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Geophysical Prospecting, 2007, 55, 835–852 doi:10.1111/j.1365-2478.2007.00640.x

Resolution analysis of geophysical images: Comparison between pointspread function and region of data influence measures

Carlyle R. Miller∗ and Partha S. RouthDept. of Geosciences, Boise State University, 1910 University Drive, Boise, ID 83725-1535, USA

Received June 2006, revision accepted September 2006

ABSTRACTPractical decisions are often made based on the subsurface images obtained by invert-ing geophysical data. Therefore it is important to understand the resolution of theimage, which is a function of several factors, including the underlying geophysicalexperiment, noise in the data, prior information and the ability to model the physicsappropriately. An important step towards interpreting the image is to quantify howmuch of the solution is required to satisfy the data observations and how much ex-ists solely due to the prior information used to stabilize the solution. A procedure toidentify the regions that are not constrained by the data would help when interpretingthe image. For linear inverse problems this procedure is well established, but for non-linear problems the procedure is more complicated. In this paper we compare twodifferent approaches to resolution analysis of geophysical images: the region of datainfluence index and a resolution spread computed using point spread functions. Theregion of data influence method is a fully non-linear approach, while the point spreadfunction analysis is a linearized approach. An approximate relationship between theregion of data influence and the resolution matrix is derived, which suggests thatthe region of data influence is connected with the rows of the resolution matrix. Thepoint-spread-function spread measure is connected with the columns of the resolutionmatrix, and therefore the point-spread-function spread and the region of data influ-ence are fundamentally different resolution measures. From a practical point of view,if two different approaches indicate similar interpretations on post-inversion images,the confidence in the interpretation is enhanced. We demonstrate the use of the twoapproaches on a linear synthetic example and a non-linear synthetic example, andapply them to a non-linear electromagnetic field data example.

I N T R O D U C T I O N

Geophysical inversion is inherently ill-posed and non-unique.To solve ill-posed geophysical inverse problems, we routinelyincorporate prior information into our solution. This stabi-lizes the solution, but also complicates the appraisal processbecause of the bias introduced into the solution. If the end goalof our data analysis involves decisions based on our solution,it is of great importance that we take steps to quantify how

∗E-mail: [email protected]

much information came from data and how much came fromprior information.

For different applications it is difficult to produce a uni-fied interpretation goal; however, an important question is todetermine which regions of the image are influenced by a pri-

ori information and which regions are influenced by the data.Although different ways of incorporating a priori informa-tion are sometimes considered to be subjective (Friedel 2003),regularization an essential step in obtaining a stable solutionthat is physically meaningful. Therefore, we usually cannotneglect imposing a priori information through the regulariza-tion procedure in the inverse problem. However, if we can

C© 2007 European Association of Geoscientists & Engineers 835

836 C.R. Miller and P.S. Routh

develop some procedure to separate out the regions that arenot constrained by the data, then it would greatly simplify theinterpretation and provide a robust way of connecting the in-terpretation with the physical experiment that generated thedata. Oldenburg and Li (1999) used two reference models ina non-linear DC resistivity inversion to determine the regionsthat are honoured by the data and generate a filter functioncalled depth-of-investigation curves to interpret the invertedimage. Alumbaugh and Newman (2000) used 50% spread cri-teria from the point spread functions to highlight the regionsof the model that are better resolved. In seismic tomography,some commonly applied approaches with a similar goal are theimpulse test and the checkerboard test. In these approaches,the raypaths of the real data are used on a synthetic impulsemodel or checkerboard model to generate the data. Subse-quently, the inverted images are compared with the syntheticmodel in order to understand the model resolution (Shearer1999).

In the depth-of-investigation approach of Oldenburg andLi (1999), the index is computed using the models obtainedfrom two inversion runs with different reference models. Theimage region that is most influenced by the data has a lowdepth-of-investigation index and we refer to this region asthe region of data influence. The advantage of this approachis that it can be applied to non-linear problems without anylinearization assumption. Alternatively, the resolving capabil-ity of the geophysical data can be obtained by computingcolumns of the resolution matrix, called point spread func-tions. We examine spread measures computed using pointspread functions and compare them with the region of datainfluence index. Both of these methods have previously beenused in the context of appraisal analysis (Oldenburg and Li1999; Alumbaugh and Newman 2000). Oldenborger (2006)applied the region of data influence approach to the 3Delectrical-resistivity-tomography experiment and showed thatit is a better approach to examining resolution, compared tosensitivity measures.

In this paper, we show the similarity of the two approaches.From a practical point of view, if two different approachesindicate similar interpretations on post-inversion images, theconfidence in the interpretation is enhanced. The examplespresented in this paper are one-dimensional (1D). We recog-nize that in reality the earth is three-dimensional (3D), butthe generic procedure of the approaches presented here willbe valid for problems in any dimensions. For computationalease, we use 1D examples throughout the paper. The paperis presented as follows: Using a linear problem we examinethe relationship and the mathematical connections that exist

between the two appraisal tools. We illustrate the relationshipwith a linear synthetic example and a non-linear synthetic ex-ample, and we apply it to a non-linear field example usingcontrolled source electromagnetics.

R E S O L U T I O N M E A S U R E S

Linearized resolution analysis

Insight into the resolution analysis problem can be gained byconsidering a linear inverse problem, dobs = Gm + ε. Thelinear problem has been studied in detail in previous literature(Backus and Gilbert 1968, 1970; Oldenburg 1983; Snieder1990; Parker 1994; Alumbaugh and Newman 2000) and herewe present the necessary equations to illustrate the variouscomponents of the resolution measure. In a linear problem,the objective function to be minimized is

φ =∥∥∥Wd(dobs − Gm)

∥∥∥2+ β‖Wm(m − m0)‖2, (1)

where G ∈RN×Mis forward modelling operator, Wd ∈ RN×N isthe data weighting matrix, Wm ∈ RM×M is the model weight-ing matrix used to impose constraints in the solution such assmoothing, m ∈ RM×1 is the model vector, m0 ∈ RM×1 is thereference model vector, dobs ∈ RN×1 is the data vector and β

is the regularization parameter that controls the relative con-tribution of the data misfit and the model norm. Denotingthe Hessian by H = (GTWT

d WdG + βWTmWm), the recovered

model is given by

m = H−1GTWTd WdG m + H−1GTWT

d Wd ε + H−1WTmWm m0.

