a discussion on depth of investigation in geophysics and ... · october 20, 2015 1. october 20,...

October 20, 2015

SEG New Orleans18-24 October 2015

A Discussion on Depth of Investigationin Geophysics and AEM Inversion Results

Ted Asch, Aqua Geo FrameworksJared Abraham, XRI GeophysicsTrevor Irons, University of Utah

October 20, 2015

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Outline

1. What is the Depth of Investigation (DOI)?

2. Approaches to Calculating the DOI:• Theoretical Approaches to the DOI• Christiansen-Auken Model/Inversion-Space Approach to the DOI• Oldenburg-Li Approach to the DOI• Stochastic Modeling Approach

3. Example of DOI Determination Using Multiple Approaches

4. Discussion and Conclusions

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SEG New Orleans24 October 2015Motivation

Clients always ask: “Well, how deep can you see? How deep can you trust the results?”

• Usually someone is most interested in certain properties, and usually changes in properties, down to, or specifically at, a certain depth which is going to be, for that particular project, the “depth of interest”. Proper sampling protocols require that the depths investigated must be well below the depth of interest.

• Therefore, geophysical surveys must be designed appropriately to meet all these requirements.

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Note Difference between Resolution and Detection of a property. Resolution: Properties of interest accurately determined.Detection: While true value of a property not accurately recovered, there is still some

sensitivity, in the data, to changes in that property.

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SEG New OrleansSEG New Orleans24 October 201524 October 2015What is the DOI?

The DOI usually refers to the depth below which acquired surface (or airborne) data are insensitive to the physical properties of the earth. That is, at what depth(s) have/do the geophysical techniques used in an investigation

become insensitive to variations in the subsurface?

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0.55V

IADOIση

=

Theoretical Approaches to Calculating the DOI

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• Nabighian, M. N., and Macnae, J. C., 1989. Time-domain electromagnetic prospecting methods in Nabighian, M. N.. Ed.. Electromagnetic methods in applied geophysics. 2: Soc. Expl. Geophys.

• Spies, B.R., 1989, Depth of investigation in electromagnetic sounding methods: Geophysics, 54 (7), p.872-888.

0

2ρδµ ω

=0

2 tρδµ

=FDEM TDEM

For a TDEM near-zone sounding: source-receiver separation < DOI

Skin Depths:

ηV = Base Noise Level

σ = Averaged conductivities at sounding location

e.g. SkyTEM 508

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Theoretical Approaches to Calculating the DOI

Spies (1989) points out that the practical limitation on the depth of investigation for TDEM systems is a function of:• System transmitting power (the

‘moment’)• Time after turn-off at which the signal

levels have decayed to ambient background noise levels

• The resistivity/conductivity of the earth.

Earth Model

DOI

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Recent Approaches to Calculating the

DOI

Deduce the DOI in the inverted model space in combination with the data space.

Examine the numerical inversion results as an earth model

The numerical resolution displayed

The observed sensitivities of the model to the acquired data.

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( )( )

loglog

ii j

j

dG

m∂

=∂

1*

Ni j

i ij

j

Gds

t= ∆

=∑

*

, 1

j

j ii M

S s= −

= ∑


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Per Christiansen and Auken (2012):

𝐺𝐺𝑖𝑖𝑖𝑖, the Jacobian, represents the sensitivity to the i th data point 𝑑𝑑𝑖𝑖from the j th model parameter 𝑚𝑚𝑖𝑖 :

Sum each column of 𝐺𝐺ij , and normalize by the standard deviation of the data points, ∆𝑑𝑑𝑖𝑖 Produce an error normalized sensitivity s of each of the model parameters j in the model Normalize by the thickness, tj, of each layer

Calculate the Cumulative Sensitivities, S, for a given sounding by summing up the individual normalized sensitivities.

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• Christiansen, A.V., and Auken, E., 2012, A global measure for depth of investigation: Geophysics, 77 (4), WB171–WB177.

Jacobian

Normalized Sensitivity

Cumulative Sensitivity

Recent Approaches to Calculating the DOI

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From Christiansen and Auken (2012):

S represents the sensitivity to the model that was reached during the inversion. It is from an interpolation of the elements of 𝑆𝑆𝑖𝑖 that the DOI values at set “global” thresholds are selected.

SensitivityCumulative Sensitivity

Note that the formulation used by Christiansen and Auken (2012) does not incorporate sensitivity to nearby equivalent models.

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Create a DOI Index, R(x,z), by first inverting multiple models with very different starting halfspace resistivities and then comparing the results to determine what parts are data-driven and what parts are model-driven.

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• Oldenburg, G. and Y. Li, 1999, Estimating depth of investigation in DC resistivity and IP surveys: Geophysics, 64, 403–416, doi: 10.1190/1.1444545.

The DOI Index represents the degree of agreement between the different inversion results, m1 and m2. The starting reference model halfspace values are m1r and m2r.

1 2

1 2

( , ) ( , )( , )r r

m x z m x zR x zm m

−=

−



Alternative Methodology for Determining the DOI (Oldenburg and Li, 1999)

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Three inversion runs with very different reference model halfspaces (4000 ohm-m, 400 ohm-m, and 40 ohm-m) produce three models. • Where the model results are similar,

the inversion features are data-driven.

• Where the models vary, usually at depth, the inversion features below the DOI depth are model-driven.

• The depth at which the data-driven features end is defined to be the DOI.

