Optimal nonlinear Bayesian experimental design: an application to amplitude versus offset experiments

Transcription

1 Geophys. J. Int. (23) 155, Optial nonlinear Bayesian experiental design: an application to aplitude versus offset experients Jojanneke van den Berg, 1, Andrew Curtis 2,3 and Jeannot Trapert 1 1 Faculty of Earth Sciences, Utrecht University, PO Box 821, 358 TA, Utrecht, the Netherlands. E-ail: J.vandenberg@phys.uu.nl 2 Schluberger Cabridge Research, High Cross, Madingley Road, CB3 EL Cabridge, United Kingdo 3 Departent of Geology and Geophysics, Edinburgh University, Grant Institute, West Mains Road, Edinburgh EH9 3JW, United Kingdo Accepted 23 May 5. Received 23 May 6; in original for 22 October 22 1 INTRODUCTION Finding an optial geoetry, or design, of a practical experient often eans finding the design which axiizes the expected postexperiental inforation of particular odel paraeters of interest. This is equivalent to iniizing the expected post-experiental uncertainties in those odel paraeters. Thus, experiental design requires an understanding of the relationship between data and postexperiental odel paraeter uncertainties (Box & Lucas 1959; Atkinson & Donev 1992; Curtis 1999a,b; Curtis & Spencer 1999; Curtis & Maurer 2). Consider a linearized proble in which the forward proble of estiating data d given a odel paraeter vector, and the associated inverse proble solution are, respectively, d = G (1) = ( G T G ) 1 G T d = Ld, (2) where G is a atrix of derivatives of d with respect to calculated at a reference odel, atrix L is defined in eq. (2), and the atrix inversion in eq. (2) represents the classical leastsquares solution and ust be replaced by = G T (G G T ) 1 Now at: Institute for Marine and Atospheric Research, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, the Netherlands. SUMMARY When designing an experient, the ai is usually to find the design which iniizes expected post-experiental uncertainties on the odel paraeters. Classical ethods for experiental design are shown to fail in nonlinear probles because they incorporate linearized design criteria. A ore fundaental criterion is introduced which, in principle, can be used to design any nonlinear proble. The criterion is entropy-based and depends on the calculation of arginal probability distributions. In turn, this requires the nuerical calculation of integrals for which we use Monte Carlo sapling. The choice of discretization in the paraeter/data space strongly influences the nuber of saples required. Thus, the only practical liitation for this technique appears to be coputational power. A synthetic experient with an oscillatory, highly nonlinear paraeter data relationship and a siple seisic aplitude versus offset (AVO) experient are used to deonstrate the ethod. Interestingly, in our AVO exaple, although overly coarse discretizations lead to incorrect evaluation of the entropy, the optial design reains unchanged. Key words: Bayesian, design, experiental, inversion, nonlinear, survey. d if the proble is under-deterined. In the presence of null spaces, the inverse operator L needs to be replaced by a ore general expression including regularization (Tarantola 1987). To be consistent with geophysical literature we refer to as the odel and call G the forward function (which is fixed for any particular experiental design). Note that this differs fro terinology in statistical experiental design literature (thus in any references cited in this paper) where the odel usually includes both G and. Uncertainties in the data are projected into the odel paraeter space as LC d L T, where C d is the covariance atrix, so any ethod for perforing experiental design ust alter either G or the uncertainties in d. For a linear proble (where G is constant with respect to, hence we can write siply G), ost ethods for experiental design are based on optiizing the eigenvalue spectru of (G T G). That is, the spectru should have as large eigenvalues and be as flat as possible (Curtis 1999a,b). This is illustrated in Fig. 1, which shows the geoetry for a siple 1-D experient with one source and one receiver. The ai of the experient is to choose the best offset x for the estiation of the single slowness of the half-space below the surface fro the traveltie d of a direct wave between source and receiver. For this exaple a slowness of 1/15 s 1 was used as the true slowness of the ediu. Fig. 2 shows odel-data relationships between and d for 5 and 1 offsets. The plots show the projection of an uncertainty in a easured datu d of ±.1 seconds into the odel paraeter C 23 RAS 411

2 412 J. van den Berg, A. Curtis and J. Trapert offset s = 1/15 s 1 Figure 1. Geoetry of a siple exaple of an experiental design. The theory is given by d = x, where d is the tie [s], is the slowness [s/] and x is the offset [], or the distance between the source and the receiver. The experiental ai is to find the optial offset for retrieving the true slowness of 1/15 s 1. space. These figures show that the function relating odel paraeters and data d is steeper for the larger offset. For a constant data uncertainty this results in a saller uncertainty region around the true odel paraeter value. So, for this siple experient, the larger offset is recoended for ore accurate odel paraeter estiates, and generally, as this proble is linear, the longer the path through the ediu the ore accurate the results are likely to be (Johnson & Leone 1977; Squires 1985; Atkinson & Donev 1992). The exaple above illustrates that the ai of linear experiental design can usually be thought of as increasing gradients G such that post-experiental uncertainties are iniized. Therefore, G ust be estiated and axiized appropriately prior to conducting the experient. In the above 1-D exaple, the single eigenvalue of G T G is the gradient squared. Hence, axiizing the gradient