Regularized extremal bounds analysis (REBA): An approach to quantifying uncertainty in nonlinear geophysical inverse problems

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1 GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L03304, oi: /2008gl036407, 2009 Regularize extremal bouns analysis (REBA): An approach to quantifying uncertainty in nonlinear geophysical inverse problems Max A. Meju 1,2 Receive 21 October 2008; revise 17 December 2008; accepte 24 December 2008; publishe 11 February [1] Geophysical measurements are ban-limite in nature, contain noise, an bear a nonlinear relationship to the subterranean features being sample. These result in uncertainty in ata interpretation an necessitate a regularization approach. Although moel uncertainty an non-uniqueness can be reuce by combining measurements of funamentally ifferent physical attributes of a subsurface target uner investigation or by using available a priori information about the target, quantifying non-uniqueness remains a ifficult task in nonlinear geophysical inversion. This paper evelops a new theoretical framework for regularize extremal bouns analysis (REBA) of moel uncertainty using the most-squares formalism. The nonlinear most-squares metho allows for combining ata constraints an their associate uncertainties in an objective manner to etermine the moel bouns. By making the assumption that the relevant bouning moels must sample the same unerlying geology, it is propose that structural issimilarity between the extreme moels can be quantifie using the cross-proucts of the graients of their property fiels an shoul serve as an objective measure of interpretational uncertainty in multiimensional inversion. Citation: Meju, M. A. (2009), Regularize extremal bouns analysis (REBA): An approach to quantifying uncertainty in nonlinear geophysical inverse problems, Geophys. Res. Lett., 36, L03304, oi: /2008gl Introuction [2] The overall goal of geophysical inversion is to make quantitative inferences about the subsurface from remotely sense ata. However, inirect inferences of subsurface parameters are subject to uncertainty owing to the typically ban-limite observations, measurement error, geological heterogeneity an theoretical moelling error. Formally, moel uncertainty an non-uniqueness can be reuce by combining ifferent measurements [Gol tsman, 1975; Gallaro an Meju, 2003, 2004, 2007] or by using available a priori information [Jackson, 1979], but there is no theory to quantify the uncertainty in nonlinear geophysical inverse problems, an hence, assessing the impact of uncertainty in large-scale multiimensional inversion remains an outstaning research problem. Many approaches have been propose to quantify this uncertainty. Stochastic methos can be use for this purpose, but their applicability is limite to 1 Department of Environmental Science, Lancaster University, Lancaster, UK. 2 Now at Subsurface Technology Department, Petronas Research Sn Bh, Kajang, Malaysia. Copyright 2009 by the American Geophysical Union /09/2008GL relatively small-scale inverse problems [Snieer, 2004]. An alternative approach is to construct solutions that are extremal in some sense an use their ifferences to characterise moel uncertainty. There are two en-member philosophies in the latter approach. [3] One philosophy stipulates that non-uniqueness is equivalent to invariance of ata constraints with respect to changes in the earth moel an uses Lie group methos to explore the null-space so as to characterise the invariance or symmetry associate with an inverse problem [Vasco, 2007]. It relies on a linearization with respect to a group parameter about an ientity transformation rather than the conventional linearization about a particular subsurface moel, an the ata fit is maintaine while the solution is moifie so as to extremize some aspect of an earth moel. The other philosophy relies on linearization about a specific earth moel an generates extreme moels that satisfy a threshol ata fit etermine by the uncertainties in ata constraints [Meju, 1994; Kalscheuer an Peersen, 2007]. In this particular approach, the emphasis so far has been on the influence of observational errors in nonlinear extremal inversion [e.g., Meju an Hutton, 1992; Meju an Sakkas, 2007], neglecting the other potential sources of uncertainty in subsurface parameter estimates such as moel parameterization an preictive uncertainty of the overly simplifie iscrete moels use