(2)

In a more compact notation we can represent the estimatedmodel in equation (2) by m = R m + η + γ , where R is theresolution matrix,

R = (GTWT

d WdG + βWTmWm

)−1GTWT

d WdG. (3)

R provides a linear mapping between the estimated modelparameters, m and the true model parameters, m. For a non-linear problem given by dobs = F(m) + ε, the estimated solutionfor the model perturbation is given by

δm = H−1GTWTd WdG δm + H−1GTWT

d Wd(Q(δm)

− P(δm) + ε) + H−1WTmWm δm0,

(4)

where G is the sensitivity matrix, δm = m(k+1) − m(k) is themodel perturbation, δm0 = m0 − m(k) is difference betweenthe reference model and the model estimate at the kth iter-ation and Q(δm), P(δm)are the higher-order terms, typicallyneglected in the linearized analysis. When the higher-order

C© 2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 835–852

Resolution analysis of geophysical images 837

terms are not negligible, bias is introduced into the solution ofthe estimated model, in addition to the bias introduced by thereference model. For our analysis we will neglect the higher-order terms and carry out the interpretation using linearizedanalysis, using R in equation (3) as resolution mapping.

Point spread functions

The columns of the resolution matrix are called the pointspread functions. A point spread function describes how adelta-like perturbation in the model manifests itself in theinverted result. If a particular point spread function is wideand/or has large side lobes, then the corresponding model ispoorly resolved at that location and the resolution ‘width’ isbroader than the cell dimension. This introduces the idea thatspread of these point spread functions can be used as a measureto assess the resolving capability of the data and the prior in-formation. We note that the prior information can effectivelybe treated as data in the inverse problem, so the point spreadfunction and hence the spread will be influenced by the priorinformation.

Spread criterion for the point spread function

To compare the resolution capability of the data and priorinformation in different regions of the model, point spreadfunctions can be compared directly with each other. This canbe useful especially in large-scale problems where the ques-tion about resolvability is limited to few specific regions ofthe model. Although the point spread function comparison isa valid procedure and has the capability to convey a signif-icant amount of detail, the process can be tedious when weare faced with comparing thousands of cells throughout theentire model instead of a few cells within a specific region ofinterest. Here, we develop a spread criterion to summarize theinformation contained in the point spread function to a singlenumber.

In general, there are many possibilities when choosing thespread criterion, i.e. spread definition is non-unique. The ideais very similar to the definition of attributes computed fromtime–frequency maps (Barnes 1993; Sinha et al. 2005). Foreach cell, this number can be used as a diagnostic measureto compare the resolving capability of the inverse operatorin different regions of the model. We propose the followingdefinition of the spread (Routh and Miller 2006):

S(rk) =[∫

W (r, rk) (p(r, rk) − (r, rk))2 dr

α + ∫p2(r, rk)dr

]1/2

, (5)

where W(r, rk) is a distance weighting matrix that increasesthe spread value when side lobes exist away from the regionof interest. We choose the weighting matrix as W(r, rk) =1 + |r − rk|λ where λ ≥ 1. For our examples we choose λ =1, and (r, rk) = 1 when r = rk and (r, rk) = 0 for r �=rk. The point spread function p(r, rk) is for the cell whosecentre is at rk, and α is a small threshold parameter chosenso that the denominator in the spread equation is not equalto zero when the energy of the point spread function is zero.This can occur in regions of insufficient data coverage. If thepoint spread function peak is shifted from the region of in-terest, the chosen spread criterion will generate large spreadvalues. In subsequent sections we demonstrate the use of thepoint spread function and spread criterion to interpret invertedmodels obtained from a synthetic linear inverse problem, asynthetic non-linear inverse problem and a field example withfrequency-domain controlled-source electromagnetic data.

Averaging kernels

The rows of the resolution matrix are the averaging or Backus–Gilbert kernels (Backus and Gilbert 1970). These kernels showhow the value of a specific model parameter is obtained by av-eraging the model in the entire domain. If the averaging kernelis a delta-like function at a certain location, then the contri-bution to the estimated value is obtained from that regionand the contribution to the average value from other regionsof the model is insignificant. If we remove the effect of thereference model in equation (2) from the inversion then theaverage value of the model parameter can be represented bythe following equation:

〈m〉 = m − H−1WTmWm m0 = R m + η. (6)

The average value of the model parameter is equal to the pro-jection of the Backus–Gilbert averaging kernel (i.e. row of R)on to the true model plus the noise contribution. The averagevalue is obtained by subtracting the bias due to the referencemodel as shown in equation (6). Since 〈m〉k is obtained by pro-jecting the averaging kernel on the true model, we denote itas the average value of the model estimate. Thus three quan-tities: the average value 〈m〉k, the averaging kernel rk = R(k,:)and the noise contribution ηk, should be jointly interpreted.The uncertainty of the average value of a specific model param-eter is associated with the noise contribution and the spread ofthe averaging kernel. We note that the point spread functions,which are more informative about the inverse operator, arefundamentally different from Backus–Gilbert kernels and wedemonstrate this difference using a linear example.



Region of data influence

For the regularized solution given by equation (2), in regionsunconstrained by the data, the estimated model will revert tothe reference model. This is the fundamental idea behind thedepth-of-investigation index introduced by Oldenburg and Li(1999). Thus, by inverting the same data (with the same noiseassumption) using two different reference models, a regionof data influence (RDI) can be constructed as proposed byOldenburg and Li (1999):

RDI = |m2 − m1|∣∣mref2 − mref

1

∣∣ . (7)

In regions of the model that are unconstrained by the data, thisratio will go to unity, whereas in the regions constrained bythe data, this ratio will approach zero. In subsequent sections,we construct this resolution measure for three examples bycalculating this ratio over the entire model domain. In theory,the reference model can be a heterogeneous or a homogeneousmodel; however, in most cases a homogeneous reference modelis used (Oldenburg and Li 1999). The advantage of the re-gion of data influence measure is it does not depend on thelinearization assumption for non-linear problems. However,for linearized iterative algorithms, the reference models can-not be far from the true reference model, due to convergenceproblems.

R E L AT I O N S H I P B E T W E E N T H ER E S O L U T I O N M AT R I X A N D R D I

In this section we examine the connection between the reso-lution matrix and the region of data influence function. Weexamine the relationship using a linear inverse problem sincethe mathematical operations are simpler. The estimated modelin equation (2) can be written in an alternative form, given by

m = m0 + R (m − m0) + H−1GTWTd Wdε. (8)

In the region of data influence approach, we use two differ-ent model estimates m2, m1 for the corresponding referencemodels m2

0, m10. The difference between the estimates can be

represented by

m2 − m1 = (m2

0 − m10

) + R2(m − m2

0

) − R1(m − m1

0

)+ [

H−12 − H−1

1

]GTWT

d Wdε, (9)

where R2 and R1 are the resolution matrices obtained by in-verting the data using the respective reference models m2

0, m10.

The kth element of the model difference is

(m2 − m1)k = (m2

0 − m10

)k + (

rk2

)T(m − m2

0

) − (rk

1

)T (m − m1

0

)+

[(hk

2

)T − (hk

1

)T]

GTWTd Wdε, (10)

where rk2, rk

1 are the averaging kernels, i.e. the kth row of theresolution matrix R, and hk

2, hk1 are the kth row of the in-

verse Hessian matrices, H−11 =H−1(β1) and H−1

2 = H−1(β2),respectively.

To analyze equation (9) or equation (10), we consider asimplified case where the model weighting matrix is onlycomposed of the smallest component, i.e. there are no smooth-ing or flatness terms and therefore Wm is a diagonal ma-trix. For computational ease, we simply consider the casewhere weighting is uniform and we use the identity matrix formodel weighting. With this simplification, the resolution ma-trix in equation (3) can be written as R = (GTG + β I)−1GTG,where G = WdG. Thus for two different reference models,the difference in the resolution matrix is in the regulariza-tion term, i.e. R1 = R(β1) = (GTG + β1 I)−1GTG and R2 =R(β2) = (GTG + β2 I)−1GTG. Representing ε = Wdε, equa-tion (9) can be written as

m2 − m1 = (m2

0 − m10

) + (R2 − R1)m − (R2m2

0 − R1m10

)+ [

H−12 − H−1

1

]GTε (11)

Equation (11) shows the mathematical connection betweenthe region of data influence and the resolution matrices.Some insight can be gained by examining each term in equa-tion (11) in detail. We use singular value decomposition (SVD)to analyse equation (11). Decomposing G = U�VT usingSVD, equation (11) can be written as (see Appendix A)

m2 − m1 = (m2

0 − m10

) + V(

�2

�2 + β2− �2

�2 + β1

)VTm

− V(

�2

�2 + β2VTm2

0 − �2

�2 + β1VTm1

0

)

+ V(

�

�2 + β2− �

�2 + β1

)UTε. (12)

Analysis of the noise term [H−1(β2) − H−1(β1)] GTε

This term is the contribution of the noise on the recoveredmodel parameters. Physically GTε is the projection of the noiseon to the model space and the pre-multiplication by the in-verse Hessian provides a scaling. The goal in this analysis is tosee how this term contributes to the region of data influence



measure. The SVD representation of this term from equation(12) is given by

[H−1 (β2) − H−1 (β1)

]GTε = V

(�

�2 + β2− �

�2 + β1

)UTε.

(13)

Consider the case when Λ � (β2, β1), then the terms insidethe bracket cancel each other. For the case Λ → 0, the termhas insignificant contribution as well. We consider anothercase when the reference models are close enough such thatregularization parameters differ by a small amount, i.e. β2 =β1 + δβ. Substituting this in equation (13), the difference inthe noise term is given by

[H−1 (β2) − H−1 (β1)

]GTε

= V

(�

�2 + β1

(1 + δβ

�2 + β1

)−1

− �

�2 + β1

)UTε. (14)

We expand the inverse power series and neglect terms of sec-ond and higher orders. Therefore to the first order, equa-tion (14) is given by

[H−1 (β2) − H−1 (β1)

]GTε

= V(

�

�2 + β1

(1 − δβ

�2 + β1

)− �

�2 + β1

)UTε

= − V(

�δβ

(�2 + β1)2

)UTε. (15)

The quantity inside the bracket can be small when the denom-inator is much larger than the numerator. For Λ → 0 andΛ � β1 the diagonal matrix in equation (15) vanishes. ForΛ ≈ β1 the diagonal matrix reduces to δβ/β3

1 which is alsoa small quantity if δβ/β1 < 1. Therefore we would expectthat the contribution of this term to the overall region of datainfluence measure is small.

Analysis of the resolution matrix projected on to referencemodels (R2m2

0 − R1m10)

The SVD representation of this term from equation (12) isgiven by

(R2m2

0 − R1m10

) = V(

�2

�2 + β2VTm2

0 − �2

�2 + β1VTm1

0

).

(16)

We apply the same argument that the reference models areclose enough so that we can write β2 = β1 + δβ, and we

obtain

(R2m2

0 − R1m10

) = V(

�2

�2 + β1

)VT

(m2

0 − m10

)

− V

(�2δβ(

�2 + β1)2

)VTm2

0. (17)

We can write the following equation to the first order by ne-glecting the second term in equation (17); we obtain

(R2m2

0 − R1m10

) ≈ V(

�2

�2 + β1

)VT

(m2

0 − m10

). (18)

Equation (18) shows that to a first order, the term (R2m20 −

R1m10) is the projection of R1 on to (m2

0 − m10).