in Fig. 2 is equivalent to axiizing the eigenvalue of G T G. In a linear proble G is constant over all reference odel paraeters for any particular experiental design ξ. Hence, it does not atter at which odel paraeter values the expected postexperiental uncertainty is estiated, since the sae estiates for post-experiental uncertainty will be obtained whatever is chosen. This is not true in nonlinear or even pseudo-linear probles. In such situations G varies as a function of and the true odel paraeter values are unknown. This leads to errors in the estiated post-experiental uncertainties if a single erroneous reference odel is used (Curtis & Spencer 1999). This proble is usually dealt with using classical nonlinear estiates for the quality of an expected design ξ (Box & Lucas 1959; Johnson & Leone 1977; Ford et al. 1989; Atkinson & Donev 1992; Chaloner & Verdinelli 1995; Curtis & Spencer 1999): (ξ) = φ(ξ, )ρ() d. (3) M In this equation φ(ξ, ) is called a quality easure and usually consists of soe easure of eleents or eigenvalues of the for- Projection of uncertainties around d for a linear G.12.1 ward operator G estiated at odel paraeter value, where this easure reflects the expected quality of a specific design ξ (the ost coonly used easure is φ(ξ, ) = det (G T G ) for design ξ). That is, φ(ξ,)reflects the expected post-experiental uncertainty of odel paraeter estiates if contains the true odel paraeter values and the odel-data relationship is approxiately linear around (within the data uncertainties). So, instead of using one reference odel paraeter value to estiate the post-experiental uncertainties, in eq. (3) a distribution of reference odels ρ() is used where ρ() ebodies the prior knowledge about the likelihood of possible odel paraeter values being the true values. In this way the quality easure (ξ) is an average easure of expected inforation over the entire feasible portion of the odel paraeter space. There is a proble with this approach arising fro the definition of φ(ξ, ), which is generally a gradient-based easure. Curtis & Spencer (1999) showed that in the case of a truly nonlinear situation, an error is coitted whenever odels within disconnected regions in the odel paraeter space ight fit the easured data to within their given uncertainties. We illustrate this with the following exaple, extended fro Curtis & Spencer (1999). Fig. 3 shows two 1-D sawtooth functions with different periods of oscillation. Let G be the derivative atrix (in this case a single value) [df/d ], where f is the sawtooth function and we ignore points where G is not defined, let φ(ξ, ) be the deterinant of G T G, the classical experiental design easure (Box & Lucas 1959; Ford et al. 1989; Atkinson & Donev 1992; Chaloner & Verdinelli 1995). In linear probles, this deterinant is constant with respect to and represents a easure of the extent to which easureent uncertainties propagate into expected postexperiental odel paraeter uncertainties, siilar to the slowness exaple above. Let us now exaine the use of this easure in the nonlinear sawtooth exaple. Data uncertainties of plus or inus.2 around a datu d = 2 are projected into the odel paraeter space using the linearized local gradient ethod around a single odel paraeter value in each plot in Fig. 3. This is visualized by the error bar projections flanked by solid lines in the figures. As the derivative of the right function is larger than the derivative of the left function, the corresponding projected uncertainty is saller in the right plot. In each plot individually, identical values for φ(ξ, ) are found for any odel paraeter value, since the agnitude of G T G is constant at all points on the sawtooth functions (ignoring the extrea). However, the quality easure (ξ) calculated using eq. (3) gives (ξ) = 1 for the left figure and (ξ) = 4 for the right figure. This is because, for fixed data uncertainties, the linearized odel Projection of uncertainties around d for a linear G.12.1 d.8.6 d.8.6 d.4.2 d x x 1 3 Figure 2. Model-data relationships for the geoetry shown in Fig. 1. In the left figure an offset of x = 5 has been used, for the right figure x = 1. The figures show the change in gradient G as a function of the offset. The dashed line and the dash-dotted lines represent the projection of the observed datu d with an observational uncertainty of ±.1 into the odel paraeter space. C 23 RAS, GJI, 155,

3 Optial nonlinear experiental design 413 Figure 3. Sawtooth functions with 1 period (left) and 2 periods (right) between = and = 1. In both figures a datu d = 2 ±.2 is projected into the odel paraeter space using the linearized local gradient ethod, shown by the error bar projections between the solid lines. The union of all of the grey error bar projections onto the odel paraeter space represents the true odel paraeter uncertainty in each case. paraeter uncertainties bounded by the solid lines are saller in the right figure than in the left figure. So, according to the classical estiate for design quality, eq. (3), the experiental design producing the right figure would be a better one than the design producing the left figure. However, the uncertainties described above are local, linearized approxiations and account only for one of several possible regions in odel paraeter space that fit the easured data. In fact, the true uncertainty in odel paraeter space is given by the union of all of those possible regions, represented by the set of all vertical grey regions in Fig. 3. As a result, the true post-inversion uncertainty for both figures is exactly equal (the su of the uncertainties in the odel paraeter space in each case is.8). So in contrast to the result fro classical design theory above, the design producing the left figure is probably a better design: the solution to the inverse proble of estiating odel paraeters given any easured data d is easier to calculate and represent, since it is less fragented than in the right figure (Curtis & Spencer 1999). When designing a nonlinear experient (ξ) is usually axiized. In the above case this would ean that the final design would produce neither of the two functions in Fig. 3, but instead would produce a sawtooth with as any extrea as possible. Siilarly to above, this would fragent the region of odel paraeter space fitting any observed data, and in reality would provide no reduction at all in post-experiental odel paraeter uncertainties. Maxiizing (ξ) according to classical easures for the quality of a design would therefore result in a ore difficult inverse proble to solve with no expected gain in inforation. Hence, classical nonlinear (linearization-based) experiental design easures are not robust in nonlinear situations. 