to escribe the unerlying physical processes in complex geological meia. It will be better to also take into account our moelling errors an a priori prejuices in the extremal inversion process. [4] Incorporating information about geological noise, theoretical moelling errors, an uncertainty of a priori ata into a non-linear inversion scheme has been attempte before using a Bayesian approach where it is assume that all uncertainties can be escribe by multiimensional Gaussian probability ensities [e.g., Snieer, 2004]. To this author s knowlege, an equivalent formalism for combining these uncertainties has not been evelope for nonlinear eterministic inversion. The objective of this paper is to evelop a theoretical framework for quantifying moel uncertainty in large-scale nonlinear geophysical inversion using a eterministic approach. Given the numerous sources of uncertainty from the ata acquisition stage to moel interpretation, a relevant question is, can one erive a composite measure of uncertainty an objectively quantify moel uncertainty in nonlinear geophysical inversion? Alternatively, one may ask what is the relative impact of the ban-limite observations, measurement errors, geological heterogeneity an nonlinearity of geophysical processes on the inverse problem? While each of these factors separately influences moel resolution, it is hypothesise here that they can be combine in the inversion to characterise moel nonuniqueness. L of5

2 [5] In this short paper, I evelop an objective eterministic framework that can be use to aress some of the issues of uncertainty analysis in large-scale nonlinear geophysical inversion. First, I formulate a regularize extremal bouns analysis (REBA) approach for non-uniqueness appraisal of conventional regularize least-squares inversion moels. Next, a metho is propose for interpretative assessment of moel uncertainty that is base on the physical attributes of the REBA moels. Finally, the implementation an possible extension of the metho are iscusse. 2. Methoology for Regularize Most-Squares Extremal Bouns Analysis [6] The regularize extremal bouns analysis (REBA) problem is state briefly as: Given an optimal least-squares moel (m ls ) erive from a regularize multiimensional inversion of remote geophysical observations, fin, on account of observational an theoretical (incluing iscretization) errors, solution equivalency an non-uniqueness, other moels that satisfy a specifie threshol misfit, q ms an are consistent with the constraints impose by the ata uncertainties an any reliable a priori information, h. [7] The above statement is mathematically equivalent to extremizing the function m T b, subject to the constraint q ¼ ð fðmþþ T W T W ð fðmþþ þ t 2 m T L T Lm þ e 2 ðm m o Þ T ðm m o Þ q ms ; ð1þ assuming that we incorporate reliable a priori information h in the regularize least-squares inversion moel, an m o = m ls. Here, b is a parameter projection vector which allows us to set the search irections, f(m) is the nonlinear forwar moel response vector, are our observe ata, W is a iagonal matrix whose elements are the reciprocals of the observational errors (s 1, s 2,..., s n ), e 2 is a amping factor in conventional parlance, an L an t 2 are the Laplacian operator an the regularization parameter use for generating m ls, respectively. [8] Following Meju [1994], we note that t 2 I an e 2 I (I is the ientity matrix) may be interprete as the inverses of the iagonally-ominant covariance matrices of our moel parameterization (a proxy for the iscretization an regularization errors) an our a priori information, h, respectively. If we efine the inverse of the covariance matrix of our observe ata as C 1 = W T W, an for notational consistency also efine the matrices C 1 m = t 2 I an C 1 h = e 2 I, then our composite weighting matrix (C 1 cmp ) of known or assume uncertainties is of the form Ccmp 1 ¼ C C 1 m C 1 h : ð2þ Thus, in effect, we have incorporate in equation (1) information about observational error (C 1 ), a proxy for iscretization an theoretical errors (C m 1 ), an uncertainty of our a priori constraints about the subsurface (C h 1 ). [9] For the REBA problem, we wish to extremize the objective function n ¼ m T b þ 1=2m ð fðmþþ T C 1 ð fðmþþ þ m T L T C 1 þ ð m m oþ T C 1 h ðm m o Þ q ms o: ð3þ Using Taylor series expansion of f(m) about moel m o,the equivalent linearize function to extremize is ¼ m T o b þ xt b þ 1=2m y T C 1 y xt y y T C 1 Ax þ xt Ax þ mt o LT C 1 o þ 2m T o LT C 1 m Lx þ xt L T C 1 m Lx þ xt C 1 h x q ms ð4þ where y = f(m o ), A is the Jacobian or sensitivity matrix containing the first-orer partial erivatives of the moel response with respect to the moel parameters, an x = m m o are the unknown parameter perturbations. [10] Differentiating the above function (4) with respect to x gives =x ¼ b þ 1=2m 2 y þ 2AT C 1 Ax þ2l T C 1 o þ 2L T C 1 m Lx þ 2C 1 h x : ð5þ Assembling like-terms an setting (5) equal to zero for the extrema, we have that b þ 1=m Ax þ LT C 1 m Lx þ C 1 h x 1=m y LT C 1 o ¼ 0: ð6þ Therefore, m L þ C 1 h x ¼ y LT C 1 o mb ; ð7þ which are the regularize most-squares normal equations from which we obtain the solution for the parameter perturbations x ¼ m L þ 1 C 1 h y LT C 1 o mb : ð8þ Nonlinearity is ealt with using an iterative formula m kþ1 ¼ m k þ m L þ C 1 h y LT C 1 k mb where A an y are evaluate at m k an the iteration is begun at k = 0. [11] Alternatively, we can obtain the irect estimate of m. Recall that m = m o + x. Replacing x with m m o in the regularize normal equations yiels m L þ C 1 h m ¼ y LT C 1 o mb þ m L þ C 1 h mo ¼ y þ AT C 1 Am o þ C 1 h m o mb ð10þ 1 ð9þ 2of5

3 Figure 1. A flowchart of the REBA algorithm. The options for estimating the plus an minus solutions iteratively using equations (8) an (9) or irectly using equation (11) are inicate by the numbers 1 an 2 respectively. from which we obtain the solution m L þ 1 C 1 h m ¼ C 1 h m o mb ð11þ where * =[y + Am o ]. [12] When the quaratic constraint q = q ms is satisfie, we have that n m ¼ ðq ms q ls Þ= b T m L þ o 1b 0:5: C 1 h ð12þ There are thus two (plus an minus) solutions for m as long as q ms is greater than q ls. [13] A flowchart of the REBA algorithm is given in Figure 1. In the iterative solution process, the sign of m is first kept positive until the constraint q = q ms is satisfie whereupon it is reverse an the operation repeate to etermine the other moel boun. For a given search irection, the plus an minus solutions (m p an m m ) may be regare as the upper an lower bouns for the specifie q ms. It may be stresse that there are three search possibilities epening on the projection vector b: (1) setting b k = 1 an all other coefficients equal to nought will yiel the extreme parameter values in the search irection corresponing to m k ; (2) setting all the coefficients of b equal to unity will yiel the plus an minus solution envelopes for m ls, an (3) ranomly varying the vector of coefficients b will approximate the ranom walk approach. Note that a compensating relationship exists between the parameters when m T b is extremize; the first term in square brackets on the right-han-sie of equations (8) an (11) is the regularize inverse an operates on the first two terms in braces to yiel the constraine least-squares solution for all the parameters an on the last term to yiel an aitional solution pertaining only to the extremize parameter. It will thus provie a much greater parameter range than woul be obtaine by simply varying one parameter whilst keeping the others fixe. We expect that poorly constraine parameters will yiel a relatively wie range between the lower an 3of5

4 upper solution bouns. However, the set of moels that satisfy the quaratic constraint q = q ms may contain unrealistic moels, as gauge by the known values of the physical parameters an theoretical consierations. Fortunately, those unrealistic moels that violate our a priori assumptions or theoretical consierations are easy to ientify an may be eliminate from the interpretation process. The question that remains for the reasonable moels is how to measure their egree of non-uniqueness an is tackle next. 3. Methoology for Interpretative Analysis of Moel Uncertainty [14] If non-uniqueness is equivalent to the invariance of ata constraints with respect to changes in the earth moel [Vasco, 2007], we posit that moel variation accompanying changes in ata constraints woul be a proxy for moel certainty or uniqueness. For the REBA solution envelopes, we expect some variations in amplitue with irection in the constructe physical property istributions for the general (three-imensional) case. Consiering the amplitue ata, heterogeneity can be characterise by the ifference between the plus an minus solutions or simply as the eviations of the elements of m p (x, y, z) an m m (x, y, z) from those of the appraise least-squares moel m ls (x,y,z). If we then consier