Analysis of the resolution matrix projected on to the truemodel (R2 − R1) m

The SVD expansion of this term from equation (12) is givenby

(R2 − R1) m = V(

�2

�2 + β2− �2

�2 + β1

)VTm. (19)

Writing β2 = β1 + δ β and applying the first-order approxi-mation, we obtain

(R2 − R1) m = −V

(�2δβ(

�2 + β1)2

)VTm. (20)

Using a similar argument to that used in equation (17), thecontribution of the term (R2 − R1) m to the region of datainfluence measure is also negligible. Thus, if we combine theresults of equations (15), (17) and (20), we obtain

m2 − m1 = (I − R1)(m2

0 − m10

) − V

(�2δβ(

�2 + β1)2

)

× VT(m − m2

0

) − V(

�δβ

(�2 + β1)2

)UTε. (21)

Equation (21) is the expression connecting the region of datainfluence and the resolution matrices when the regularizationperturbation between the two inversions is of first order, i.e.the change in the regularization parameter between the twoinversions using the two reference models is not too large, suchthat O(‖δβ‖2) → 0. Neglecting the contribution of the secondand the third terms in equation (21) and denoting R = R1, weobtain

m2 − m1 ≈ (m2

0 − m10

) − V(

�2

�2 + β1

)VT

(m2

0 − m10

)= (I − R)

(m2

0 − m10

). (22)



This approximate relationship shows that the difference be-tween the two inversions in the estimated models is the pro-jection of the blurring operator (I − R) on to the differencedreference models. For a linear problem, it formally establishesthe connection between the region of data influence (RDI)measure and the resolution matrix. In terms of each element,equation (22) can be written as

(m2 − m1)k ≈ (m2

0 − m10

)k − (

rk)T (

m20 − m1

0

). (23)

If the rows of the resolution matrix rk = (0, . . . 0, 0, . . . ,0), then RDI = 1, indicating poor resolution. In the re-gions where rk = δk = (0, . . . 0, 1, 0, . . . , 0), the differencein equation (23) approaches zero, thus RDI = 0, indicat-ing good resolution. A further simplification can be madeif we assume that the two reference models are constantand equal to two different homogeneous half-space values.This is commonly used in the region of data influence (RDI)approach (Oldenburg and Li 1999). Writing (m2

0 − m10) =

αe where e is a vector of ones and α is a constant equal to thedifference of the half-space values, then RDI for the kth cellis given by (m2 − m1)k/α ≈ ∑

(I − R)k, i.e. it is equal to thesummation of the kth row of the blurring function (I − R).Thus the RDI of the kth cell can be written as

RDIk ≈∑

j

(1 − rkj

), (24)

Figure 1 Plots of the model, kernels and data for the linear example. The top plot shows the true model. The horizontal axis is distance and thevertical axis is amplitude. The centre plot shows the Gaussian data kernels used for the example. The bottom plot shows the true data and theobserved data, contaminated with 2% standard deviation random noise.

where rkj is the jth element in the kth row of the resolutionmatrix.

We note that the region of data influence measure givenin equation (7) is the ratio of the difference between the twoinverted models to the difference between the two referencemodels. In the limit that the two reference models are closeenough, the region of data influence measure can be mathe-matically stated as the derivative of the inverted model withrespect to the change in the reference model. Therefore to de-rive the relationship between the region of data influence andthe resolution matrix, we assumed that the two reference mod-els are close enough so that the difference in regularization isnot significant. Equation (22) was obtained using SVD anal-ysis by considering the model weighting as diagonal. For ageneral model weighting matrix, a similar expression can bederived as shown in Appendix B.

An important point to note is that the region of data influ-ence measure is dependent on the averaging kernels, i.e. rowsof the resolution matrix and not the point spread functions.So the spread from the point spread functions can potentiallyprovide a different resolution measure. In the next section, us-ing a linear example, we provide insight into the relationshipbetween the region of data influence and resolution matrixand we demonstrate the similarities and differences betweenthese two measures.



L I N E A R E X A M P L E

We consider a 1D, linear inverse problem of the form dj =(gj, m), where the data kernels are blurry Gaussian functionscommonly used in computer scale vision problems, given byg j = Aexp( −(x−xj )2

4σ2 ), where A = 2.0 and σ = 0.026 for ourparticular example. Using a synthetic model vector with 100parameters, we generated 30 data observations and contami-nated these data with Gaussian random noise.

Figure 1 shows the true model, the Gaussian data kernelsand the observed data. Next, we inverted the same data withtwo different objective functions, one designed to recover thesmallest model and the other designed to recover a combina-tion of the smallest and flattest models. This was accomplishedby beginning with a general model objective function whichis a combination of the smallness and flatness components,given by φm = αs ‖Ws(m − m0)‖2 + αx ‖Wx(m − m0)‖2 whereWs is a smallest model weighting matrix that is effectively adiagonal matrix, and Wx is the flattest model weighting ma-trix which is the discrete form of the first-derivative operatorto impose smoothness in the solution. αs and αx are the con-trol parameters chosen to weigh the relative contribution ofsmallness versus flatness. An L-curve criterion (Hansen 1992)was used to select the value of the regularization parameterfor each inversion.

Figure 2 For this set of plots, we consider a model objective function containing only a smallness component (αs = 1.0; αx = 0), thus the modelweighting matrix is the identity matrix. The top panel shows the comparison between the true and recovered models for constant reference modelvalues of 0.8696 (Inv-1) and 1.15 (Inv-2). The second panel shows the point spread functions (PSFs) and Bachus–Gilbert kernels for Inv-1 andInv-2 at the 10th cell (x = 0.10). The third panel shows a similar comparison at the 95th cell (x = 0.95). The fourth panel shows the differencein the inverse Hessian matrix for inversion 1 and inversion 2 at the 10th and 95th cells. The bottom panel shows the L-curve (φd vs. φm) usedto select the regularization parameter.

Inversion with only the smallest component

First, we consider only a smallest model component by choos-ing αs = 1, αx = 0. The choice of smallest model weightingproduces a diagonal matrix and the inversion has the abilityto recover the reference model when data kernels are non-informative.

Figure 2 shows a comparison of the two inverted modelsusing the different reference models (m1

ref = 0.8696, m2ref =

1.15) and plots of point spread functions and Bachus–Gilbertkernels from different portions of the model. From Fig. 2, wesee that the resolution in the 10th cell is significantly betterthan that in the 95th cell. The flat point spread function andaveraging kernel for the 95th cell clearly demonstrate the lackof resolution. The third panels in Fig. 2 show the differencein the inverse Hessian matrix for the 10th (x = 0.1) and 95th(x = 0.95) cells. In each case, this difference is small becauseof the small difference in the regularization parameter. Thebottom plots in Fig. 2 show the L-curves used in selecting theregularization parameters for the two separate inversions.