2 THEORY AND METHOD To be able to construct a easure for experiental design that will work for any nonlinear experient, a fraework without linearization is now introduced. We use a Bayesian approach for odel paraeter inference in which probability density functions (p.d.f.s) represent a given state of inforation. According to Tarantola & Valette (1982), the solution to an inverse proble is given by the posterior, or post-experiental p.d.f., ρ(d, )θ(d, ) σ (d, ) =, (4) µ(d, ) where ρ(d, ) represents the prior knowledge on data d and odel paraeters, θ(d, ) represents the inforation about the physics relating data and odel paraeters, and µ(d, ) is called null inforation and represents an objective reference state of iniu inforation (Tarantola & Valette 1982; Tarantola 1987). We will adopt the following convention (which differs fro that of Tarantola & Valette 1982). We include within θ(d, ) the entire (uncertain) relationship between the actual data easureents recorded and the odel paraeters. Thus, θ(d, ) includes both (i) the relationship between and idealized, noise-free data, and (ii) the relationship between these idealized data and the data values actually recorded in the experient. Then, ρ(d) includes only inforation about the actual data values recorded (and not on their assued uncertainties). Thus, θ(d, ) represents the physical relationship between data and odel paraeters including all uncertainties over which we have soe influence through the experiental design, and ρ(d, ) contains only a priori inforation over which we have no control, other then through θ(d, ). In contrast, Tarantola & Valette (1982), include relationship (i) within θ(d, ) and relationship (ii) within ρ(d). In designing an experient we ai to axiize the inforation about odel paraeters that are expected to be contained within σ (d, ). Therefore, it is necessary to be able to quantify the inforation content of a p.d.f. The entropy of any rando vector X ay be defined in relation to Shannon s easure for inforation (Shannon 1948); see also Tarantola & Valette (1982), Shewry & Wynn (1987) and Sebastiani & Wynn (2), Ent(X) = I ( f (x)) + c = f (x)log(f (x)) dx, (5) X where f (x) is the p.d.f. of X, and I is the inforation content of a p.d.f. as defined by Shannon (1948). The easure of inforation I is equal to inus the entropy, except for a constant c assuing a unifor null distribution. When designing an experient a data set fro that experient is not available, so we set ρ(d) = µ(d), assuing that the prior distribution can be decoposed as ρ(d, ) = ρ(d)ρ() and that the null distribution can be decoposed siilarly. The post acquisition inforation on odel paraeters is described by the arginal posterior distribution, σ () = σ (d, ) dd. (6) D C 23 RAS, GJI, 155,

4 414 J. van den Berg, A. Curtis and J. Trapert d d σ(d) σ(d) ρ() ρ() Figure 4. Uncertainty distributions in the data space corresponding to a fixed unifor uncertainty distribution ρ() in the odel paraeter space for two linear forward functions with different gradients. The left figure has a sall data space uncertainty and a low value of Ent(d ξ), the right figure has a large data space uncertainty and a high value of Ent(d ξ). One ight expect to be able to design an experient such that inforation expected to be in σ () is axiized. However, θ(d, ) can often be decoposed as θ(d, ) = θ(d ) µ(), i.e. θ(d, ) incorporates no additional inforation on. Therefore, using ρ(d) = µ(d), we often obtain fro eq. (4) that σ (d, ) = ρ() θ(d ) and hence σ () = ρ(). This does not vary with the experiental design ξ and hence cannot be used to deterine ξ. Instead, experients can be designed by axiizing the inforation expected to be contained in the conditional posterior p.d.f., σ (d, ) σ ( d) = (7) σ (d) where σ ( d) represents the probability of being the true value for the odel paraeter given any data easureent d. In eq. (7), σ (d) is the arginal posterior distribution on the data d: σ (d) = σ (d, ) d. (8) M Prior to conducting an experient, σ (d) ebodies all inforation about what data are likely to be recorded during the experient. In those frequent cases when σ (d, ) = ρ() θ(d ), we see that σ (d) siply contains prior inforation on the odel paraeters projected into the data space through the physical relationship θ(d ). A quality easure for nonlinear (nl) experiental design can then be defined as, nl (ξ) = Ent( d, ξ)σ (d) dd, (9) D where Ent is the entropy function and Ent( d, ξ) represents the aount of inforation contained in the conditional p.d.f. σ ( d) about the odel paraeters given a particular data easureent d recorded using experiental design ξ. This easure of inforation is weighted by the likelihood that data easureent d will be obtained when perforing the experient, σ (d). Integration over all possible data easureents d results in nl (ξ). nl (ξ) above and (ξ) fro classical nonlinear ethods (equation 3) have siilar for. The ost iportant difference between nl (ξ) and (ξ) is that the forer requires no linearization