the spatial variations in the moels, an important geological constraint is that if a true subterranean bounary or transitional zone exists, it shoul be sense by the correct moels in the same or opposite irection. Thus the geometrical attributes of the extreme moels, an especially the egree of structural issimilarity, may be use as another quantitative measure of uncertainty in moel appraisal. For geometrical assessments, a criterion to apply is that the cross-proucts of the graients of the m p property fiel an the m m property fiel must be zero at an important bounary or target zone. This is a reasonable geological frame of reference. To measure the structural issimilarity of the resulting extreme moels, we may simply calculate the cross-proucts of the graients of the physical property fiels of the relevant plus an minus solutions. [15] If rm p (x, y, z) enotes the vector fiel of the graient of the moel erive using the plus solution an rm m (x, y, z) enotes the graient of the moel corresponing to the minus solution, we efine the crossgraients function [Gallaro an Meju, 2003, 2004] for the general case as tðx; y; z Þ ¼ rm p ðx; y; zþrm m ðx; y; zþ: ð13þ The vector fiel of the cross-graients function is given by ~tðx; y; ; @m ð14þ an the cross-graients conition for structural similarity between the plus an minus moels requires each of the three components of this function to be zero, i.e., t (x, y, z) = 0. Their eparture from zero at any given location in the moel (m(x, y, z)) may be taken as an interpretative measure of structural uncertainty at that location. Note that the unerlying assumption here is that m p an m m must change in some way from the original m ls in orer to attain the threshol ata fit, q ms. Inee, this is obvious in equations (8) an (11) which show that the other parameters are moifie also whilst that specifie by the projection vector b is being minimise or maximise. Consequently, if the quantity rm ls (x, y, z) rm p (x,y,z)orrm ls (x, y, z) rm m (x, y, z) is zero when the quaratic constraint q = q ms is satisfie, then the solution is eeme to be structurally nonunique since it is geometrically invariant with respect to changes in the ata constraints. 4. Discussions an Conclusion [16] There is no establishe theory for quantifying uncertainty in nonlinear geophysical inversion an this paper attempts to establish a simple framework for assessing moel non-uniqueness when using a eterministic approach. It was shown that the regularize most-squares formalism provies a suitable framework for nonlinear uncertainty analysis. The regularize extremal bouns analysis (REBA) technique presente here is new an the mathematical formulations incorporate information about observational errors, moel regularization errors an the available a priori ata. The flexibility of moel search in any irection stipulate by a parameter projection vector means that our approach can be use to explore the peculiarities of moel uncertainty. By ranomly varying the coefficients of this projection vector, the REBA scheme can also mimic the ranom search methos. It is philosophically ifferent from the newly emerging Lie group metho where the ata fit is maintaine while the solution is moifie (by moving in the null space) so as to minimize or maximize some aspect of the appraise moel [Vasco, 2007]. It is also ifferent from the more common eterministic approach of simply taking the square root of the iagonal elements of the regularise inverse in equations (8) an (11) as the parameter uncertainty estimates for an optimal constraine least-squares moel. [17] Although not emonstrate here, the implementation of the REBA scheme shoul be straightforwar (Figure 1). For instance, assuming an average error level of 10 percent in our fiel an theoretical ata for an optimal least-squares moel (m ls ) with the minimum achievable resiual error (q ls ), we woul seek the range of moel parameters that will yiel a threshol resiual sum of squares error (q ms ) given by q ms =q ls 1.1 which is consistent with the 10 percent ata error level. Interpreting the resulting extreme moels using the propose cross-graients criterion is geologically attractive an physically intuitive. By making the assumption that the relevant solutions from amongst all the possible moels that satisfy the quaratic constraint q = q ms must sample the same unerlying geological structure (a common frame of reference), structural similarity can be quantifie as suggeste above an use as a measure of interpretational uncertainty. The conition that t(x, y, z) = 0 implies mathematically that rm p (x, y, z) is parallel (or anti-parallel) to rm m (x, y, z) at any given position in the sample subsurface. Its geological implication is that if a true bounary or transitional zone exists, it shoul be sense 4of5