The top left-hand plot in Fig. 3 shows the two inverted mod-els as well as the true model. Note that both estimated modelsclearly revert back to their respective reference models beyonddepth of x = 0.75. The middle left-hand plot in Fig. 3 showsthe region of data influence measure computed using two



Figure 3 For this set of plots, αs = 1.0 and αx = 0, as in Fig. 2. The top left-hand plot shows a comparison between the recovered models andthe true model. The middle left-hand plot shows a comparison of the region of data influence (RDI) computed using equation (7) (red) and theregion of data influence computed using the approximation in equation (24) (blue). The bottom left-hand plot shows the point spread function(PSF) spread. The three plots on the right show a graphical view of the terms in equation (10). The top right-hand plot shows the GTWT

d Wdε

term. The middle right-hand plot shows the norm of the difference in the rows of the inverse Hessian matrices. The bottom right-hand plotshows that the contribution of the term [H−1

2 − H−11 ] GTWT

d Wdε to the region of data influence is minimal.

separate methods, one using the exact method of equation (7)and the other using the approximation from equation (24).Note that although there are differences between the two re-gion of data influences, both indicate good resolution up toa depth of x = 0.75 and a loss in resolution beyond this dis-tance. The same can be said for the point spread functionspread value shown in the bottom left-hand plot of Fig. 3.The three right-hand panels in Fig. 3 provide a graphical viewof some of the terms from equation (10). The top right-handpanel in Fig. 3 shows the GTWT

d Wdε term. Note that whereresolution is good, this quantity is non-zero, but where resolu-tion is poor, this quantity goes to zero. The middle right-handplot in Fig. 3 shows the norm of the difference in the rows ofthe inverse Hessian matrices. Note that the difference is smalleverywhere because of the diagonal model weighting and thesmall difference in the regularization parameters. The bottomright-hand plot in Fig. 3 shows that when these two quanti-ties are multiplied, the result is small everywhere, i.e. [H−1

2 −H−1

1 ]GTWTd Wdε≈ 0, and makes a minimal contribution to the

difference in the model estimates in equation (10).

Inversion with flattest and smallest component

Next we consider a more general case where the modelobjective function contains both a smallness component and a

flatness component so that the model-weighting matrix is nolonger a diagonal matrix. We chose αs = 10, αx = 1 for ourcomputations.

Figure 4 first shows a comparison of the two inverted mod-els using the different reference models (m1

ref = 0.10, m2ref =

10) and plots of point spread functions and Bachus–Gilbertkernels from different portions of the model. From Fig. 4, wesee that the resolution in the 10th cell is again significantly bet-ter than the 95th cell. The flat point spread function for the95th cell clearly demonstrates a lack of resolution. However,the averaging kernel for the 95th cell shows a peak, indicat-ing that the average value is non-zero, although this peak isshifted well to the left of the region of interest. This clearlydemonstrates the difference between point spread functionsand Bachus–Gilbert kernels. In the regions where there is noresolution, there is no purpose in considering the credibilityof the average value and hence these regions should be re-moved from any further interpretation. The bottom plots inFig. 4 show the L-curves used in selecting the regularizationparameters.

The top left-hand plot in Fig. 5 shows the two invertedmodels as well as the true model. Note that both estimatedmodels attempt to revert back to their respective referencemodels beyond approximately x = 0.75. However, in the



Figure 4 For this set of plots, we consider a model objective function containing both a flatness (αx = 1) and a smallness (αs = 10) component,thus the model weighting matrix is not the identity matrix. The top panel shows the comparison between the true and recovered models forconstant reference model values of 0.10 (Inv-1) and 10 (Inv-2). The second panel shows the point spread functions (PSFs) and Bachus–Gilbertkernels for Inv-1 and Inv-2 at the 10th cell (x = 0.10). The third panel shows a similar comparison at the 95th cell (x = 0.95). The fourth panelshows the difference in the inverse Hessian matrix for inversion 1 and inversion 2 at the 10th and 95th cells. The bottom panel shows the L-curve(φd vs . φm) used to select the regularization parameter.

Figure 5 For this set of plots, αx = 1 and αs = 10, as in Fig. 4. The top left-hand plot shows a comparison between the recovered models andthe true model. The middle left-and plot shows a comparison of the region of data influence (RDI) computed using equation (7) (red) and theregion of data influence computed using the approximation in equation (24) (blue). The bottom left plot shows the point spread function (PSF)spread. The three plots on the right show a graphical view of the terms in equation (10). The top right-hand plot shows the GTWT

d Wdε term.The middle right-hand plot shows the norm of the difference in the rows of the inverse Hessian matrices. The bottom right-hand plot shows thatthe contribution of the term [ H−1

2 − H−11 ] GTWT

d Wdε to the region of data influence is minimal.



case of inversion 2 where the reference model equals 10, thesmoothing imposed by the model objective function does notallow for the inverted model to revert completely back to thereference model. The middle left-hand plot in Fig. 5 showsthe region of data influence measure computed using two sep-arate methods, one using the exact method of equation (7)and the other using the approximation from equation (24).There are minor differences between the two region of datainfluences, but both indicate good resolution up to approxi-mately x = 0.75 and a loss in resolution beyond this distance.The region of data influence does not reach a value of unitybecause the imposed model smoothing does not allow the in-verted result to revert fully back to the reference model inthe region where the resolution is poor. The bottom left-handplot of Fig. 5 shows the point spread function spread value.The point spread function spread looks similar to the spreadcomputed in the smallest model case suggesting that the re-gion of data influence and the point spread function spreadare two different resolution measures. Note that both the re-gion of data influences ratio and the spread indicate a loss inresolution beyond x ∼0.75, corresponding to the distance atwhich the recovered models for the two inversions diverge totheir respective reference models, i.e. beyond this location, therecovered models are insensitive to the available data.

