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Geophysically motivated inverseproblems for the classroomC. W. Groetsch aa Department of Mathematical Sciences , University ofCincinnati , Cincinnati, OH 45221‐0025, USAPublished online: 09 Jul 2006.

To cite this article: C. W. Groetsch (1995) Geophysically motivated inverse problems for theclassroom, International Journal of Mathematical Education in Science and Technology, 26:3,379-388, DOI: 10.1080/0020739950260307

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INT. J. MATH. EDUC. SCI. TECHNOL., 1995, VOL. 26, NO. 3, 379-388

Geophysically motivated inverse problems for theclassroom

by C. W. GROETSCH

Department of Mathematical Sciences, University of Cincinnati, Cincinnati,OH 45221-0025, USA

(Received 4 January 1994)

Two model inverse problems inspired by geological prospecting are proposedas vehicles for classroom demonstrations of mathematical validation of basicphysical intuition, investigation of non-uniqueness and instability of solutions,and illustration of the power of good numerical software.

1. IntroductionThe current spirit of reform in mathematics education has excited a good deal

of interest in the integration of science, mathematics and computing in theundergraduate curriculum. There is a fairly general agreement that we should fostermodel building, kindle physical intuition, encourage conjecturing, and developanalytical thinking in our students. Moreover, we should exploit the capability ofmodern computers and software to promote insight as well as to extract answers. Butmathematics is driven by problems; and finding meaningful and interestingproblems in pursuit of these goals is no easy task.

Our aim in this note is to report on our experience using two problems (onediscrete and the other continuous) from what might be called 'pre-geophysics' in themathematics classroom. Such earth-related problems have a special appeal forstudents who often ask for 'down-to-earth' applications. Indeed, there has beenrecent interest in using the geosciences as the motivating theme in elementarymathematics courses [1,2].

The problems we discuss are inverse problems of remote sensing. Even indiscrete inverse problems solutions are often not unique [3] and in continuousinverse problems solutions are usually unstable with respect to small perturbationsin the data [4]. In addition to illustrating these mathematical notions, the problemswe present are useful vehicles for encouraging mathematical conjectures based onphysical intuition and displaying the power that modern numerical software(particularly, MATLAB [5]) can bring to bear on the analysis of difficultmathematical problems.

2. Locating a gravitational point sourceOur first inverse problem, involving a discrete gravitational model, requires

finding two numbers: the mass and location of a subterranean gravitational pointsource. The mathematics involved is at the level of elementary algebra and the

0020-739X/95 $1000 © 1995 Taylor & Francis Ltd.

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380 C. W. Groetsch

Xj

Figure 1.

Figure 2.

primary goal is to integrate physical and mathematical reasoning and to exploreuniqueness of solutions. This is done by first carrying on a dialogue with the studentswhich is based entirely on primitive physical intuition. This dialogue points the wayto a mathematical solution and mathematical validation of the results conjecturedon the basis of physical intuition provides students with a satisfying illustration ofthe interplay of mathematical and physical reasoning.

Suppose a gravitational point source, say a nugget of gold, lies one distance unitbelow the flat surface of the earth. We treat a much simplified model in which weassume all other gravitational sources are purely homogeneous so that thegravitational anomaly generated by the point source will be considered to be the onlytrue gravitational effect. Moreover, we assume a one-dimensional flat earthgeometry, so that the surface of the earth is a line (Figure 1). At various sites on thesurface we take, using a delicate mass-spring scale, measurements of the verticalcomponent of the gravitational attraction caused by the point source ([see 8, Chapter6] for a more complete description of gravity prospecting). Our goal is to determinetwo numbers, the position x and the mass m of the nugget.

Before taking up the mathematics, or even the exact nature of the physics of theproblem, I begin by exploring the problem with my students using primitivephysical intuition alone. My first question is, 'Will a single measurement suffice tosolve the problem?' Normally, the response is, 'No, because we are trying todetermine two independent numbers'. I press the students for a less mathematical,more physical, explanation and they usually produce an illustration like Figure 2 toshow that many different configurations can account for a single gravitymeasurement.

My next question is, 'Do two measurements at distinct sites suffice?' After a bitof thought, the response is usually, 'Not necessarily', and a situation as illustrated

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Geophysically motivated inverse problems 381

9 9

Figure 3.

Figure 4.

in Figure 3 is cited. Astute students point out that two equal observations at distinctsites indicates that the source is located midway between the sites (Figure 4).Furthermore, it is sometimes suggested that in the case of unequal measurementsat distinct sites it should be possible to locate the source and determine its mass ifwe know a priori that the nugget lay between the sites.

