2010 joint timelapse grl

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Structural joint inversion of timelapse crosshole ERT

and GPR traveltime data

Joseph Doetsch,1 Niklas Linde,2 and Andrew Binley3

Received 13 September 2010; revised 21 October 2010; accepted 26 October 2010; published 21 December 2010.

[1] Timelapse geophysical monitoring and inversion arevaluable tools in hydrogeology for monitoring changes inthe subsurface due to natural and forced (tracer) dynamics.However, the resulting models may suffer from insufficientresolution, which leads to underestimated variability andpoor mass recovery. Structural joint inversion using crossgradient constraints can provide higherresolution modelscompared with individual inversions and we present thefirst application to timelapse data. The results from asynthetic and field vadose zone water tracer injectionexperiment show that joint 3D timelapse inversion ofcrosshole electrical resistance tomography (ERT) andground penetrating radar (GPR) traveltime data significantlyimprove the imaged characteristics of the point injectedplume, such as lateral spreading and center of mass, as wellas the overall consistency between models. The jointinversion method appears to work well for cases when onehydrological state variable (in this case moisture content)controls the timelapse response of both geophysicalmethods. Citation: Doetsch, J., N. Linde, and A. Binley (2010),Structural joint inversion of timelapse crosshole ERT and

GPR traveltime data, Geophys . Res . Lett. , 37, L24404,

doi:10.1029/2010GL045482.

1. Introduction

[2] Time

lapse geophysical monitoring and inversion arevaluable tools in a wide range of application areas, such ashydrogeology, seismology, volcanology, landslide studies,and reservoir management. By inverting for temporal chan-ges in geophysical properties it is possible to focus onresolving changes in state variables, such as water content andpore water salinity. The quality and resolution of timelapseinversion results may also improve compared with staticinversions as modeling and observational errors are generallysmaller [e.g., LaBrecque and Yang, 2001]. Timelapseinversion results are, unfortunately, also resolutionlimited,leading to models that might be physically implausible or theresolved scales might be larger than those of interest [e.g.,DayLewis et al., 2005]. Well known problems include thedifficulty of recovering the injected mass of tracer or waterfrom timelapse inversion results [e.g., Binley et al., 2002a]and significant smearing in the horizontal directions[Singha and Gorelick, 2005] that reduce the value of geo-

physical timelapse models in quantitative flow and trans-port studies.

[3] Structural joint inversions of geophysical data acquiredunder static field conditions provide geometrically similarmodels and improve model resolution compared with indi-vidual inversions [e.g., Gallardo and Meju, 2004; Lindeet al., 2008]. We focus here, for the first time, on theapplicability of a structural joint inversion approach to timelapse data. Structure is imposed by penalizing deviationsfrom cases when the gradients for different geophysicalproperties of the total model updates from backgroundmodels point in the same or opposite directions. Thesebackground models are obtained by inversion of the dataacquired prior to any perturbation. We investigate the meritsof this crossgradientconstrained joint inversion using asynthetic and field vadose zone waterinjection experiment,both of which employ timelapse crosshole electrical resis-tivity tomography (ERT) data and firstarrival groundpen-etrating radar (GPR) traveltimes. Under these conditions, thetimelapse changes in both data types are related solely tovariations in moisture content.

2. Methods

2.1. TimeLapse Inversion Strategy

[4] The first step of our timelapse inversion strategy is toobtain background (and initial) models of the logarithm ofelectrical resistivity (me,0) and radar slowness (mr,0) byinverting data sets acquired prior to any perturbations. Thedata are inverted following an Occams type inversion bypenalizing differences from a homogeneous model as definedby an exponential covariance model [Linde et al., 2006]. Wethen use a difference inversion approach to invert the timelapse data [e.g.,LaBrecque and Yang, 2001] inwhich we, in asimilar manner, penalize deviations from me,0 and mr,0.

[5] For the ERT inversions, we use an error model con-sisting of a systematic contribution ees that is the same forall timelapse steps, and a random observational error ee,p(p = 0, 1, 2, P, where Pis the number of time steps) that isdifferent for each timelapse data set [e.g., LaBrecque andYang, 2001] but assumed to stem from the same zeromeanGaussian distribution. The observed data at time 0 are thus

dobse;0

g me;0

ees ee;0; 1

with the forward response g(me,0). The main contribution tothe background residual

re;0 dobse;0

g me;0

ees ee;0 2

is the systematic error ees, which is a combination of mod-eling errors and systematic measurement errors due toground coupling problems or geometrical errors. It is largely

1Institute of Geophysics, ETH Zurich, Zurich, Switzerland.2Institute of Geophysics, University of Lausanne, Lausanne,

Switzerland.3Lancaster Environment Centre, Lancaster University, Lancaster,

UK.