The three plots on the right side of Fig. 5 provide a graphicalview of some of the terms from equation (10). The top right-hand plot in Fig. 5 shows the GTWT

d Wdε term. Note that,similarly to the smallest model case, where resolution is good,this quantity is non-zero, but where resolution is poor, thisquantity goes to zero. The middle right-hand plot in Fig. 5shows the norm of the difference in the rows of the inverseHessian matrices. Note that the difference is small when reso-lution is good and large when resolution is poor, which is theexact opposite of the GTWT

d Wdε term shown in the top right-hand plot. The bottom right-hand plot in Fig. 5 shows thatwhen these two quantities are multiplied the result is smalleverywhere, i.e. [H−1

2 − H−11 ] GTWT

d Wdε ≈ 0, and makes aminimal contribution to the difference in the model estimatesin equation (10) and thus, by extension, makes a minimal con-tribution to the region of data influence measure.

N O N - L I N E A R E X A M P L E S

We further demonstrate the use of the point spread functionspread and the region of data influence ratio with examplesfrom controlled-source electomagnetics. The particular phys-ical experiment considered here is controlled-source audio-frequency magnetotellurics (CSAMT). CSAMT is a frequency-

domain electromagnetic sounding experiment which has beenapplied to a number of problems, such as minerals explo-ration, geothermal exploration, groundwater investigationsand structural analysis (Zonge and Hughes 1991). For bothexamples, the CSAMT source considered is an earthed elec-tric dipole of finite length with a frequency range of 0.5 to8192 Hz.

Synthetic CSAMT example

Using a five-layer, 1D earth conductivity model, containing ashallow resistive region and a deeper conductive region, wecomputed the electric and magnetic field data at a single re-ceiver location. The transmitter for this example is 1 km inlength and is orientated in the east–west direction. The receiverlocation is 2 km north of and centred on the transmitter. Notethat this receiver location will be in the strong signal zone forthe Ex, Hy, and Hz components and in the null zone for the Ey

and Hx components. For this reason, the results shown for thisexample primarily consider the Ex, Hy and Hz components.

The frequency range for the synthetic experiment was from0.5 Hz to 8192 Hz. Data were computed for five field com-ponents (Ex, Ey, Hx, Hy, Hz). These data were then contami-nated with noise, with a standard deviation based on 5% of theobserved amplitudes and 2◦ for phases (Fig. 6). We invertedthese data for an 80-layer earth conductivity model using anon-linear Gauss–Newton approach (Routh and Oldenburg1999).

Figures 7 and 8 show some point spread functions and av-eraging kernels selected from the conductive and resistive re-gions of the model. Although the inverted model estimates(Figs 7 and 8) vary, depending on the field data combinationchosen for inversion, it is clear that resolution is better in theconductive region compared to the resistive region, which isexpected in electromagnetic imaging experiments. Also notethat the resolution and the model recovery is better in thoseinversions that include the Ex-component.

The linearized resolution matrices for the various field datacombinations are shown in Fig. 9. Note that the resolutiondrops off for all inversions in the vicinity of 1 km depth. It isclear that resolution is significantly degraded at depth; how-ever, the inversions using the electric field component show aslightly greater resolved depth interval. This increased depthof investigation becomes more obvious when we compute thespread value for each of the inversions.

Figure 10 show that the inversions containing only magneticfield data have a diminished depth of investigation when com-pared to the inversions containing the electric field component.



Figure 6 Electric and magnetic field data versus frequency with random noise added. The standard deviation is based on 5% of observedamplitudes and 2◦ for phases. The horizontal axis for all plots is frequency (Hz). The vertical axis for the top left-hand plot is electric fieldamplitude (µV/m); the vertical axis for the top right-hand plot is magnetic field amplitude (µA/m); the vertical axis for the bottom two plots isphase (◦).

Figure 7 Model estimates and selected point spread functions and averaging kernels for the inversion of Ex (left-hand column), Hy (centrecolumn), and ExHy (right-hand column). The horizontal axis for all plots is depth (m). Row (a) shows the model estimate (blue) and the truemodel (red) for the three separate inversions. The vertical axis is electrical conductivity (S/m). For remaining plots (rows b,c,d,e), the verticalaxis is dimensionless amplitude of R. Rows (b) and (c) show the point spread functions and the averaging kernels for the conductive region,respectively. Rows (d) and (e) are the point spread functions and the averaging kernels for the resistive region, respectively.



Figure 8 Model estimates and selected point spread functions and averaging kernels for the inversion of Hz (left-hand column), ExHyHz (centrecolumn), and Dxy (right-hand column). The horizontal axis for all plots is depth (m). Row (a) shows the model estimate (blue) and the truemodel (red) for the three separate inversions. The vertical axis is electrical conductivity (S/m). For remaining plots (rows b,c,d,e), the verticalaxis is dimensionless amplitude of the resolution matrix. Rows (b) and (c) show the point spread functions and the averaging kernels for theconductive region, respectively. Rows (d) and (e) are the point spread functions and the averaging kernels for the resistive region, respectively.

Figure 9 Linearized resolution matrices from inversion of various field component combinations. Horizontal and vertical axes are log10(depth)in metres. The colour bar on the right is common to all of the figures, and corresponds to the dimensionless amplitude of the resolution matrix.



Figure 10 Point spread function spread value at the final iterationof the inversions. The horizontal axis is depth (m). This plot clearlyshows a distinction in the depth of investigation for those inversionsincluding the Ex-component and those using only the magnetic fielddata.

If we choose an iso-spread value (here we have chosen a valueof 3), we can then compute the depth of investigation for thevarious inversions (Fig. 11). In fact, when we do this, we seethat there is diminished resolution both in the near surfaceand at depth so that we are able to define an interval over

Figure 11 The top panel shows the spread curves for the ExHyHz (blue) and HxHyHz (black) field inversions. The horizontal line indicates aniso-spread value of 3 chosen to examine the depth of investigation of the fields. Point spread functions within the conductive layer are shown inthe middle (ExHyHz) and bottom (HxHyHz) panels. The horizontal axis for all three plots is depth (m), and the vertical axis is dimensionlessamplitude.

which we may then interpret the results of our inverted models(Fig. 12).