The intuitive physical discussion above is entirely qualitative, but it may bequantified by the introduction of Newton's law of gravitational attraction. Oncequantified, the results obtained by intuitive physical reasoning can be validated bythe application of some elementary algebra.

We assume that our scale is a simple weighted spring of mass one so that thegravitational force that the mechanism measures is, by Newton's law, ymjr2, wherey is the gravitational constant, r is the distance from the observation site to the pointsource, and m is the mass of the source. If the source is located s units from a referenceposition and a measurement /Xj of the vertical component of gravitational force isobserved at site xj on the surface (Figure 1), then

m

(1)

Settinggj = (fijly)213 and M = m2/3, the equation relating the measured quantity g, atsite Xj to the unknown mass and position of the point source may be written

-s)2+l) = M (2)

Clearly, a single observation gx at site xi is insufficient to determine M and 5.Suppose then that observations gi and g2 are made at sites x\, x2, respectively, withxi <x2. From equation (2) we have

(gi -gi)s2 + (2g2x2 ~ 2gixi)s + (gix2-g2xl) = 0 (3)

Consider first the case of equal observations: g\ =g2. In this case equation (3) is

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382 C. W. Groetsch

uniquely solvable giving s = (x\ + xi)l2 and M (hence, m) is obtained from equation(2) verifying the student's intuition in the case of equal observations at distinct sites.

Take now the case of unequal observations g\ =£ gi at distinct sites X\<X2. In thiscase equation (3) has two solutions

_g\x\ gixi ± Vg^g2(x2 *i)

g\~gi

The solution s+ corresponding to the positive root may be written

s+ = (V5i*i + v£*2)/(V# + Vft)

and hence this root lies in the interval [3:1,3:2]. The other root, S-, is

Vg~2s-=xi+—= —(x2-xi). (5)

Vgi ~ Vgi

From equation (5) we see that 5- <x\ \ig\ >gi, whiles- > xi \igi>g\. In eithercase, we find that there is a unique solution in [a: 1,3:2] and hence there is a uniquesolution of our problem if we know a priori that the source lies between themeasurement sites, again confirming the physical hunch of our students.

Now is the time to ask the students for a method for solving the inverse problem.The discussion above makes it clear that two observations are insufficient. In general,at least three measurements will be required. The students themselves will soon comeup with a recipe for finding the golden nugget. Make observations at two distinctsites. If the measurements are equal, then the source lies midway between theobservation sites, as shown above. If unequal measurements are obtained at distinctsites, saygi <g2 at sites x\ < x2, move from the site at which the larger measurementwas made (e.g. x2) in the direction away from the site at which the smallermeasurement was obtained until reaching a site, say 3:3, at which a measurementg3<g2 is obtained (it may take many trials, but eventually this condition will beobtained). We are now assured that the source lies between 3:1 and 3:3 and the methoddescribed in the previous paragraph will determine the location and mass of thesource.

3. Identifying a distributed densityThe main mathematical point in the investigation of the discrete gravity

prospecting problem of the previous section was non-uniqueness of solutions. In thissection we take up a continuous analogue of the discrete gravity problem. As withmost continuous inverse problems, the principle concern with this problem isinstability of solutions [4]. The continuous problem, which involves identifying adistributed mass density, requires more advanced mathematics than the discreteproblem. I have used it in a linear algebra class for in-service secondary schoolteachers to illustrate instability in linear problems and to display the utility ofMATLAB [5] numerical software.

Consider a continuous distribution of mass along a segment OS$<1 one unitbelow the flat surface. We wish to identify this distribution by taking measurementsof the vertical gravity force along the segment 0 £ 3: < 1 on the surface. If we denotethe mass density at point s by w(s), then that portion, A^(x), of the vertical gravitymeasurement at point 3: on the surface, engendered by the mass element zv(s)As atpoint s on the subsurface is given, as in (1), by

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Geophysically motivated inverse problems 383

Afi(x) = y ((* - s)2 + 1) ~ mw(s)As

The usual limiting process leads us to the following relationship betweenobservations n(x) of vertical gravity force on the surface and the unknownsubterranean density w(s):

y\ ((x-Jo

(6)

Integral equations such as equation (6) have not been welcomed in theundergraduate curriculum. This is understandable as the theory of such equationsis rather advanced, analytical methods of solution are few, and numerical methodsrequire intense computation. However, the advent of powerful user-friendlynumerical software has made it possible to painlessly carry out the calculationsinvolved in numerical solutions. The principal idea behind numerical solutions isdiscretization of equation (6) and solution of the resulting linear algebraic system.Integral equations therefore provide excellent vehicles for illustration of quadraturerules and the use of liner algebra in non-trivial problems. Moreover, the linearsystems that result from discretization of equations of the form of equation (6) havetheir own surprises, as we shall see.