5 by the correct moels in the same or opposite irection. Such a criterion has been applie for the ifferent purpose of linking uncorrelate physical property fiels in joint inversion [e.g., Gallaro an Meju, 2003, 2004, 2007; Tryggvason an Line, 2006; Line et al., 2006, 2008]. Note that other measures of structural issimilarity can be efine an use for appraising the REBA moels. For instance, a normalise variant of equation (13) can be use [cf. Gallaro an Meju, 2004, equation (1)] but care must be taken to ensure that the aopte function has no problems of iscontinuity or singularity, for example where rm p (x, y, z) or rm m (x, y, z) vanish. [18] In conclusion, this paper presents a way to objectively quantify uncertainty in nonlinear geophysical inversion. The propose metho can incorporate in the inverse calculations, information about observational errors, a proxy for moelling errors an any available a priori ata or prejuices. However, it shoul be borne in min that fiel observations are by their nature incomplete, inconsistent an insufficient, hence reliable a priori information an/or combination of multi-physics ata are necessary to reuce moel uncertainty. The metho presente here can be use for appraising moels constructe by joint inversion of multi-physics ata sets. [19] Acknowlegments. The work escribe in this paper was complete uring a campus visit to Stanfor University in April 2008 for which the author thanks the Geophysics Department. The constructive comments by an anonymous reviewer helpe improve the clarity of this paper. References Gallaro, L. A., an M. A. Meju (2003), Characterization of heterogeneous near-surface materials by joint 2D inversion of c resistivity an seismic ata, Geophys. Res. Lett., 30(13), 1658, oi: /2003gl Gallaro, L. A., an M. A. Meju (2004), Joint two-imensional DC resistivity an seismic travel time inversion with cross-graients constraints, J. Geophys. Res., 109, B03311, oi: /2003jb Gallaro, L. A., an M. A. Meju (2007), Joint two-imensional crossgraient imaging of magnetotelluric an seismic travel-time ata for structural an lithological classification, Geophys. J. Int., 169, Gol tsman, F. M. (1975), Statistical theory for the interpretation of geophysical fiels (in Russian), Izv. Earth Phys., 1, (translate bym. N. Pillai) Jackson, D. D. (1979), The use of a priori ata to resolve non-uniqueness in linear inversion, Geophys. J. R. Astron. Soc. Lonon, 57, Kalscheuer, T., an L. B. Peersen (2007), A non-linear truncate SVD variance an resolution analysis of two-imensional magnetotelluric moels, Geophys. J. Int., 169, Line, N., A. Binley, A. Tryggvason, L. B. Peersen, an A. Revil (2006), Improve hyrogeophysical characterization using joint inversion of cross-hole electrical resistance an groun-penetrating raar traveltime ata, Water Resour. Res., 42, W12404, oi: /2006wr Line, N., A. Tryggvason, J. E. Peterson, an S. S. Hubbar (2008), Joint inversion of crosshole raar an seismic traveltimes acquire at the South Oyster Bacterial Transport Site, Geophysics, 73, G29 G37, oi: / Meju, M. A (1994), Biase estimation: A simple framework for parameter estimation an uncertainty analysis with prior ata, Geophys. J. Int., 119, Meju, M. A., an V. R. S. Hutton (1992), Iterative most-squares inversion: Application to magnetotelluric ata, Geophys. J. Int., 108, Meju, M. A., an V. Sakkas (2007), Heterogeneous crust an upper mantle across southern Kenya an the relationship to surface eformation as inferre from magnetotelluric imaging, J. Geophys. Res., 112, B04103, oi: /2005jb Snieer, R. (2004), Uncertainty estimation in inverse problems: An uncertain proposition, Eos Trans. AGU, 85(47), Fall Meet. Suppl., Abstract NG33B-01. Tryggvason, A., an N. Line (2006), Local earthquake (LE) tomography with joint inversion for P- an S-wave velocities using structural constraints, Geophys. Res. Lett., 33, L07303, oi: / 2005GL Vasco, D. W. (2007), Invariance, groups, an non-uniqueness: The iscrete case, Geophys. J. Int., 168, M. A. Meju, Subsurface Technology Department, Petronas Research Sn Bh, Kajang, Malaysia. (maxwell_meju@petronas.com.my) 5of5