of the odel-data relationship. The concept of axiizing a gradient has not been used. According to Shewry & Wynn (1987), nl (ξ) + Ent(d ξ) = Ent(d, ξ) = b, (1) where b is a constant, if Ent(d, ξ) is design-independent. They deonstrate that this is the case for any geophysical probles. For exaple, the data-odel paraeter relationship θ(d, ) used for ost geophysical probles is, d = f () + ɛ (11) where ɛ is a vector of independent, rando errors, which do not depend on either the odel paraeter space or the design. It can be deonstrated that eq. (1) holds for relationships of this type. Therefore, instead of axiizing nl (ξ), the optial design can also be found by axiizing Ent(d ξ). For this easure only inforation about σ (d) is required, hence the calculation is siplified. Eq. (1) can be explained intuitively using Fig. 4. For linear exaples we showed earlier that the optial design can be found by axiizing the gradient of the forward relationship between data and odel paraeters. Consider the case where σ (d, ) = ρ() θ(d ) and θ(d ) = D( f () d) where D is the Dirac delta function and f () is a noise-free odel-data relationship. Suppose we have two designs producing different 1-D, linear forward relationships f (), and that we have a fixed unifor prior odel paraeter distribution ρ(). Distribution σ (d, ) projects this distribution through each forward relationship. We find that the corresponding posterior easureent uncertainties σ (d) are unifor and different for the two designs. For the left figure (sall gradient of f (), least inforative design), the corresponding easureent uncertainty is sall, hence we obtain a low value for Ent(d ξ). For the right figure (high gradient of f (), ost inforative design), the corresponding easureent uncertainty is large, hence we obtain a high value for Ent(d ξ). Therefore, the optial design is the one which gives the highest value for Ent(d ξ). The theory above generalizes this for nonlinear and uncertain forward relationships in θ(d, ). For ost nonlinear probles the p.d.f.s required to calculate nl (ξ) are not known analytically. Hence, for a generally applicable ethod all integrations ust be defined nuerically. After testing several ethods we opted to use Monte Carlo integrations, (Lepage 1978), g(x) dx 1 g(x) N s(x), (12) X where N is the nuber of saples taken of a function g(x), and s(x) is the sapling distribution-the probability that a saple is drawn at position x. In our ethod, saples are drawn only fro the region where σ (d, ) is non-zero. The saples are generated using ρ() in the odel paraeter space and are distributed uniforly around the forward function f () in the data space, whichever the shape of the distribution of expected easureent uncertainties. In the 1-D exaples used in this paper, the true data uncertainty region around C 23 RAS, GJI, 155,

5 Data space Paraeter space Figure 5. Paraeter-data space geoetry; this figure shows the discretization of the data space (horizontal lines). Saples are drawn only in the grey region where σ (d, ) is non-zero. For each saple, σ (d, ) is calculated using eq. (4). f () for any is Gaussian, but saples are drawn uniforly fro a region f () ± 3δ, where f () represents the forward function relating odel paraeters and data in eq. (11), see Fig. 5. δ is the standard deviation of the Gaussian uncertainty. To calculate σ (d), the data space is discretized into regular intervals (Fig. 5). Applying eq. (12) to approxiate eq. (8), the arginal σ (d) is approxiated by, σ (d) 1 N σ (d, ), (13) s() where N now is the nuber of saples inside one discretization interval of the data space and s() is the distribution as a function of of only those saples. Because s() is not always known analytically within each data interval (as f () ay not be analytic and ay be nonlinear), a nuerical approxiation to s() is ade by binning all the locations of the saples in the odel paraeter space for each data interval in data space, and noralizing the histogra to have unit volue. The obtained histogra is used as an approxiation to s() where for each individual saple a linear interpolation ethod gives the final value of s(). The procedure is illustrated for one diension in Fig. 6. Finally, using the result for σ (d) fro eq. (13), the coplete quality easure for experiental design is given by Ent(d ξ) 1 M σ (d) s(d), (14) where s(d) is a unifor distribution in the data space, as σ (d) has been approxiated using a regular discretization, Fig. 5. M is the total nuber of intervals in the data space and hence the total nuber of values of σ (d) available for the calculation of the entropy. M is directly related to the size dx of the discretization interval in the data space. This eans that the total nuber of saples T will approxiate NM. Optial nonlinear experiental design THE ENTROPY OF A SAWTOOTH FUNCTION In the introduction, an exaple with sawtooth functions was used to show that classical nonlinear design ethods fail in ulti valued probles. The entropy ethod should be capable of dealing with any proble, including those that are strongly nonlinear. This is deonstrated by calculating Ent(d ξ) for sawtooth forward functions with periods 1, 2, 5 and 1 siilar to those in Fig. 3. All sawtooth functions have an aplitude of 2.5 and for each odel paraeter value, the distribution θ(d ) is given by a Gaussian shaped uncertainty in the data space with standard deviation.1 around f () where f () is the sawtooth forward function and the Gaussian is truncated at three standard deviations fro the ean. The odel paraeter space runs fro to 1 and ρ() is unifor and equal to 1/1. The analytical solutions for the entropy Ent(d ξ) for each of these sawtooth functions are identical, and are equal to This shows that the entropy ethod correctly evaluates the designs. It does not necessarily favour designs with steeper gradients in contrast to the classical gradient based ethodology which does. In practice, entropies are calculated nuerically. There are three sources of errors in the nuerical calculations, assuing perfectly known physics: (1) the discretization interval size in the data space dx, Fig. 5, (2) the discretization interval size in the odel paraeter space d, Fig. 6, (3) the total nuber of saples T. In the following sections, these effects are investigated individually. 