Copyright 2010 by the American Geophysical Union.00948276/10/2010GL045482

GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L24404, doi:10.1029/2010GL045482, 2010

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http://dx.doi.org/10.1029/2010GL045482http://dx.doi.org/10.1029/2010GL045482http://dx.doi.org/10.1029/2010GL045482


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removed from the timelapse data by using the differences~de,pobs (for time step p), to invert for the model update dme,p,

where

~dobs

e;p dobse;p re;0 g me;0 dme;p

ee;p ee;0: 3

This formulation improves ERT timelapse inversion results,

where typically ees >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

e;0 e2

e;pq due to permanentlyinstalled electrodes and stable coupling conditions. In ourcase, we assume thatees is 5 times larger than ee,p.

[6] For the firstarrival GPR data, we assume that theconstant and systematic error contribution is smaller than theerrors associated with picking, timezero, and antennaepositioning for each timelapse data set. We thus solve fordmr,p using (at time step p)

dobsr;p g mr;0 dmr;p

er;p: 4

Our inversions for dme,p and dmr,p proceeds iteratively bydecreasing the weight that penalize deviations from me,0 andmr,0, as quantified by an exponential covariance function,until the residuals are as large as the assumed data errors[Linde et al. 2006]. Tests using models from the previous timestep as background models gave inferior results, as artifactsappeared at previously occupied positions of the plume.

2.2. Joint Inversion Strategy

[7] Coupling between the ERT and GPR timelapse up-dates dme,p and dmr,p is introduced in the inversion bycrossgradient constraints [Gallardo and Meju, 2004]. Thecrossgradients function of the model updates at timestep p

tp x;y;z rdme;p x;y;z rdmr;p x;y;z 5

is discretized with a centraldifference scheme and subse-quently linearized. Deviations from zero of the discretized

tp are heavily penalized at all discretized locations x, y, zofthe inversion domain with a constant weight for all inversionsteps. The joint inversion proceeds as for the individual in-versions, but with the additional crossgradient constraints.

[8] The assumption of structural similarity between dme,pand dmr,p, as quantified by equation (5), is valid when onlyone state variable varies with time or when the methodsemployed are sensitive to the same physical property (e.g.,electrical conductivity). The assumption holds for vadosezone tracers that have the same electrical conductivity as thepore water such that timelapse ERT and GPR data onlysense changes in moisture content. Simulations and jointinversions of field data acquired following saline tracerinjection (not shown here) reveal that dme,p and dmr,p are

not structurally similar and that the resulting inversionmodels display artifacts.

3. Results

3.1. Site Characteristics

[9] At Hatfield in the UK, a test site was developed tostudy flow and transport in unsaturated media (for detailssee the work by Binley et al. [2002a]). The dominant sublithology at the site is medium grained sandstone, but withfine and medium sandstone subhorizontally laminated on amillimeter scale (occurring in 0.20.5 m thick units, spacedat 13 m vertical intervals). Binley et al. [2002a] document a

water tracer test carried out at the Hatfield site; here we usegeophysical data from this test to illustrate our approach in afield setting. The center of mass and the spread of geophys-icallydefined plumes using individually inverted timelapsedata were previously used to characterize the hydrodynamicsat Hatfield based on individual inversions and allowedderiving fieldscale properties such as effective hydraulicconductivity [Binley et al., 2002a].

[10] For the ERT measurements, 16 stainless steel elec-trodes were installed at 0.73 m intervals between a depth of2 and 13 m in four boreholes in a trapezoidlike mannerwith sidelengths varying between 5 and 8 m. For the GPRmeasurements, two boreholes (along the xaxis) were drilledwith 5 m spacing inbetween one of the diagonals formed bythe ERT boreholes.

[11] Between 14:30 on 7 October and 13:40 on 10 October1998, 2100 l of water tracer was injected at a uniform rateof approximately 30 l/h in a borehole slotted between 3and 3.5 m depth located inbetween the two GPR bore-holes (x = 3; y = 4). To obtain a pure flow (no transport)experiment, the conductivity of the injected water was cho-sen to match the conductivity of the pore water. Multiple

ERT and GPR data sets were acquired before and after tracerinjection. We concentrate below on the timelapse data setsrecorded directly after the end of injection (day 3) and twodays later (day 5).

