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CWP-631
Probabilistic micro-earthquake location for reservoirmonitoring and characterization
Ran Xuan and Paul SavaCenter for Wave Phenomena, Colorado School of Mines
ABSTRACTMicro-seismicity is used to monitor fluid migration during reservoir produc-tion and hydro-fracturing operations. This is usually done with sparse networksof seismic sensors located in boreholes. The data used for micro-earthquakemonitoring are corrupted by noise which reduces the signal-to-noise ratio tovalues as low as 0.1. Monitoring methods based on traveltime picking of variouswave modes (P or S) cannot deal with this level of noise and require extensiveuser interaction. An alternative class of methods uses time reversal to focusmicro-earthquake information at the source position. These methods can han-dle noisier signals, but are also costlier to run. The technique advocated in thispaper exploits time-reversal within the general framework of Bayesian inver-sion. Given an assumption about the possible locations of micro-earthquakes,we use recorded data to evaluate the feasibility of micro-earthquakes occur-ring at various locations in the Earth. The method takes into account imagingimperfections due to unknown components of the model or acquisition arrayaperture. We simulate wavefields corresponding to possible sources distributedin the model and evaluate their match with the wavefield reconstructed fromreal data recorded in the field. In this regard, the method operates like a pattern-recognition procedure and can exploit a wide variety of techniques designed forthis purpose. We use simple cross-correlation to take advantage of the speedand robustness of this technique. The wavefields reconstructed at various loca-tions are used to scan over time the wavefield constructed from field data, thusour method is able to identify not only the position of the micro-earthquakesbut also their onset times. The final outcome of this automated process is amap of probabilities indicating the confidence of micro-earthquake occurrenceat various positions and times.
Key words: wave equation, imaging
1 INTRODUCTION
High-pressure fluid injected into oil and gas reservoirscauses time-invariant stress and strain changes. Whenthe stress exceeds a threshold characterizing the resis-tance of rock materials to stress, micro-seismicity is trig-gered by the release of pressure along pre-existing frac-tures or through the creation of new fractures (Maxwell& Urbancic, 2001; Duncan, 2005). Precisely locatingmicro-seismic events can be used to monitor the hy-draulic fracturing and for reservoir characterization(Rentsch, 2004).
Most location methods currently used require theidentification of seismic arrivals which involves accuratepicking of P- and S-wave arrival times. The onset timeand the coordinates of the hypocenter of micro-seismicevents are given by calculations which require accurateknowledge of the velocity model and of the physical re-lationships describing wave propagation in the subsur-face. The source is located by optimizing a misfit func-tion between measured and calculated quantities (Pu-jol, 2004; Lay & Wallace, 1995; Thurber & Rabinowitz,2000). The methods in this category assume that the ar-rival time of a specific event can be identified on seismic
204 R. Xuan & P. Sava
traces (Gajewski, 2007). However, event identificationis not always possible due to the typically low signal-to-noise ratio (SNR) characterizing micro-seismic events.Furthermore, such location methods strongly depend onthe picking accuracy of P- and S-wave arrival times andhave a low degree of automation (Rentsch, 2007).
In contrast to the methods based on picked ar-rival times, migration-based methods use the full seis-mic wavefield to locate micro-seismic events. The mainadvantage provided by the methods in this category isthat they are independent of the picking of specific ar-rivals and can locate weak events by focusing energyat the source using time-reversal (Gajewski, 2005). Thedrawback of methods in this category is the high compu-tational cost which limits their practical use. However,this drawback becomes less and less of a problem dueto the significant advances in computer hardware, whichallow for fast processing of large volumes of data.
Within the context of migration-based methods, weadvocate in this paper a new methodology which canbe used to automatically identify and locate sourcesof micro-seismicity. We use time-reversal based on nu-meric solutions to the acoustic wave-equation to recon-struct seismic wavefields in the space around the pos-sible source position and at times around the possibleonset time of seismic events. Ideally, the energy corre-sponding to a micro-seismic event focuses at some timeand some position in space. However, due to the incom-plete acquisition, inaccuracies of the velocity model andnoise present in data, this focus is hard to identify ormay not even form at all.
We address this problem by constructing at vari-ous locations in the model and at various times “ideal”wavefields which we would record given the field ac-quisition geometry and the velocity model in the re-gion under investigation if a micro-earthquake sourcewould be located at that position and time. Then, weemploy a pattern matching procedure to compare thereconstructed wavefields based on field data with oursimulated wavefields and identify the most probable lo-cations and onset times of micro-seismic events. Thisprocess exploits the kinematic properties of the focusedwavefields and takes advantage of the distortions causedby the incomplete acquisition in the area under investi-gation.