3.1 Discretization of the data space As a sawtooth function is essentially a sequence of linear functions, σ (d ) can be calculated analytically and is equal to σ (d ) =.1( er f (5 2( d )) + er f (5 2(2.5 + d ))) (15) where erf is the errorfunction. σ (d) is evaluated at different values corresponding to idpoints of the discretization intervals, renoralized, and the corresponding entropy Ent(d ξ) is calculated nuerically fro those analytical values as in eq. (14). Thus, the dependencies on the nuber of saples N and the discretization interval size of the odel paraeter space d are reoved fro the proble and it can be seen easily which discretization interval size in the data space dx is needed for a good approxiation of the entropy. In Fig. 7 (left) the entropy is plotted as a function of the discretization interval in the data space dx. Fig. 7 (right) shows the sensitivity of the entropy to uncertainties in the value of the distribution σ (d ) (since σ (d ) is approxiated nuerically). Clearly, even large uncertainties do not have a significant influence on the calculation of the entropy. Also, the effects are not particularly sensitive to the size of the discretization interval below dx =.25. For discretizations larger than dx =.3 it is not expected that any nuber of saples T is sufficient, since even with an analytically known σ (d ) the entropy can not be evaluated correctly. 3.2 Discretization of the odel paraeter space The influence of the nuber of saples N and the discretization interval d was checked. Assuing that the sapling distribution C 23 RAS, GJI, 155,

6 416 J. van den Berg, A. Curtis and J. Trapert s() Approxiation of nuerical s() s() s() s() Figure 6. The sequential steps in estiating the sapling distribution s(). First the saples are drawn fro the analytical sapling distribution (in this figure a boxcar function (top left)). Then a histogra is ade in which the locations of the saples in the odel paraeter space are binned (top right). Then this histogra is noralized and with a linear interpolation (botto left), the value of the nuerical s() for each individual saple is deduced (botto right). s() will be close to a Gaussian or a boxcar depending on whether the prior distributions are Gaussian or unifor, the tests were perfored for both a boxcar and a Gaussian distribution s(), where the Gaussian was truncated at three standard deviations fro the ean. Saples were drawn fro these analytical functions. Then, the nuerical approxiation for s() was estiated as illustrated in Fig. 6. Fig. 8 shows the average percentage difference between analytical (ana) and nuerically estiated (nu) distributions for s(), defined by ɛ = nu ana ana 1, (16) over the entire box or truncated Gaussian as a function of the discretization interval d and the nuber of saples N. Instead of the actual size of the discretization interval, the quantity ( ax in )/d is used. So, if ( ax in )/d equals 5, there are five Entropy as a function of discretization interval size dx 1.8 Ent(d ξ) Entropy as a function of discretization interval size dx 1.69 Ent(d ξ) intervals within the boundaries of the box or Gaussian. For highly nonlinear probles note that the saples ay occupy ultiple disconnected regions in the odel paraeter space. ( ax in ) should then be replaced by local values ( ax local in local ). When calculating the entropy for a particular design, the sapling distribution s() is estiated for each discretized data-interval. Hence, the nuber of saples N considered in Fig. 8 is not the sae as the total nuber of saples T required to calculate the entropy. It is the nuber of saples N required for the calculation of σ (d ) using eq. (13) for each specific value of d. Fig. 8 shows two iportant results. First, both figures look very uch alike. However, in the case of a truncated Gaussian the errors are larger. This is probably due to the way the ean percentages were calculated. Eq. (16) shows that sall values for the analytical s() (low probability regions in odel paraeter space) can cause larger values for ɛ due to sporadic errors in the nuerical histogra itself. Since in the case of a unifor distribution all analytical values η=% η=15% η=3% η=5% Interval size dx Interval size dx Figure 7. Left: convergence towards the analytical value for Ent(d ξ) as a function of the data space discretization interval size dx. The analytical value of the entropy is Saples fro σ (d) are taken at the center of each discretization interval and these saples are renoralized to obtain the nuerical approxiation for σ (d). Right: convergence towards the analytical Ent(d ξ) as a function of dx in the case where a rando perturbation is added to the analytical value of σ (d). η represents the axiu range of uncertainty and the actual uncertainty is a rando nuber between and η. C 23 RAS, GJI, 155,