3.2. Synthetic Example

[12] A synthetic example mimicking the Hatfield waterinjection experiment was first used to evaluate our timelapsejoint inversion. We use a FEFLOW v6.0 Richards equationsolution assuming a uniform geological media. For this wediscretized a region 8 m by 10 m (in plan) and 12 m deep into73,202 6node triangularprism linear finite elements, withspecific refinement around the tracer injection area. Thelower boundary of the region defined a water table, and henceDirichlet boundary conditions. The saturated hydraulic con-ductivity forall elements was set to 4.63 106 ms1, which isconsistent with Binley et al. [2002a]. We used the widelyadopted van Genuchten [1980] representation of unsaturatedhydraulic characteristics, with a residual saturation of 0.0025,exponentnvG= 1.964 and avG= 4.1 m

1. In order to developmore natural initial conditions we first setup a uniform satu-ration of 0.5 within the model and then ran a 20 day drainageperiod. The tracer was then imposed within the model and thetracer movement was simulated with a maximum time step of0.05 days. Synthetic GPR and ERT data were simulated usinginterpolated moisture content at days 0, 3 and 5. The bulkelectrical resistivity r was calculated as a function of satu-

ration S= / (where is porosity) using Archie

s second law[Archie, 1942]

sSn

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where rs = 66 Wm is the bulk resistivity at full saturation andn = 1.13 is Archies saturation exponent determined fromthree samples of the main lithology at the Hatfield site [seeBinley et al., 2002b].

[13] The relative permittivity was calculated using thecomplex refractive index model (CRIM) [Birchak et al.,1974]

ffiffiffi

p 1

ffiffiffiffiffis

p

ffiffiffiffiffiffiw

p

ffiffiffiffiffia

p7

DOETSCH ET AL.: STRUCTURAL TIMELAPSE JOINT INVERSION L24404L24404

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where w = 81 and a = 1 are the relative permittivities ofwater and air, respectively. The porosity = 0.32 and therelative permittivity of the sediment grains s = 5 were

obtained from lab measurements on retrieved cores [Westet al., 2003]. Radar slowness, s, was calculated from thepermittivity using s =

ffiffiffi

p/c, with c the speed of light in a

vacuum.[14] Forward solvers were used to calculate electrical

resistances and radar traveltimes for these models. Theelectrical responses and related sensitivities were computedusing a finiteelement solver implemented by Rcker et al.[2006], and the traveltimes and sensitivities were calculatedin the high frequency limit using a finitedifference algo-rithm [Podvin and Lecomte, 1991]. The ERT measurementscheme included a variety of fourelectrode configurationsusing electrodes in a varying number of boreholes and thedata were filtered to only include configurations with a

geometrical factor of less than 600. The multiple

offsetgathers were calculated using 0.25 m intervals betweenantenna positions over the range 011 m below ground levelfor cases when the angle between the transmitter andreceiver antennas were within 45 from the horizontal. Foreach timestep, the resulting data sets of 833 resistances and1181 multiple offset GPR traveltimes were contaminatedby Gaussian noise according to the error models ofequations (1)(4) with zero mean and standard deviationsstd(ees) = 2.5%, std(ee,0) = std(ee,p) = 0.5% and std(er,0) =std(er,p) = 0.5% + 0.5 ns. The same configurations anddata error descriptions were also used to invert the fielddata described below.

[15] The background data sets (i.e., before water injection)were inverted individually in 3D (not shown) and the timelapse data were inverted both individually and jointly in 3D

using a regular inversion grid with voxel side lengths of0.35 m. Integral scales of the exponential covariance modelused to regularize the inversion of the background data setwere 2 m in the horizontal and 1 m in the vertical direction torespect the independently observed anisotropy at the fieldsite. The integral scale chosen for the timelapse inversionwas 0.7 m in all directions, corresponding to the expectedlength scale at which the tracer plume might be resolved.Values in the range of 0.51.0 m provide similar results.After 10 iterations, all inversion models fit the data to thespecified error level with the largest possible weight to themodel regularization.

[16] The water content was calculated from the resultingmodels using equations (6) and (7) with the petrophysical

parameters mentioned above. Vertical profiles of the inferredtimelapse change in moisture content, D, from the back-ground model are shown in Figure 1. It is seen that themagnitude ofD is rather well estimated in the GPR inver-sion but markedly underestimated in the ERT inversion forboth the individual and joint inversions, which can beexplained by the more significant resolution limitations ofERT inversions [DayLewis et al., 2005].