Given the uncertainty characterizing the process ofmicro-earthquake identification, we do not construct ourresults as precise positions and onset times. Instead, weconstruct maps of probability indicating the likelihoodthat a micro-seismic source exists at a given position inspace and a given time. We construct our solutions us-ing Bayesian inversion theory which simply states thatinversion is a process which refines our state of knowl-edge. The Bayesian theory discusses about a prior stateof knowledge which incorporates our knowledge aboutour system in absence of any measurements and abouta posterior state of knowledge which combines the prior
with additional knowledge acquired from new measure-ments.
We begin the paper with a review of Bayesian inver-sion theory applied to the problem of micro-earthquakeidentification and illustrate our technique with syntheticdata examples.
2 A METHODOLOGY FOR AUTOMATICMICRO-SEISMIC LOCATION
2. 1 Bayesian inversion theory
Bayesian inversion theory uses probability density func-tions as a measurement of a given state of knowledge(Jaynes, 2003). A crucial features of Bayesian theory isthat a probability can be assigned to a hypothesis. Ac-cording to the Bayesian theory, we begin our investiga-tion with a certain state of knowledge which is indepen-dent of any subsequent measurement we make. We as-sign to any possible model acceptable to our prior stateof knowledge a likelihood which is quantifies how proba-ble a particular model is to represent reality (Tarantola,2005). The posterior combines our prior understandingof the model parameters with additional informationderived from measurements and assumptions about thephysical laws governing the process under investigation.This approach allows us to incorporate any a priori in-formation in the data, parameters and theory. As dis-cussed by Tarantola (2005), the posterior state of infor-mation is obtained using the Bayes theorem
σ(m,d) = kρ(m,d)Θ(m,d)
µ(m,d), (1)
where m represents the model, d represents the data,ρ(m,d) represents the a priori state of information,Θ(m,d) represents the theoretical state of information,µ(m,d) represents the homogeneous state of informa-tion, k represents a normalization constant, and σ(m,d)represents the a posteriori state of information.
According to equation 1, the a posteriori stateof information σ(m,d) is computed by combining thetheoretical function Θ(m,d) with the prior informa-tion ρ(m,d). The solution to the inversion problem isgiven by σM(m) which represents the posterior marginalprobability on the model space and is obtained by in-tegrating the a posteriori probability density functionover data:
σM(m) =
ZD
σ(m,d)dd , (2)
where D is the data space. When the uncertainty as-sociated with the kernel function is negligible, σM(m)is
σM(m) = νρM(m)ρD(G(m),dobs) , (3)
where ρM(m) is the prior state of information on themodel space, ρD(d) is the prior information on the data
Probabilistic micro-seismic imaging 205
(a)
(b)
Figure 1. (a) The velocity model used for the kernel function and (b) the velocity model with random variability used tosimulate data. The random velocity in panel (b) is the sum of the velocity in model (a) with correlated Gaussian noise witha magnitude of 40% of the background. The background velocity model is sampled every 2 m and ranges between 1900 and2400 m/s. The acquisition array consists of 10 receivers placed at x = 0 km with a spacing of 40 m. We simulate an explosiveseismic source at x = 0.136 km and z = 0.333 km.
206 R. Xuan & P. Sava
space, and ν is a normalization constant. A summary ofBayesian inversion theory is presented in Appendix A.
We apply the Bayesian inversion theory to solvethe problem of micro-earthquake identification. The lo-cations corresponding to micro-seismic events are pro-vided by incorporating information from recorded seis-mograms with prior information on the events, e.g. pos-sible locations and onset time of the events, as well asphysical relations between the model and data.