Figure 13 shows the region of data influence for the variousdata combinations. In this case, the region of data influencecurves provide very similar information to that provided bythe spread curves in Fig. 10. We see that the inversions in-cluding the Ex-component have a depth of investigation thatis better than those inversions considering only the magneticfield components and that resolution is poor for all the in-versions within the padding region of the inverted model. Werecognize that the choice of an iso-spread in the case of pointspread function spread and the choice of an index value inthe case of region of data influence are subjective, and thesechoices will ultimately affect which regions of the model willbe interpreted. Once the choice is made it provides the demar-cation between the good and bad regions of the model resolvedby the data. In spite of this subjectivity, the flexibility of hav-ing these measures provides a useful interpretation capabilitythat is of practical value.

Field CSAMT example

In this section we present the analysis of the point spreadfunction spread and the region of data influence measurefor a real data set. CSAMT data were collected in order to



Figure 12 The top panels show the point spread function spread curves for the ExHyHz (blue) and HxHyHz (black) field inversions. Thehorizontal dashed lines show the chosen iso-spread value of 3. The vertical dashed lines show the depth intervals resolved by the data, based onthis choice of iso-spread value. The bottom panels show the truncated models. The horizontal axis for all four plots is depth (m), the verticalaxis for the upper plots is dimensionless amplitude, and for the lower plots it is electrical conductivity (S/m).

Figure 13 Region of data influence curves for various field componentinversions. The horizontal axis is depth (m). The region of data influ-ence plot, like the spread plot (see Fig. 10), clearly shows a distinctionbetween the depths of investigation for those inversions including theelectric field data and those only using the magnetic field data.

characterize a geothermal prospect in south-western Idaho,USA. The source was a 1400 m long earthed electric dipole,located approximately 6 km from the receiver line (Fig. 14).Here, we present results of inversion of the amplitude ofthe x-component of the electric field for 56 receiver stations.These data were measured at 25 frequencies ranging from 1 to

8192 Hz. The reference model used was a 100 �m half-space. To calculate the region of data influence ratio, wealso inverted these data with reference models of 10 �m and1000 �m. In each case, we inverted for a 1D, 60-layer earthmodel with layer thickness increasing logarithmically withdepth, using a noise assumption of 5% of the observed electricfield amplitudes.

Figure 15 shows the generally good agreement between theobserved and predicted data. The majority of these data werefit within a χ2 misfit criterion of one. Results for those datathat were not fitted within this criterion were discarded fromfurther analysis. Figure 16 shows the inverted model to a depthof 1200 m, and the spread and region of data influence plotsfor the entire model depth. Note that for the region of themodel that we are interested in, i.e. from approximately 100to 500 metres depth, the resolution is good according to boththe region of data influence and the spread plot, so we areable to interpret this portion of the model with improved con-fidence, compared to the very shallow or very deep regionsof the model. Both the spread plot and the region of datainfluence plot show a well-resolved interval between 10 and1000 metres. This is the depth interval hypothesized to con-tain the primary flow conduits for the geothermal system, thusthe more conductive regions within the conductivity imageare likely to correspond to the location of geothermal fluids.



Figure 14 Map view of CSAMT survey in south-west Idaho, USA. We show results from transmitter N, and receiver line 900N, indicated bythe thick red lines on the map.

Figure 15 Observed (top plot) and predicted (bottom plot) electric field amplitude data for Line 900N. The colour bar on the right is commonto both plots, and corresponds to log10(|Ex|) (µV/m).

Unfortunately we do not have any borehole information toverify the result but the presence of hot fluids in surface springsindicates the presence of geothermal fluids in the subsurface.

C O N C L U S I O N S

It is important to demonstrate the credibility of the subsurfaceimages obtained from geophysical data, particularly when in-

ferences and decisions are made, based on the images. There-fore, in addition to computing the final images, further analy-sis should be carried out using various appraisal tools to eval-uate which parts of the model are more believable than others.Even at a very simplistic level, this knowledge is useful for aid-ing model interpretation. Depending on which specific ques-tions need to be answered based on the geophysical images, theappraisal analysis can be appropriately formulated. We have



Figure 16 Model and resolution for Line 900N. The top panel is theconductivity image to 1200 m depth with a colour bar that corre-sponds to log10(σ ) (S/m). The middle panel is the point spread func-tion spread value for the entire depth interval with the colour barcorresponding to the dimensionless spread amplitude. The bottompanel is the region of data influence (RDI) index for the entire depthinterval with the colour bar corresponding to dimensionless region ofdata influence amplitude. The horizontal axis for all three plots is thedistance (m) along the receiver line.

examined and compared the region of data influence measure,introduced by Oldenburg and Li (1999), and the spread, com-puted using linearized resolution analysis with point spreadfunctions. If different appraisal tools provide similar answersthen confidence in the model interpretation is enhanced.

In both region of data influence and point spread functionspread measure, the threshold between good and bad resolu-tion must be chosen. Although this is sometimes subjective,the value added by generating these measures far outweighsthe subjectivity of choosing the threshold value. To gain in-sight into these two measures, we demonstrated the connec-tion between them, i.e. the point spread function spread andthe region of data influence, using a linear inverse problem.Although the connection is better understood for linear prob-lems, we demonstrated via a numerical example that similarconclusions hold for a non-linear CSAMT inversion problem.An important difference is that the region of data influenceseems to depend more strongly upon the averaging kernels(rows of resolution matrix) than on the point spread function(columns of resolution matrix). Synthetic examples demon-strate that the point spread function spread has more detailed

information about the resolution problem than the region ofdata influence. Although there are some differences in bothapproaches, overall they indicate similar information aboutbetter resolved regions both for synthetic and field examples.Using different appraisal tools for a problem can help us gainadditional insight into the results of our inversions. This canlead to increased confidence in our solutions and help us tomake better informed decisions.