7 % difference between nuerical and analytical s() for a box 6 Optial nonlinear experiental design ( ax in ) d Nuber of saples N % difference between nuerical and analytical s() for a Gaussian ( ax in ) d Nuber of saples N Figure 8. The percentage difference between analytical and nuerical s() as a function of discretization and nuber of saples N, when the analytical s() is a boxcar function (top figure) or a truncated Gaussian (botto figure). The saples are drawn according to the analytical s(), then counted into a histogra. This histogra is noralized and assued to be an approxiation for s() as shown in Fig. 6. Then, the nuerical value for s() for each individual saple is deduced by linearly interpolating the histogra. The percentage difference shown here is the average percentage difference over the entire box or truncated Gaussian for one single run of the algorith. are equal, this tendency to ephasize areas with low probability has no effect. Second, Fig. 8 shows that it is ore iportant to have sufficient saples inside an interval than to have a large nuber of intervals. The errors due to undersapling can becoe very large, while errors due to a coarse discretization are relatively liited even if only 1 or 2 intervals are used. The accurate estiation of s() is iportant for the calculation of the entropy, but since this is calculated as an integration, it is expected that up to a certain liit errors in s() ay be cancelled if the errors are rando. 3.3 The nuber of saples T We now fix a discretization and estiate the required total nuber of saples T. The discretization in the data space is set to dx =.1; Fig. 7 shows that deviations fro the analytical value are then negligible. Taking into account the results in Fig. 8, the discretization of the odel space is fixed to ( ax in )/d = 5, for each discretization interval in the data space. Fro hereafter a sawtooth function with a period of 1 is used. Fig. 9 (top) shows the percentage difference between the analytical and the nuerical values for entropy as a function of the nuber of saples T. The solid line is the average percentage difference over 5 runs, the dashed lines are the iniu and axiu percentage differences over those 5 runs. Soe artefacts are clearly visible. For very low nubers of saples T the entropy converges rapidly towards the analytical value. But after approxiately 5 saples the curve starts to diverge, and beyond roughly 3 saples the entropy converges properly towards the analytical value. Also, the error around the average percentage difference increases after 3 saples before decreasing again after approxiately 4 saples. Since the data C 23 RAS, GJI, 155,

8 418 J. van den Berg, A. Curtis and J. Trapert % difference between analytical and nuerical Ent(d ξ) as a function of T Nuber of saples T % difference between analytical and nuerical Ent(d ξ) as a function of T Nuber of saples T Figure 9. Percentage difference between the analytical and nuerical solution for Ent(d ξ) for the sawtooth exaple with period 1, as a function of the nuber of saples T. A Gaussian uncertainty with standard deviation.1 was assued as data uncertainty around the sawtooth forward function f (). The solid line is the average percentage difference over 5 runs, the dashed lines are the iniu and axiu percentage differences over 5 runs. The analytical solution is The entropies were calculated with dx =.1 in the data space and d =.2 in the odel paraeter space (top), and dx =.5 and d =.5 (botto), equivalent to ( ax in )/ d = 1. space discretization interval dx =.1 and the data are running fro approxiately 2.9 to 2.9 (which eans roughly 55 6 discrete intervals), it is likely that the forer artefacts are due to undersapling. Using 6 saples T, for instance, eans that for the calculation of each value for σ (d ) only 1 or 2 saples N are available, which is clearly too few (Fig. 8). The local iniu lies roughly at the point where each interval on average has 1 saple N. The local axiu is roughly at the point where each interval has on average 4 5 saples N. The test was therefore repeated with dx =.5 and ( ax in )/d = 1. Figs 7 (left plot) and 8 show that in general this will degrade accuracy, but Fig. 7 (right plot) shows that large uncertainties in the histogras only have a inor effect on the calculation of the integral. The advantage of ore coarse discretizations is that fewer saples are required. Fig. 9 (botto) shows the results for a sawtooth with period 1. The figure shows that with this discretization, only about 1 saples T are required to get within 5 per cent of the analytical solution. Fig. 1 shows that with certain discretizations only 2 saples are required for the sawtooth to approxiate the entropy to within 5 per cent of the analytical value far fewer than the nuber of saples suggested by Fig. 9. This shows the iportance of choosing a sensible discretization. 3.4 Results We calculated the entropy of σ (d ) with dx =.5 in the data space, d =.5 in the odel paraeter space and 5 saples for 4 sawtooth functions with periods 1,2,5 and 1. Since these sawtooth functions have different gradients, this eans that ( ax in )/d varied for each sawtooth. However, the discretization d and the nuber of saples T should be sufficient to estiate the value of Ent(d ξ) correctly for each sawtooth function. Fig. 11 shows that indeed all sawtooth functions have the sae entropy and are very close to the analytical value of We can thus use the entropy criterion in a fully nuerical ipleentation to obtain correct results in situations where the standard Bayesian design theory fails. The ost iportant factor in this calculation is the choice of discretization, since this strongly influences the nuber of saples, and hence the calculation tie required. 4 AN AVO EXAMPLE 4.1 Theory and geoetry We present a detailed exaple using aplitude versus offset (AVO) data. Using the aplitudes of waves generated by a surface source reflected at a specific depth d and recorded at the surface again (Fig. 12), it is possible to estiate the velocity α2 of the layer below the reflector, for given assuptions on the other odel paraeters. The design proble is to choose the optial source-receiver distance which is expected to give the ost accurate post-inversion estiate for α2. Aki & Richards (198) approxiate the reflection coefficient for P waves at a single interface by 1 α R P = 2 cos 2 i α 4β2 p 2 β β (1 4β2 p 2 ) ρ ρ, (17) where α is the average P-wave velocity and equal to (α1 + α2)/2, β is the average S-wave velocity and equal to (β1 + β2)/2, ρ is the average density and equal to (ρ1 + ρ2)/2, i equals (i 1 + i 2 )/2, α eans (α2 α1), and β and ρ are defined siilarly. p is the slowness given by Snell s law: p = sin i 1 α1 = sin i 2 α2. (18) Rewriting eq. (17) yields (Yilaz 21), [ ] ] 1 α R P = 2 (1 + tan2 i) [4 α β2 β α 2 sin2 i β [ )] 1 + (1 4 β2 ρ 2 α 2 sin2 i ρ. (19) If we assue β = cα with soe constant c (c = 1/ 3 for a Poisson ediu) and ρ =, this equation siplifies to ( 1 [ R P = 1 + tan 2 i ] α 4c 2 sin i) 2 2 α. (2) This is a nonlinear equation as a function of one odel paraeter α2, given α1, c and i. Instead of using angle i, one usually works C 23 RAS, GJI, 155,