The model parameters used for the micro-earthquake location problem are m = {x, t}, wherex = {x, y, z} describes the position of a micro-seismic inspace and t describes the onset time. The data d repre-sent the wavefields reconstructed from recorded seismo-grams in the space around the most probable micro-earthquake source locations and onset times. In thiscontext, it is irrelevant whether the data are acquiredin a borehole, on the surface, or using a combinationof borehole and surface arrays. The data parametersD(x, t) are voxels of the 4-D wavefield. Each model hasonly one datum because there is one wavefield charac-terizing one micro-seismic event in space and time. Thekernel function G(m) describes the theoretical relation-ship between the data and model. As discussed earlier,we evaluate data for all possible models by taking intoconsideration the velocity in the region under investi-gation and the acquisition geometry used for the fieldexperiment. In our case, the kernel function is based onthe two-way acoustic wave-equation and is constructedusing the following steps:
• First, for a given model (m) characterized by theonset time t and location x, we propagate waves forwardfrom the source using the velocity c(x) according to thetwo-way acoustic wave-equation
1
c2U = ∇2U + f(m, t) , (4)
where f(m, t) is the considered source wavelet andU(x, t) represents the seismic wavefield simulated for-ward in time.• Second, the forward wavefield is recorded at re-
ceiver coordinates x0 arbitrarily located in space:
R(x0, t) = U(x = x0, t) , (5)
where R(x0, t) represents the recorded seismogram.• Third, the recorded seismogram is time reversed
and propagated backward in time from the acquisitionarray to reconstruct the wavefield at all locations inspace and all times within the window considered forour prior model. This step is also based on the two-wayacoustic wave-equation
1
c2D = ∇2D + R(x0,−t) , (6)
where D(x, t) are the data parameters as explained ear-lier.
For a given model, we refer to the data calculated using
this procedure as the predicted data Dpre. In contrast,we refer to the wavefields reconstructed from field seis-mograms as the observed data Dobs.
We illustrate our method with a 2D synthetic ex-ample. We assume that we use the velocity shown inFigure 1(a) as the background velocity to construct thepredicted data. We consider the velocity shown in Fig-ure 1(b) as the true velocity which is used to simulatethe field seismograms. We simulate data by consideringa micro-seismic source at mt = {xt, zt, tt}, where thesubscript t indicates the “true” position of the source.The wavefield simulated for this event is calculated us-ing,
1
vU = ∇2U + f(mt, t) , (7)
where v(x) is the velocity shown in Figure 1(b). Thesimulated synthetic seismogram is shown in Figure 2.We assume that the velocity v is unknown during theinversion process.
2. 2 The prior information on the model
We assume that the hypocenter and the onset time ofmicro-seismic events may be located with equal proba-bility within a certain space domain and time interval.Thus, we represent this prior knowledge on the modelspace as a boxcar function in space and time. The priorprobability ρM(m) = 1 within this region of space andtime and 0 elsewhere. We uniformly sample the modelspace within this region. The number of models locatedin this space is nx × nz, where nx and nz represent thenumbers of models along the x and z axes, respectively.In our example we sample the model parameters every4 m, and some of the locations for which we simulatedata using our procedure are shown in Figure 3. By se-lecting a finite number of possible source positions, weboth exploit our knowledge about the true source lo-cation and reduce computational time. The assumptionwe make here is that the data (reconstructed wavefieldcorresponding to the recorded seismograms) character-ize with equal probability all models within our selectedmodel space, as illustrated in Figure 1(a).
2. 3 The theoretical information on modeland data
For all models within our selected region of space,we compute the corresponding wavefield by the for-ward/backward procedure outlined in the preceding sec-tion. In our case, we assume that each model corre-sponds to one dataset and that all simulated data areequally probable to explain the recorded data. This con-struction, which corresponds to the illustration in Fig-ure 1(b), applies the kernel functions G linking theoret-ically individual members of the model and data spaces.
Probabilistic micro-seismic imaging 207
Figure 2. The synthetic seismogram simulated using the velocity model shown in Figure 1(b). We consider an absorbingsurface. The source used to simulate this event is a Ricker wavelet with a dominant frequency of 50 Hz. A total of 2000 timesteps with a time increment of 0.2 ms are calculated.
Figure 3. A schematic representation of the a priori information on the model space. We assume that the micro-earthquakesoccur in the region indicated by the dense dots between x = 0 − 0.35 km and z = 0.25 − 0.35 km. A similar discrete set ofpossible onset times are defined along the time axis.
208 R. Xuan & P. Sava
We assemble all those calculations in a database of wave-fields which is used to compare with data reconstructedfrom the “field” seismograms. Figure 4 illustrates thewavefield database at all locations indicated by dots inFigure 3. The size of the cubes in Figure 4 is two wave-lengths in space and two periods in time.