A C K N O W L E D G E M E N T S

We thank Zonge Engineering and Prof. Paul Donaldson forpermission to show the CSAMT data. C.M. is funded by agraduate fellowship from the Inland Northwest Research Al-liance (INRA). We thank Prof. Doug Oldenburg for an excel-lent review and particularly for his suggestion to look into thesmallest model example that provided insight into the prob-lem and resulted in an improved manuscript. We also thankan anonymous reviewer who helped to bring more clarity tothe paper. This research was funded by NSF EPSCOR award# EPS0132626 and EPA award #EPA-X-96004601.

R E F E R E N C E S

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A P P E N D I X A

In this appendix we present the necessary equations to repre-sent the singular value decomposition of the inverse Hessianand the resolution matrices. Denoting G = WdG and assumingthe model weighting matrix as diagonal, the inverse Hessianmatrix is given by H−1 = (GTG + β I)−1 and the resolutionmatrix R = (GTG + β I)−1GTG. We denote ε = Wdε and de-compose the matrix G = U�VTusing SVD. The expression forinverse Hessian can be obtained by solving Hx = b using SVD.The SVD representation of the equation (GTG + β I) x = b isgiven by (VΛ2 VT + β I) x = b. The estimated solution is givenby x = VVT x = V(�2 + β I)−1 VT b. Thus we can denote theinverse Hessian by H−1 =V (Λ2 + β I)−1 VT. Therefore thedifference in the inverse Hessian is given by

H−12 − H−1

1 = V(

1�2 + β2

− 1�2 + β1

)VT. (A1)

To clarify the notation, the matrix inside the brackets in equa-tion (A1) is a diagonal matrix such that 1/Λ2 + β = diag

(1/Λ21 + β, 1/Λ2

2 + β, 1/Λ2M + β.). In equation (A1) if we

consider Λ > (β2, β1), the difference in singular spectrum isnegligible. However, when Λ → 0 the difference in the singularspectrum is not zero and therefore the difference in the inverseHessian will not be zero. This insight into the inverse Hessianis useful; however for the noise term in equation (11), we areinterested in the projection of the inverse Hessian on to theadjoint operator, i.e. H−1GT = (GTG + β I)−1GT. For exam-ple, the noise term is given by H−1GTε = (GTG + β I)−1GTε.Writing it using SVD, we obtain H−1GTε = V(�/.�2 + β) UT ε.Thus the difference in the noise terms between the two inver-sions is given by

(H−1

2 − H−11

)GTε = V

(�

�2 + β2− �

�2 + β1

)UT ε. (A2)

To obtain the expression for the resolution matrix, we solve(GTG + β I) x = GTGb using SVD. The resulting solution isgiven by x = VVT x = V(�

2/.�2 + β )−1 VT b, such that we can

represent the resolution matrix as R = V(�2/.�2 + β )−1 VT.

Thus the difference in the resolution matrix R2 − R1 is givenby

R2 − R1 = V(

�2

�2 + β2− �2

�2 + β1

)VT. (A3)

A P P E N D I X B

In this section we derive the relationship between the regionof data influence and the resolution matrix when the modelweighting matrix is more general than diagonal weighting.Although the derivation using the SVD for diagonal modelweighting is insightful, in reality the prior information used ininverse problems is often far more complicated than a diagonaloperator. The region of data influence (RDI) in equation (7)can be considered as an approximation of the derivative whichcan be stated as: determine the change in the inverted modelgiven a change in the reference model. This allows us to write

RDI =∣∣∣∣ δmδm0

∣∣∣∣ , (B1)

where δm0 is the perturbation in the reference model that re-sults in the perturbation of the inverted model δm. To proceedfurther, we consider equation (8) and write it in an expandedform, given by

m = m0 +(GTG + β WT

mWm

)−1GTG (m − m0)

+(GTG + β WT

mWm

)−1GTε. (B2)

Next we perturb the reference model m0 → m0 + δm0. Thisresults in the perturbation of the regularization parameterβ → β + δβ and consequently it changes the inverted modelfrom m → m + δm. Mathematically this can be representedby

m + δm = m0 + δm0 +(GTG + (β + δβ) WT

mWm

)−1

× GTG (m − m0 − δm0)

+(GTG + (β + δβ) WT

mWm

)−1GTε. (B3)

In a more compact notation, we can write equation (B3) as

m + δm = m0 + δm0 + (H + δH)−1 GTG

× (m − m0 − δm0) + (H + δH)−1 GTe, (B4)



where H = (GTG + β WTmWm) and δH = δβWT

mWm. The in-verse Hessian operation in equation (B4) can be written as (H+ δH)x = b. The solution is given by x = (I + H−1δH)−1H−1b.To first order the solution can be written as x ≈ (I −H−1δH)H−1b = H−1b − H−1δH H−1b by neglecting the higher-order terms in the series expansion. Applying this expressionto equation (B4), we obtain

m + δm = m0 + δm0 + H−1GTG (m − m0 − δm0)

− H−1δH H−1GTG (m − m0 − δm0)

+ H−1GTε − H−1δH H−1GTe. (B5)

By rearranging terms, writing R = H−1GTG and substitutingδH = δ β WT

mWm, we obtain

m + δm =(m0 + R (m − m0) + H−1GTε

)+ (δm0 − Rδm0) .

− δβ H−1WTmWm R (m − m0 − δm0)

− δβ H−1WTmWm H−1GTε.

(B6)

Using equation (B2), equation (B6) can be simplified to obtain

an expression in terms of the inverted model perturbation,given by

δm = (I − R) δm0 − δβ H−1WTmWm

×(R (m − m0 − δm0) + H−1GTε

). (B7)

We can write δβ H−1WTmWm = (δβ/β)H−1βWT

mWm = (δβ/β)(I − R). Thus equation (B7) can be further simplified into theform,

δm = (I − R)(

δm0 − δβ

β(R (m − m0 − δm0) + H−1GTε)

).

(B8)

If the ratio (δβ/β) in equation (B8) is small, then we obtainthe following relationship,

δm ≈ (I − R) δm0. (B9)

Thus for a general model weighting matrix we also observethat the approximate relationship between the region of datainfluence and the resolution matrix is the same as that obtainedin equation (22).


resolution analysis of geophysical images: comparison between … · 2020. 3. 27. · introduction...

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