9 Nuber of saples T needed for 5% accuracy Optial nonlinear experiental design d.3.3 Figure 1. Nuber of saples T required to reduce the axiu percentage difference between the nuerical and analytical value for the entropy over 5 runs to within 5 per cent of the analytical value for the entropy (1.645). This for a sawtooth with period 1 with a Gaussian uncertainty with standard deviation.1 in the data space. For discretizations dx larger than.3 it is not expected that any nuber of saples is sufficient, Fig. 7. The corresponding axiu d is then also.3 (( ax in )/d = 1). Ent(d ξ) for sawtooths as a function of period with dx=.5 and 5 saples Period of the sawtooth Figure 11. Ent(d ξ) with dx =.5 in the data space, d =.5 in the odel paraeter space and 5 saples T for 4 sawtooth functions with periods 1, 2, 5 and 1, with an aplitude of 2.5 and a Gaussian uncertainty with standard deviation.1. Each dot represents one run and the solid line connects the averages of 5 such points. with offsets x, Fig. 12. Assuing a horizontal interface and a known depth d, the offset translates into i as ( ) x i 1 = arctan (21) 2d ( ) α2 i 2 = arcsin α1 sin i 1. (22) The aplitude data are given by A 1 = A R P, where A is the aplitude at the source and A 1 is the easured aplitude at the receiver and where we assue that the data have been adjusted to reove the effect of geoetrical spreading, so that A is 1, and A 1 = R P. The aplitude-data can be calculated given specific values for α1, α2, the constant c and the depth of the reflector d..2 dx source α1,β1,ρ1 α2,β2,ρ2.1 x i i1 i2 receiver Figure 12. Geoetry of a single interface with a P wave source and a P wave reflection and transission. The distance or offset between source ( ) and receiver ( ) is given by x, the depth of the reflector is d, the incident angle is i 1 and assuing a horizontal reflector this is equal to the angle i 1, the transission angle is given by i 2. The characteristics of the ediu are the velocities of the P (α 1,2 ) and S (β 1,2 ) wave velocities and the densities (ρ 1,2 ). 4.2 Results We investigated two experiental design probles, specified by α1 = 275 s 1 (23) α2 = [32, 33] s 1 (first), α2 = [3, 45] s 1 (second) (24) c = 1 3 (25) d = 5, (26) where [a, b] s 1 eans that the prior inforation on α2 isgiven by a unifor distribution between α2 = a s 1 and α2 = b s 1, d C 23 RAS, GJI, 155,

10 42 J. van den Berg, A. Curtis and J. Trapert.5 1 Ent(d ξ) as a function of offset dx=.2,t=5 dx=.1,t=5 dx=.1,t=1 dx=.1,t=1 1.5 Ent(d ξ) Figure 13. Aplitude data as a function of the odel paraeter α2 for 4 different offsets, x = 5, x = 1, x = 15 and x = 2. The velocity of the top layer is 275 s 1 and the depth of the reflector is 5. The functions are calculated using dx =.1, d = 1dx and 1 saples T. and θ(d, ) = U[ f (), δ], where f () is the absolute value of R P, eq. (2), and δ =.1 represents the standard deviation of a Gaussian expected easureent uncertainty. Fig. 13 shows the forward function f (), as a function of the odel paraeter α2, for the offsets x = 5, x = 1, x = 15 and x = 2. The axiu gradient is of the order of.1. The results of the previous section indicate that for the calculation of σ (d ) in each data-interval, ( ax in )/d = 1 does not cause large errors. After checking this was also true for the AVO exaple, all tests in this section were perfored using d = 1dx, where d is the discretization interval size in odel paraeter space and dx is the discretization interval size in the data space. It is ipossible to copare the results of the tests to an analytical solution, as this is a nonlinear exaple and the required integrals cannot be calculated analytically. Hence, the experiental design proble was calculated for different discretizations and several nubers of saples. Fig. 14 shows the entropy as a function of offset for dx =.2 and 5 saples, dx =.1 and 5 saples, dx =.1 and 1 saples, and dx =.1 and 1 saples. Two things are clear fro this figure. First, coparing the curves for dx =.1 and dx =.1 we see that the entropy shows no significant changes at sall discretization interval sizes. Second, while it is clear that for large discretization interval sizes, the values for the entropy do not converge to the correct value, this does not change the results of the experiental design proble: the overall shape of the entropy-curves reains roughly the sae and the desired axiu entropy design always occurs at the sae offset. This is an indication that it is safe to use a coarser discretization since the errors due to undersapling are uch larger and less predictable than the errors due to too coarse a discretization. The optial offset is found to be at approxiately 15 fro the source. Fig. 13 shows that this optial offset is found where the data are ost sensitive to changes in the odel paraeters. This suggests that we could have used a gradient based ethod. This is probably due to the fact that this was not a strongly nonlinear exaple (Fig. 13). The design proble is therefore repeated with a different ρ(). In the first exaple ρ() was unifor between α2 = Offset in eters Figure 14. Ent(d ξ) as a function of the offset in etres for