The kernel functions constructed with this proce-dure are functions of the velocity, the acquisition geom-etry and the source type. We refer to the predicted datacalculated from the models defined by ρM(m) for thefixed velocity model, layout of receivers and source typeas the reconstructed wavefield database. This databaseonly needs to be calculated once as long as we keep thesubsurface velocity and acquisition geometry fixed.
If two sources are located at the same position buthave different onset times, the refocused fields of thesetwo events are the same. It is sufficient to compute datafor one onset times and then compare these simulatedwavefields with the wavefields reconstructed from fieldseismographs at various times. Thus we can not onlyidentify the location of the source, but we can also iden-tify the onset time.
In an ideal case, if the sources are completely sur-rounded by receives, the wavefronts reconstructed bytime reversal should focus into a point at the sourcelocation and at the onset time. However, typical acqui-sition arrays are limited to a relatively small region ofspace. In the example shown in Figure 3, 10 receivers areplaced along a borehole located at x = 0 km. For lim-ited aperture of receivers, the wavefronts do not focusat the source location.
As illustrated in Figure 5, the cross-range resolu-tion Cr of the refocused field at t = 0 s increases whenthe range decreases, but the range resolution does notchange much. The relationship between the cross-rangeresolution Cr and the range L is Cr = λ0L/a, whereλ0 is the wavelength and a is the array aperture (Born& Wolf, 2003). The wavefront shape at t = 0 s is con-trolled by the range and the cross-range for a fixed arrayaperture and wavelength. It is apparent not only thatthe focus is incomplete, but that the shape of the re-constructed wavefields changes in space correspondingto the relative positions to the acquisition array as wellas the velocity distribution in space.
2. 4 The prior information on the data
As discussed earlier, we refer to data as reconstructedwavefields at all locations in space and within a giventime window. We loop over the models provided byρM(m) and look for the times and locations in spacewhere data reconstructed from recorded seismogramsmatch the simulated data at locations where we assumethat micro-seismic events are located. We can concep-tually describe the process as measuring the similaritybetween multi-dimensional images. Various methodscan be used to compare the similarity between the
predicted data and the observed data. These methodsinclude cross-correlation, sum-of-squared differences,and ratio image uniformity, which are three of themost common area-based methods used in medicalimage registration to compare the images of the samemodality (Zitova & Flusser, 2003). In this work, weuse normalized cross-correlation (Ncc) to measure thesimilarity between theoretical wavefields and wavefieldsreconstructed from field seismograms. By definition, wecompute normalized cross-correlation by the expression
Ncc(m) =
x+MxPx−Mx
t+MtPt−Mt
(Dobs −Dobs)(G(m)−G(m))sx+MxPx−Mx
t+MtPt−Mt
(Dobs −Dobs)2x+MxPx−Mx
t+MtPk=t−Mt
(G(m)−G(m))2
,
(8)where the overline denotes average values, Dobs are theobserved data, and G(m) are the predicted data. Thesize of the averaging window is (2Mx +1)××(2Mt +1),where x = {x, y, z}. The normalized cross-correlation isevaluated at the locations of every model m. The val-ues of Ncc are between −1 and +1. Using this equation,we pinpoint the model which generates the highest Ncc.In terms of Bayesian inversion, this process of identifi-cation of the most probable model corresponds to theconjunction of the prior and theoretical probabilities,as indicated in Figures 1(c)-1(d). The size of the win-dow used in the weighting function affects the value ofNcc(m) and its identification power.
Four snapshots of the wavefield reconstructed fromthe seismogram given by Figure 2 are shown in Fig-ures 6(a)-6(d).
The reconstructed wavefields are symmetric aboutto the borehole axis, which creates an ambiguity inthe identification of micro-earthquake sources. In gen-eral, eliminating this ambiguity required additional az-imuthal coverage, e.g. from additional boreholes or fromthe surface. In our synthetic example, we eliminate thisambiguity by the choice of the prior information.
We assume that there is no error associated withthe theoretical information, but that errors are intro-duced by the measurement uncertainty of the observeddata. We also assume that the error is distributed ina Gaussian distribution and we give the highest prob-ability to the model which generates the smallest datamisfit. The probability density function based on thenormalized cross-correlation given by equation 8 is
σM(m) = ρM(m)e− (1−Ncc)2
2Cd , (9)
where Cd is the measurement uncertainty. The largerthe value of Cd, the flatter the distribution of σM(m).When Cd is small, σM(m) tends toward a delta func-tion. We expect that for the distribution of σM(m), theradius of the contour with a value of 0.9 is half the wave-length because the location error of the Ncc method is
Probabilistic micro-seismic imaging 209
Figure 4. Seismic wavefield database. The center of each cube is the source location of the predicted data.