Before proceeding to a numerical method, we point out that the most significantfeature of any solution of an equation of the form of equation (6) is its instability.This instability is a consequence of the Riemann—Lebesgue theorem [6]. For eachx, the function

is integrable, and hence by the Riemann-Lebesgue theorem

r1

((x-s)2+ir3/2sinmds->0Jo

as n—> oo. An additive perturbation of a solution w(s) of equation (6) having the formA sinnj, where A is an arbitrarily large amplitude, has, for large n, a negligible effecton the right-hand side jx(x). To put it another way, arbitrarily small perturbationsin pi can correspond to arbitrarily large perturbations in w. This sort of instabilityis the curse under which practitioners in many fields must work if they are confrontedwith continuous inverse problems.

A reasonable approach to approximating solutions of equation (6) is to discretizethe integral by use of an approximating Riemann sum. For example, we couldpartition the interval [0,1] into TV subintervals of width h= 1//V and sample atmidpoints of the subintervals giving

N

where sj = (2/ — l)/i/2, j = 1, 2 , . . . , N. The values w(s/) of the unknown density canthen be approximated by the numbers Uj which solve the system

*.-* i ) 2 +l)" 3 / 2 «y = fc i = l ,2 N (7); = i

where 6,- = /i(*,-), #i = (2i — 1)^/2, i = 1, 2 , . . . , N. In this way the integral equation (6)is replaced by the approximating system of linear algebraic equations

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384 C. W. Groetsch

Au = b (8)

where A is the NX N symmetric matrix with entries

the vector b represents gravity measurements at the points Xi, and the solution u ofequation (8) is meant to approximate the values of the true density w at the pointss\,..., SN. The matrix A is positive definite and hence problem (8) will have a uniquesolution for each right-hand side. This solution can be obtained in a few key strokeswith the MATLAB command

u = A\b.

Our primary interest in this section is to investigate, through MATLABsimulations, the instability of the inverse gravimetric problem. For this we need asimple test case. We will take the case of a uniform unit mass density w(s) = 1 for0 £ x ̂ 1. Substitution in equation (6) gives the corresponding gravity effects (forsimplicity we take y = 1):

f ((x-sV+U~3l2ds = 1 ~X + - (Q)J \\x s> T l> as \<x— i y + I!1'2 \x2 + II1'2

(the computation of this definite integral is a nice student exercise on integration bytrigonometric substitution).

I have used the test problem (9) as a model problem for the numerical solutionthe gravity prospecting problem in linear algebra and numerical analysis courses. Iprovide the students with a simple MATLAB program (see Appendix) whichdiscretizes the integral operator by use of the mid-point rule and discretizes theright-hand side by sampling at the mid-points. The students are then asked to solvethe discretized problem for various numbers N of subintervals, starting withrelatively small values (at first we take ep = 0 in the program; more on this later). Forexample, the commands

u = A\b; (10)

results in an approximation to the true solution w(s)=l using a five-pointdiscretization. We should not expect miracles with this low-dimensional approxi-mation and indeed if we plot the approximate solution u against the true solutionusing the command

p\ot(.t,u,t,w) (11)

we obtain the result in Figure 5.Students agree that the results are not very satisfactory, but they are not

particularly alarmed as only five discretization points were used and all of their priorexperience convinces them that a finer discretization will give better results. Thestudents are then asked to repeat the simulation using 100, rather than 5,subintervals. A plot of the computed solution appears in Figure 6.

This result in itself is sufficient to perplex most students, but it must be pointedout that in any real situation the data are subject to measurement errors and thereforea fairer test of the solution procedure would replace the true right-hand side withan approximate version containing small random errors. This is accomplished by the

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Geophysically motivated inverse problems 385

Figure 5.

0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-250

parameter ep in the function Geo (see Appendix). Geo adds uniformly distributedrandom errors in the range [ — ep, ep] to each sampled value of the discrete right-handside. For example, invoking Geo with ep = 0-01 (roughly 1 % error) and N= 100 leadsto the plot in Figure 7 which represents the discretely sampled right-hand side ofproblem (9) with uniform random errors in the range [ — 0-01,0-01].

Using these somewhat more realistic data the students are asked to solve for theunknown density as in commands (10) and plot the results as in command (11). Atypical plot is given in Figure 8.

