversion theory: Smoothness-constrained least squares in ResxDinv• Inversion practices: How to get a good inversion result?

Program

Partially based on Spanish lecture notes:

http://www.ags-cloud.dk/Wiki/tiki-download_wiki_attachment.php?attId=322&page=W_GeotomoNotes&download=y

Recap: Short live demo

RATCMIX.dat

Pseudosections

Pseudosection: data

Pseudosection: model

Actual model section

What is a pseudosection?

4 electrode positions (x,y) ->1 position (x,y)

Pseudosection: Simple example

Pseudosection in ResXDInv

Median depth of investigation(mediana de la profundidad de investigación)- Defined in Edwards, 1977

A bit more advanced, but essentially just a method to map from 4 electrode positions to depth

Array configurations and pseudosectionsDispositivos y pseudosecciónes

Pseudosections of a block model

Pseudosections of a block model

Take-home message:Pseudo-sections are not very representative of the actual resistivity distribution.Inversions are needed!

Pseudosections

Pseudosection: data

Pseudosection: model

Actual model section

Starting model Forward calculation Comparison of forward calculation and measureddata, is stopping criterionreached?

Update starting model

YesDone!

Inversion theory: Smoothness-constrained least squares in ResxDinv

No

Forward calculations• Subdivide model domain into a

number of polygons with constant resistivity each

• Approximate Poisson’s Equation for a static electric field using finite differences or finite elements

• Calculate potential field “for each” measurement and find modelled potential difference for each potential electrode pair in each measurement.

• Convert modelled potential difference to modelled apparent resistivity

potencialConductividad=1/Resistividad

Starting model Forward calculation Comparison of forward calculation and measureddata, is stopping criterionreached?

Update starting model

YesDone!

Inversion theory: Smoothness-constrained least squares in ResxDinv

No

Comparison of forward calculation and measured datavector de discrepancia

(vector with residuals) vector del 

resistividadaparenteobservados

(vector with observedapparentresistivities)

vector del resistividadaparentecalculados

(vector with modeled apparentresistivities)

Minimize thisnumber

Starting model Forward calculation Comparison of forward calculation and measureddata, is stopping criterionreached?

Update starting model

YesDone!

Inversion theory: Smoothness-constrained least squares in ResxDinv

No

Update starting model (simple approach, Gauss-Newton)vector de discrepancia

(vector with residuals) vector del 

resistividadaparenteobservados

(vector with observedapparentresistivities)

vector del resistividadaparentecalculados

(vector with modeled apparentresistivities)

Minimize thisnumber

Matriz Jacobiana(Jacobian Matrix)I.e: If I change resistivity of model block j, how much would resistivity at measurementi change?

vector del ”resistivitiesat eachblock”

Problems with Gauss-Newton

UnstableStrong lateral/vertical resistivity variation. Needs dampingBetter approach: Marquardt-Levenberg with smoothness constraints

Marquardt-Levenberg with smoothness constraintsvector de discrepancia

(vector with residuals) vector del 

resistividadaparenteobservados

(vector with observedapparentresistivities)

vector del resistividadaparentecalculados

(vector with modeled apparentresistivities)

Minimize thisnumber

Matriz Jacobiana(Jacobian Matrix)I.e: If I change resistivity of model block j, how much would resistivity at measurementi change?

vector del ”resistivitiesat eachblock”

Smoothness constraints

Tends to minimize spatial resistivity differences:• L2-norm: “smooth”• L1-norm: “robust”, “blocky”

Inversion practices: How to get a good inversion result?

Knud

2 most important factors in obtaining good inversion results

• Correct inversion settings!

• Good data quality!

The importance of selecting the correct inversion parameters – 2 inversions from exactly the same dataset!

The importance of selecting the correct inversion parameters – 2 inversions from exactly the same dataset!

• L2 norm/smooth model constraint

• L1 norm/robust model constraint

You need to use knowledge about the local geology to determine which result is most correct!

Good data quality

• Starts in the field: Good equipment, well charged batteries, correct procedures etc.

• But some effort can be done in the office as well

Assessing the quality of an inversion – DOI (Depth of investigation)Is a measure of to which degree the inversion result is constrained by the measured data as opposed to the starting model.

Where qm1 and qm2 are the starting model resistivities, and q1(x,z) and q1(x,z) are the inversion results. Low value = data driven result, high value = constrain driven result.

Assessing the quality of an inversion – DOI (Depth of investigation)Must be run and loaded as a separate inversion:

Assessing the quality of an inversion – DOI (Depth of investigation)

Further reading• Res2DInv and Res3DInv manuals https://ags-

cloud.dk/AGS/Geotomo%20Manuals/

• Dr. M. H. Lokes electrical survey course notes http://www.ags-cloud.dk/Wiki/W_GeotomoNotes

• Workflow guides for Arhus Workbench http://www.ags-cloud.dk/Wiki/W_GeotomoGuides

• Spanish lecture notes: http://www.ags-cloud.dk/Wiki/tiki-download_wiki_attachment.php?attId=322&page=W_GeotomoNotes&download=y

Questions?