210 R. Xuan & P. Sava
Figure 5. Schematic of the range and cross-rangeparameters characterizing the resolution of the re-focused field. The regular polygons represent re-ceivers and the circle represents the source loca-tion. The shortest distance between the source andthe array is called range L. The shortest distancebetween the source and the line which is perpendic-ular to the center of the array is called cross-rangeC.
X
Z
L
C
(a) (b)
(c) (d)
Figure 6. Four snapshots of the observed data. The wave field defocuses when it moves away from the source location. It ishard to choose between (a) and (b) by manual picking.
Probabilistic micro-seismic imaging 211
(a) (b)
Figure 7. (a) Normalized cross-correlation. (b) Probability distribution based on the normalized cross-correlation value. Themeasurement uncertainty controls the width of the probability distribution around the maximum value.
Figure 8. One snapshot of the probability density function, where the dot is the hypocenter of the micro-earthquake. Theouter contour has probability of 0.8 and the contour interval is 0.05.
212 R. Xuan & P. Sava
less than half of the wavelength. Figures 7(a)-7(b) showa comparison between the normalized cross-correlationand the associated probability given by equation 9. Onesnapshot of Figure 7(b) is shown in Figure 8 and theradius of the contour with a value of 0.9 is about 20 m.We can conclude that a micro-seismic event is locatedwithin this contour with a probability of 0.9. We com-pute normalized cross-correlation by comparing the pre-calculated wavefield database with the observed data,as shown in Figure 7(a). The maximum Ncc value is0.991 at x = 132 m, z = 334 m, t = 0.044 s. How-ever, the Ncc value is 0.985 at x = 136 m, z = 334 m,t = 0.042 s. These two values are very close to each otherand it would be unwise to pick one without taking theother into consideration. Therefore, instead of selectinga single model as the answer, we construct a probabil-ity distribution characterizing various models and theirlikelihood to represent sources of micro-earthquakes.
3 SYNTHETIC EXAMPLE
In the preceding section, we demonstrate micro-earthquake location in the case when only one seismicevent occurs in a region during a time interval. However,a more common situation is that multiple micro-seismicevents occur more-or-less at the same time and overlapin space during a given time interval. In this section wedemonstrate our methodology on another example withmultiple overlapping events. The motivation for such anexperiment derives from field observations about howfractures propagate during fluid injection. By spatiallymapping the signals emitted by micro-earthquakes, wecan monitor the extensions of fractures (Keppler et al.,1982; Shapiro et al., 2006). Numerous field observationsindicate that the seismic zone surrounds the fracturesand forms an elongated pattern (Phillipsa, 1998; Rut-ledge & Phillips, 2001; House, 1987).
In this numeric experiment, we assume that themicro-seismic events are distributed along a fracture lo-cated at z = 0.326 km, as indicated in Figure 9. Theopening of this fracture simulates high-pressure steaminjected into the borehole located at x = 0 km. The trig-gering time of micro-seismic event varies linearly withthe distance from the injection point. The differencesbetween the onset times of nearby events is greater thanone period. For simplicity, we assume that the mag-nitude of all micro-earthquakes is the same, and thatthe sources are characterized by Ricker wavelets with adominant frequency of 50 Hz. Figure 10 shows the simu-lated seismogram characterizing all the events. The seis-mogram is contaminated with two types of noise. Thefirst type of noise is due to model heterogeneities whichwe assume that we do not know in the imaging process.The second type of noise is corresponding to the exper-iment being run in a noisy environment. The frequencyband of the random noise is 0−140 Hz overlapping withthe micro-earthquake band. The SNR for this example
is as low as 0.1. We assume that measurement uncer-tainties are 0.64 because of the low SNR. We simulate20 micro-earthquakes and attempt to locate all of themwith our methodology, as seen in Figures 11(a)-11(h).The contours shown indicate the probability of earth-quakes occurrence at various times and range from 0.8to 0.95 with a contour interval of 0.05.