Students find this plot to be disconcerting, to say the least. Most of theirexperience has led them to believe that fine discretizations will lead to acceptableresults, but now they are confronted with errors on the order of 1% in the data

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386 C. W. Groetsch

0-95

0 0-1 0-2 03 04 05 06 0-7 08 09 1

075

-0-5

0-3 0-4 05 0-6 0-7 0 8 0-9

Figure 8.

producing errors on the order of 1018% in the solution! The culprit of course is theinherent instability in problem (6). A fine discretization simply means that theunstable continuous problem (6) is accurately mirrored in the discrete problem (8).MATLAB itself provides a quantitative indicator of this instability in terms of thecondition number of the matrix A. In fact, invoking the MATLAB function cond(A)results in a number on the order of 1018. This means that relative errors incomponents of the right-hand side can be magnified by the solution process by afactor on the order of 1018 [7].

Is all lost? Is it possible to extract any useful information from the noisy data?The key is to mitigate the effect of the small eigenvalues of A (whose reciprocals are

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Geophysically motivated inverse problems 387

105

0-95

0-85

0-75-

0 01 02 03 04 05 06 07 08 09 1

Figure 9.

1002

1-001

0-999 -

0-998 •

0-997 -

0-996 -

0-9940-1 0-2 0-3 0-4 0-5 0-6 07 08 09

very large eigenvalues of A '). In the solution process these eigenvalues are invertedleading to large amplification of certain error components in the noisy data. Sincethe matrix is positive definite the eigenvalues can be moved away from zero by addinga small positive multiple of the identity to A and solving, instead of equation (8),the regularized problem

Au + au = b (12)

where a > 0 is a regularizationparameter. For example, solving problem (12) witha = 0-0001 and using the same noisy data vector b that produced the approximatesolution in Figure 8 gives the regularized solution plotted in Figure 9.

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388 Geophysically motivated inverse problems

Although the exact quantitative results are not immediately impressive, note thatthe approximate solution of Figure 9 has maximum error of about 25%, a very farcry from the roughly 1018% error in Figure 8! Solving the regularized problem (12)with noise-free data (ep = 0) results in the plot of Figure 10, where the maximumerror is on the order of ^% in comparison to the 25 000% error in the unregularizedsolution of Figure 5!

Although the choice of the regularization parameter a is a nontrivial matter (see[4] for a bit more on choice of regularization parameters), these numerical examplesprovide dramatic illustrations of the power of simple mathematical analysis coupledwith modern software to rescue some information from seemingly hopelesssituations brought on by the mathematical instability of continuous inverseproblems.

AcknowledgmentsThis work was supported in part by the National Science Foundation

(USE-9154172) and NATO (CRG-930044).

Appendix

function [A,b,t,w]=Geo(n,ep);% Discretization of first kind integral equation for one-% dimensional geophysical prospecting problem.% True solution: w(s)=l% Right hand side=(l-x)/((x-l)A2+l)A(l/2) + x/t% t=vector of sampling points% ep=amplitude of uniform random error%% Initializations

h=l/n; t=h*([l:n]'-0-5); w=ones(n,l);rand('uniform');

%% Integral is discretized by mid-point rule and collocation at% mid-points.%

A = h*((t*ones(l,n) - ones(n,l)*t').A2 + ones(n,n)).A(-3/2);%% Discretization of right-hand side%

b=(ones(n,l)-t)./sqrt((t-ones(n,l)).A2 + ones(n.l));b=b + t./sqrt(t.A2 + ones(n.l));b=b+ep*(ones(n,l)-2*rand(n,l));

References[1] GAL-EZER, J., and ZWAS, G., 1992, The UMAP J., 13, 93.[2] SCHAUFELE, C, and ZUMOFF, N., 1993, Earth Algebra (New York: Harper-Collins).[3] GROETSCH, C. W., 1993, The College Math. J., 24, 210.[4] GROETSCH, C.W., 1993, Inverse Problems in the Mathematical Science (Wiesbaden, Verlag

Vieweg).[5] MATLAB, 1987, The Mathworks, Inc., 21 Elliot St., South Natick, MA 01760.[6] HARDY, G. H., and ROGOSINSKI, W. W., 1956, Fourier Series, third edition (Cambridge:

Cambridge University Press).[7] JOHNSON, L., and RIESS, R., 1982, Numerical Analysis, second edition (Reading, MA:

Addison-Wesley).[8] BURGER, H. R., 1992, Exploration Geophysics of the Shallow Subsurface (Reading, MA:

Addison-Wesley).

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