the AVO exaple with a Gaussian uncertainty in the data space with a standard deviation of.1. Velocity of the top layer is 275 s 1, the depth of the reflector is 5 and the odel paraeter space runs fro 32 s 1 to 33 s 1. The entropy is calculated using four different discretizations and nubers of saples T. In all cases d = 1dx has been used. 32s s 1 and α2 = 33 s 1. For the second exaple, ρ()is set to be unifor between α2 = 3 s 1 and α2 = 45 s 1, a uch larger interval. Fig. 15 shows the relationship between data and odel paraeters for four different offsets, x = 5, x = 1, x = 15 and x = 2. Clearly, this proble is ore strongly nonlinear. The discontinuity in the curves is at the velocity where the angle of incidence reaches the value for the critical angle of incidence. The resulting values for Ent(d ξ) are shown in Fig. 16. Again, the axiu value is at approxiately 15, but this is no longer Aplitude Aplitude data as a function of α2 for different offsets x=5 x=1 x=15 x= =α2 Figure 15. Aplitude data as a function of the odel paraeter α2 for 4 different offsets, x = 5, x = 1, x = 15 and x = 2. The velocity of the top layer is 275 s 1 and the depth of the reflector is 5. The functions are calculated using dx =.1, d = 1dx and 1 saples T. C 23 RAS, GJI, 155,

11 Ent(d ξ) Ent(d ξ) as a function of offset dx=.2,t=5 dx=.1,t=5 dx=.1,t=1 dx=.1,t=1 dx=.1,t=3 Optial nonlinear experiental design 421 For the exaples shown in this report, a siple search with unifor steps throughout the design-space was used. For larger, highdiensional probles, it is strongly recoended to use an efficient search-algorith in order to keep the coputational cost as low as possible. The required coputational cost appears to be the only practical liitation to the application of this ethod. For the practical application of this theory it is iportant to realize that the exaples as discussed in this paper are with one odel paraeter and one receiver. More receivers autoatically increases the diensionality of the proble. Further research is necessary to apply this theory to realistic experients with ore odel paraeters and ore receivers Offset in eters Figure 16. Ent(d ξ) as a function of the offset in etres for the AVO exaple with a Gaussian uncertainty in the data space with a standard deviation of.1. Velocity of the top layer is 275 s 1, the depth of the reflector is 5 and the odel paraeter space runs fro 3 s 1 to 45 s 1. The entropy is calculated using five different discretizations and nubers of saples T. In all cases d = 1dx has been used. obvious fro the gradients in Fig. 15. Otherwise, the results are coparable with the previous exaple; selection of the axiu entropy design is unaffected by the range of discretizations considered here. 5 CONCLUSIONS An entropy-based ethod for nonlinear experiental design has been presented. In principle, this ethod is applicable to all experiental design probles, but in particular those nonlinear probles where classical nonlinear ethods for experiental design fail. The ain difficulty in applying this ethod lies in choosing a sensible discretization for odel paraeter and data spaces without knowing the degree of nonlinearity of the proble in advance. This choice strongly influences the nuber of saples required. Our results suggest that it ay be better to choose a coarser discretization to obtain final designs, since the errors associated with undersapling are larger and less predictable than the errors associated with too coarse a discretization interval size. The AVO exaples above suggest that the final experiental design reains unaffected by a slightly too coarse discretization. However, further research is necessary to deterine to what extent this rule of thub reains true. For the synthetic sawtooth exaple, this liit is visible in Fig. 7 (left), since no nuber of saples is sufficient to estiate the entropy for discretizations dx larger than.3. REFERENCES Aki, K. & Richards, P.G., 198. Quantitave Seisology: Theory and Methods, San Francisco: Freean. Atkinson, A.C. & Donev, A.N., Optiu Experiental Design, Oxford Science Publications. Box, G.E.P. & Lucas, H.L., Design of Experients in Non-linear Situations, Bioetrika, 46, Chaloner, K. & Verdinelli, I., Bayesian Experiental Design: A Review, Statistical Science, 1, Curtis, A., 1999a. Optial Experient Design: Cross-borehole Toographic Exaples, Geophys. J. Int., 136, Curtis, A., 1999b. Optial Design of Focused Experients and Surveys, Geophys. J. Int., 139, Curtis, A. & Maurer, H., 2. Optiizing designs of geophysical experients and surveys: Is it worthwhile?, EOS, Trans. A. geophys. Un., 81, 2, Curtis, A. & Spencer, C., Survey Design Strategies for linearized nonlinear inversion, 69th Annual Int. Meeting, Soc. of Expl. Geophys., Expanded Abstracts, pp Ford, I., Titterington, D.M. & Kitsos, C.P., Recent Advances in Nonlinear Experiental Design, Technoetrics, 31, Johnson, N.L. & Leone, F.C., Statistics and Experiental Design in Engineering and the Physical Sciences, Wiley series in probabability and atheatical statistics. Wiley, New York. Lepage, G.P., A New Algorith for Adaptive Multidiensional Integration, Journal of Coputational Physics, 27, Sebastiani, P. & Wynn, H.P., 2. Maxiu Entropy Sapling and Optial Bayesian Experiental Design, J. R. Statist. Soc. B, 62, Shannon, C.E., A Matheatical Theory of Counication, Bell Syste Tech. J., 27, Shewry, M.C. & Wynn, H.P., Maxiu Entropy Sapling, Journal of Applied Statistics, 14, Squires, G.L., Practical Physics, 3rd edn, Cabridge Univ. Press, Cabridge. Tarantola, A., Inverse Proble Theory: Methods for Data Fitting and Paraeter Estiation, Elsevier, Asterda. Tarantola, A. & Valette, B., Inverse Probles = Quest for Inforation, J. Geophys., 5, Yilaz, O., 21. Seisic Data Analysis: Processing, Inversion and Interpretation of Seisic Data, Society of Exploration Geophysics. C 23 RAS, GJI, 155,