Figure 11(a) shows the results obtained for a micro-earthquake occurring at m0 = {x, z, t} = {0.012 km,0.326 km, 0.02 s}. This point is included in the contourcorresponding to 0.95 confidence, indicating the successof our procedure. Figure 11(b) shows two events, m4
= {0.04 km, 0.314 km, 0.16 s} and m5 = {0.052 km,0.326 km, 0.24 s} located within the contour with aconfidence level of 0.9. This example shows that ourprocedure may not have the resolution to distinguishamong events that are located too close to one-another.The true location of the second events is indicated bythe contours shown in Figure 11(c). However, this ap-parent mismatch is not a big problem because both re-gions indicate the presence of micro-earthquakes andthey overlap in space and time. Other choice of param-eters, e.g. in the definition of the wavefield database,might increase the spatial resolution of the identifica-tion. This example also seems to indicate the presenceof another event located inside the contour centered atabout {x = 0.17 km, z = 0.35 km}. This is simply anartifact since no event occurs at this position and timein our simulation. Figure 11(d) shows the event m8 ={0.072 km, 0.33 km, 0.32 s} located within the contourwith a confidence value of 0.9.
Figure 11(e) shows that the event at m12 ={0.112 km, 0.33 km, 0.52 s} is not located within thecontours obtained by our inversion. In this case, ourmethod fails. This phenomenon is likely due to the factthat multiple micro-earthquakes overlap in this regioncreating false foci that masquerade as real events. Fig-ure 11(f) shows the event m14 = {0.132 km, 0.326 km,0.62 s} located within the contour with a confidencevalue of 0.9. Figure 11(g) shows the event at m15 ={0.144 km, 0.334 km, 0.68 s} located within the contourwith a confidence value of 0.8. Although we successfullyidentify this event and the experiment seems to placeit at the right position, we have lower confidence in itsposition. The lower confidence is likely due to the lowSNR in the seismogram, which is approximately 0.1. Fi-nally, Figure 11(h) shows the event at m19 = {0.212 km,0.33 km, 1.02 s} located well within the contour with aconfidence value of 0.95. This is another example of suc-cessful identification of a seismic source.
Although in this example we do not locate all ex-isting micro-earthquakes, we regard this experiment asa success because we are getting a good representationof the direction and speed of fracture propagation in adifficult setting characterized by low SNR and sparsedata acquisition.
Probabilistic micro-seismic imaging 213
Figure 9. Synthetic model. The dots located at about z = 0 km represent receivers, the other dots represent the micro-seismicevents.
Figure 10. Synthetic seismogram. A total of 7000 time steps with an increment of 0.2 ms are calculated.
214 R. Xuan & P. Sava
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 11. Snapshots of the probability distribution of micro-seismic events. The value of the outer contour is 0.8, the contourinterval is 0.05. The dots represent the locations of micro-earthquakes.
Probabilistic micro-seismic imaging 215
4 CONCLUSION
We present a method for automatic micro-earthquakelocation using Bayesian inversion theory. Our methodexploits the unique shape of wavefields partially refo-cused at the source position. This method does not re-quire picking of arrival times but relies on wavefieldfocusing obtained by time reversal. Our method notonly identifies the spatial location of the source, butalso specifies the onset time of the source. Furthermore,since we are using a probabilistic approach to micro-earthquake location, our method provides confidencemaps of micro-earthquake locations which can be usedfor risk assessment.
Synthetic data examples demonstrate the robust-ness of the method and its applicability to situationswhen micro-earthquakes occur in close succession in asmall region in space and are recorded in a noisy envi-ronment with a SNR as low as 0.1.
5 ACKNOWLEDGMENTS
This work was supported by ExxonMobil URC. We aregrateful to Mike Payne, Jie Zhang, Rongrong Lu, AlexMartinez, and Anupama Venkataraman at ExxonMobilURC for access to data, technical discussions and finan-cial support of this project.
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6 APPENDIX A
This appendix summarizes the main elements ofBayesian inversion theory. Bayesian inversion theorycharacterizes our state of knowledge in a probabilisticmanner through the use of probability density functions(PDFs) linking model and data parameters (Jaynes,2003). We can define two states of knowledge. The priorcharacterizes our knowledge about the model and dataparameters before any measurements are taken. Thisinformation is based on our general knowledge of thephenomenon under investigation and on the distribu-tion of model parameters and assumptions about relia-bility of our measurements. The posterior characterizesour knowledge about the model and data parameters af-ter measurements are made and data are processed andinterpreted. Both states of knowledge are characterizedby PDFs linking model and data parameters. Ideally,the measurements help refine the prior into the poste-rior state of knowledge which provides a tighter connec-tion between model and data parameters. In describingBayesian inversion theory, we need to discuss the fol-
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Figure A-1. (a) A priori probability density function ρ(m,d) represented by a delta function corresponding to ρM(m) which isalso represented by a step function. (b) Theoretical probability density function Θ(m,d) representing the physical relationshipgiven by G(m) and assumed to be exact. (c) Conjunction between ρ(m,d) and Θ(m,d). (d) A posteriori probability densityfunction σ(m,d) indicating that the solution to the Bayesian inversion problem is given by a single model. (e) A comparisonbetween the marginal probabilities showing ρM(m) in the upper panel and σM(m) in the lower panel.