Example of Oldenburg-Li (1999) Process for Determining the DOI

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Stochastic Model Space Analysis For Determining the DOI

• Perform a model-space search for a distribution of probable models fitting a given data set. Having searched the model space exhaustively, equivalence problems and other model-space related issues can then be included in the DOI discussion.

• The obvious drawback of this approach is the massive computational time & cost to perform the model-space search, which can make performing this analysis on large datasets untenable.

• Furthermore, the result of such an analysis is a distribution of possible models and interpreting this into actual geologic results is not straightforward.

• Not specifically a DOI investigation i.e. The model space investigation approach is fundamentally concerned with resolution and model equivalence and is not primarily concerned with determination of DOI’s, although certainly, at depth, models become spurious and extremely poorly constrained and a DOI can be inferred.

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An example of one sounding’s calculated Sensitivities and Cumulative Sensitivities.

The orange line plots the original Sensitivities and the blue line the calculated Cumulative Sensitivities.

CS = 1.2

CS = 0.6

Christiansen and Auken (2012) use Cumulative Sensitivities of 1.2 (square) and 0.6 (circle).



Example of the Determination of the DOI per Christiansen-Auken Methodology

Sensitivity


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CS = 1.2

CS = 0.6

Cumulative Sensitivity:Dashed Line: CS = 1.2Solid Line: CS = 0.6

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Determination of the DOI per Christiansen-Auken Methodology

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Determination of the DOI per Christiansen-Auken and Theoretical Methodologies

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CS = 1.2

CS = 0.6

Theoretical DOI

Spies (1989): Typical TEM dB/dt noise levels:Winter: 0.2 to 0.5 nV/m2

Summer: 2 to 10 nV/m2

A noise level of 0.5 nV/m2

was used to calculate the DOI shown here.

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An Expanded set of DOI Cumulative Sensitivity Thresholds

The orange line plots the original Sensitivities and the blue line the calculated Cumulative Sensitivities.

CS = 1.2

CS = 0.6

CS = 0.4

CS = 0.2

CS = 0.1

Determination of the DOI per Christiansen-Auken Methodology


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CS = 0.6

CS = 0.4

CS = 0.2CS = 0.1

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24 October 201524 October 2015Cumulative Sensitivity thresholds 1.2 (dashed white line), 0.6 (solid white line),

0.4 (green line), 0.2 (brown line), and 0.1 (black line).

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Cumulative Sensitivity thresholds 0.6 (solid white line) and 0.1 (black line) with available shallow and deep borehole logs.

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Theoretical DOI @ 4 nV/m2

Theoretical DOI @ 150 nV/m2

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Cumulative Sensitivity thresholds 0.6 (solid white line) and 0.1 (black line), available shallow and deep borehole logs, and theoretical DOI’s at 4 nV/m2 (red line) and 150 nV/m2 (brown line).

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18-24 October 201524 October 201540 Ohm-m Starting Halfspace

10 Ohm-m Starting Halfspace


Residual Avg Error = 0.547



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Determination of the DOI per Oldenburg-Li Methodology

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Oldenburg-Li DOI Index, R(x,z)

CS = 0.6

CS = 0.1

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Determination of the DOI per Oldenburg-Li Methodology

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Discussion on the Determination of the DOI

1. While Oldenburg and Li indicate that the range of 𝑅𝑅 should be between 0 and 1, the actual range for the calculated DOI Index data is from -0.53 to 1.98 due to the narrow band of starting models that was necessary in our study. Blue colors represent areas where the models match (low DOI Index values), and red colors areas where they don’t match (high DOI Index values).

2. Christiansen-Auken Cumulative Sensitivity DOI’s: The 0.6 DOI, the default “lower” DOI, plots well within the blue area where the models match, meaning that this DOI is set shallower in depth than where the inversion results are still data driven.

3. The Cumulative Sensitivity 0.1 DOI, however, generally lies in the yellow to orange colored zones (representing Oldenburg-Li DOI Indices of 0.2 to 0.4) just above the red colored areas, the low credibility results per Oldenburg-Li.

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Conclusions on the Determination of the DOI

1. Determinations of the DOI using a theoretical basis, model to data sensitivities, and multiple starting models have been presented.

2. The default DOI’s exported with the Aarhus Workbench inversion results, i.e. cumulative sensitivity thresholds 1.2 and 0.6, are overly conservative (i.e. shallow).

3. DOI’s using global cumulative sensitivity thresholds 0.4, 0.2, and 0.1 may allow for better discrimination between data- and model-driven results.

4. Calculation of the Oldenburg-Li DOI Index and theoretical DOI’s based on reasonable ambient noise levels serve to quantify the analysis.

5. Difference between resolution (accurate determination) and detection (sensitivity)suggests that the DOI be the line of demarcation between resolution and detection.

6. Equivalency: Best addressed through a model space-type analysis. Christiansen and Auken (2012) method does not incorporate sensitivity to equivalent models. The Oldenburg-Li (1999) approach can be thought of as a minimal model-space investigation and the full model-space investigation as an end-point of that approach.

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Questions?

Ted Asch, Ph.D., P.Gp.Aqua Geo Frameworks, LLCP.O. Box 150475, Lakewood, CO 80215Mobile: (720) 415-7312Email: [email protected]

mailto:[email protected]

a discussion on depth of investigation in geophysics and ... · october 20, 2015 1. october 20,...

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