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lowing elements: model (m) and model space, data (d)and data space, the prior state of knowledge (a prioriPDF ρ(m,d)), the theoretical relations between modeland data (theoretical PDF Θ(m,d)), and the posteriorstate of knowledge (a posteriori PDF σ(m,d)).
6. 1 Model and data spaces
The model space (M) is composed of a set of individualmodels which can be denoted by mi (i ∈ {1, 2, ..., N}).The model space represents all the models that can pos-sibly characterize the physical phenomenon under inves-tigation.
The data space D is composed of all instrumen-tal responses to the investigated models which can bedenoted by di (i ∈ {1, 2, ..., M}). The data space repre-sents all data that can possibly be recorded and char-acterize the physical phenomenon under investigation.
6. 2 A priori probability density ρ(m,d)
We describe ρ(m,d) by a joint PDF over the modeland data spaces. This prior information is independentof any measurement. Furthermore, in the special casewhen the prior information on the model space ρM(m)is independent of the prior information on the dataspace ρD(d), the a priori probability density functionρ(m,d) is (Tarantola, 2005)
ρ(m,d) = ρD(d)ρM(m) , (A-1)
where ρD(d) describes the measurement uncertainty ofthe observed data, and ρM(m) illustrates our confidencein the chosen model parameters.
Two examples of a priori probability density func-tions are shown in Figures 1(a) and 2(a). The verticalaxis represents the value of the probability that a par-ticular model characterizes the physical system underinvestigation. In Figure 1(a), ρ(m,d) = ρM(m)δ(d −Dobs), where Dobs denote the observed data, and the apriori probability in the model space is a step function.In this case, no error is associated with the observed
data. In Figure 2(a), ρ(m,d) = ρM(m)e− (d−Dobs)2
2Cd ,where ρM(m) is a tapered step function, and Cd is themeasurement uncertainty of the data. We assume thatthe data are distributed following a Gaussian distribu-tion.
6. 3 Homogeneous probability density µ(m,d)
In order to remove the effect of the discretization of themodel and data parameters, we define a homogeneousPDF µ(m,d) whose role is to balance the discrete modeland data spaces. By definition, the homogeneous prob-ability distribution states that a probability assigned toeach region of the space is proportional to the volume of
the region (Mosegaard & Tarantola, 2002). In the spe-cial case when the homogeneous probability densities onthe model and data spaces are independent, µ(m,d) isexpressed as
µ(m,d) = µD(d)µM(m) , (A-2)
where µM(m) is the homogeneous probability densityon the model space and µD(d) is the homogeneous prob-ability density on the data space.
For example, in spherical coordinates, the volume ofa standard region is dV (r, θ, ϕ) = r2 sin θdrdθdϕ. Thehomogeneous probability distribution is µ = kr2 sin θ,where k is an arbitrary constant. For 3-D Cartesian co-ordinates, dV (x, y, z) = dxdydz and µ is a constant.
6. 4 Theoretical probability density Θ(m,d)
The theoretical probability density Θ(m,d) describesthe correlations between the model and data parame-ters. The defined relationship corresponds to a physicallaw and may incorporate uncertainties associated withtheory, for example due to assumptions or simplifica-tions of the physical law or due to imperfect knowledgeof the physical parameters underlying it. The theoreticalprobability density Θ(m,d) can be expressed as
Θ(m,d) = θ(d|m)µM(m) , (A-3)
where θ(d|m) describes the probability distribution ofdata for the given kernel function and model, andµM(m) is the marginal probability defined in the pre-ceding section. In Figure 1(b), θ(d|m) = δ(d−G(m)),where G(m) is the kernel function which describes thephysical relationship between d and m. No error isassociated with this kernel function. In Figure 2(b),
θ(d|m) = e− (d−G(m))2
2Ct , where Ct describes the theoryuncertainty. The error associated with this kernel func-tion follows a Gaussian distribution.
6. 5 A posteriori probability density σ(m,d)
The conjunction between the a priori state of infor-mation ρ(m,d) and the theoretical probability densityfunction Θ(m,d), which describes the theoretical rela-tionship between model and data, provides the a poste-riori state of information σ(m,d):
σ(m,d) = kρ(m,d)Θ(m,d)
µ(m,d), (A-4)
where k is a normalization constant that serves the pur-pose of keeping the area under the graph of σ(m,d)constant. The expression for k is
k =1R
XdXρ(m,d)Θ(m,d)/µ(m,d)
, (A-5)
where X = (D, M) is the joint space of the data and themodel.
The posterior information σ(m,d) is computed
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Figure A-2. (a) A priori probability density function ρ(m,d) represented by a Gaussian function corresponding to ρM(m) rep-resented by a tapered step function. (b) Theoretical probability density function Θ(m,d) representing the physical relationshipgiven by G(m) and assumed to be uncertain. (c) Conjunction between ρ(m,d) and Θ(m,d). (d) A posteriori probability densityfunction σ(m,d) indicating that the solution to the Bayesian inversion problem is given by several models corresponding to theuncertainties of the prior and theoretical probability density functions. (e) A comparison between the marginal probabilitiesshowing ρM(m) in the upper panel and σM(m) in the lower panel.
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based on observations. In the special case when boththe model space and the data space are described inCartesian coordinates, the posterior information is pro-portional to the conjunction between the prior informa-tion and the theoretical information:
σ(m,d) ∝ ρ(m,d)Θ(m,d). (A-6)
Figures 1(c) and 2(c) show the conjunction betweenthe ρ(m,d) and Θ(m,d). In Figure 1(c), σ(m,d) ∝δ(Dobs − G(m)) which is shown in Figure 1(d). Fig-ure 2(c) gives the conjunction between the PDFs shownin Figures 2(a) and 2(b). Its physical meaning is thatthe observed data are exactly the same as the datacalculated from the kernel function. In Figure 2(c),
σ(m,d) ∝ e−(Dobs−G(m))2
2C , where C = Cd + Ct. Fig-ure 2(d) shows σ(m,d). The probability distribution inFigure 2(d) shows how well the data calculated from thekernel function explain the observed data.
6. 6 The marginal probability density σM(m)
In order to provide a solution to the inversion problem,we have to transfer the information provided by σ(m,d)to the model space. We obtain the marginal probabil-ity density in the model space σM(m) by projectingσ(m,d) onto the model space:
σM(m) =
ZD
σ(m,d)dd . (A-7)
The probability density function provided by σM(m)indicates what models satisfy at the same time our priorknowledge on the distribution of model parameters, thetheoretical relationships between model and data andthe measurement uncertainties. A comparison betweenρM(m) and σM(m) is shown in Figures 1(e) and 2(e).The distribution on σM(m) is narrower on the posteriorrelative to the prior indicating the tighter connectionintroduced by the recorded data.
6. 7 Negligible theoretical uncertainties
When we assume that no uncertainty is associated withthe kernel function G(m) which describes the physicalrelationship between the data and the model, the the-oretical probability density Θ(m,d) can be expressedas
Θ(m,d) = δ(d−G(m))µM(m) , (A-8)
where µM(m) is a constant for models parameterized inCartesian coordinates. The theoretical probability den-sity Θ(m,d) = 1 when the data equal the value calcu-lated from the kernel function and zero otherwise.
In this case, the marginal probability in the modelspace in equation A-7 can be expressed as
σM(m) =
ZD
σ(m,d)dd = k
ZD
ρ(m,d)Θ(m,d)
µ(m,d)dd ,
(A-9)
and when Θ(m,d) is given in equation A-8, withρ(m,d) = ρD(d)ρM(m) and µ(m,d) = µD(d)µM(m),we can write
σM(m) = kρM(m)
ZD
ρD(d)δ(d−G(m))
µD(d)dd , (A-10)
where ρD(d) is a function of d and the observed data,and µD(d) is a constant for data discretized in Carte-sian coordinates. Thus, equation A-7 for an ideal kernelfunction can be written as
σM(m) = νρM(m)ρD(G(m), Dobs) , (A-11)
where
ν =
ZM
ρM(m)ρD(d, Dobs)dm (A-12)
is a constant. In equation A-11, σM(m) is proportionalto the product of the prior probabilities on the modeland data spaces.
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