[17] To quantify the changes we define a plume boundary,for each model, at 1/3 of the maximum D. These plumeswere then used to calculate the mass, center of mass and thevariances of the plumes, with the resulting statistics pre-sented in Table 1. This plume definition is rather simplistic

Figure 1. Vertical profiles of moisture content change (D) since the beginning of a synthetic water injection test asinferred from individual and joint inversion of timelapse crosshole ERT and GPR traveltime data. (a) Through the injectionpoint at the end of injection (day 3), (b) 3.5 m away from the injection point and the GPR acquisition plane at the end ofinjection (day 3), (c, d) same as Figures 1a and 1b but two days after the end of injection (day 5).


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[cf. DayLewis et al., 2007], but the relative differences

between the individual and joint inversions are similar forother cutoff values, and serves here only to investigate ifthe plume definition is improved by the joint inversion. Notethat the 3D shape of the plume is heavily dependent on theregularization used as the data sets are acquired betweenpairs of boreholes.

[18] The individual GPR inversion model overestimatesthe mass (+46% (+58%) for day 3 (day 5)), whereas it ismore reasonable for the joint inversions (+5% (+13%) for

day 3 (day 5)). These results indicate that the ERT datahelp to constrain the geometry of the GPR model thatotherwise is only based on data acquired along one planeand extended in 3D based on the regularization. Theindividual (49% (56%) for day 3 (day 5)) and joint ERT(61% (51%) for day 3 (day 5)) models significantlyunderestimate the mass with no improvement for the jointinversions.

[19] The error in the center of mass from the individualGPR (0.47 m (0.20 m) for day 3 (day 5)) and ERT inversions(0.42 m (0.34 m) for day 3 (day 5)) are improved in the jointinversion (0.18 m and 0.17 m, respectively) for day 3, butless so for day 5 (0.20 m and 0.25 m, respectively). Thehorizontal variances of the estimated GPR and ERT plumesfrom the joint inversions are less overestimated (+100% onaverage) than for the individual inversions (+190% onaverage). This is due to resolution improvements of jointinversions of crosshole data that are most important in thehorizontal direction [Linde et al., 2008].

3.3. Hatfield 1998 Water Injection

[20] Our timelapse inversion methodology was thenapplied to the Hatfield field data. Radar transmission datawere acquired using Sensors and Softwares Pulse EKKOradar system with 100 MHz antennas. ERT measurementswere acquired using the DMT Resecs resistivity instrument.The background data sets were inverted individually in 3Dand the final models (not shown) are consistent with theknown geology and fit the data to the specified error levels(see previous section). The timelapse data were inverted

Table 1. Statistics of the GeophysicallyDefined Plumes for the

Synthetic Example Using CutOffs of One Third of the Maximum

Moisture Content Change (D) for Each GeophysicallyDerived

Modela

SyntheticMass[m3]

Center of Mass [m] Variance [m2]

x y z sxx2

syy2

szz2

Day 3

True 2.12 3.00 4.00 5.05 0.37 0.37 1.00GPR individual 3.06 2.54 3.89 5.07 1.36 1.05 0.61ERT individual 1.08 3.19 4.36 5.17 1.17 1.12 0.92GPR joint 2.20 2.92 4.16 5.07 0.90 0.76 0.57ERT joint 0.81 2.92 4.15 5.08 0.89 0.75 0.57

Day 5True 2.08 3.00 4.00 6.01 0.55 0.55 1.43GPR individual 3.32 2.85 3.98 6.14 1.82 1.32 1.10ERT individual 0.93 3.05 4.34 6.04 1.36 1.34 0.98GPR joint 2.39 3.13 4.20 6.14 1.14 0.92 1.04ERT joint 1.02 3.10 4.20 6.12 1.06 0.83 0.99

aThe ERT data and model weight in the joint inversions is 2.8 times thatof the GPR to assure that both models converge to the same target datamisfit.

Figure 2. As in Figure 1, but with data from the Hatfield 1998 water injection test.


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both individually and jointly in 3D for 12 iterations suchthat all inversion results fit the prescribed error level.

[21] Figure 2 shows vertical profiles of the inferred Dusing equations (6) and (7) with the specified parametervalues. These results are consistent with the syntheticexample in that the joint inversion models are more focused.

Scatter plots of the inversion results show that the scatterobserved in the individual inversions (Figures 3a and 3c) ismuch reduced and that the D magnitudes are increased inthe joint inversions (Figures 3b and 3d). The plumes definedby the fractional thresholding procedure were then used tocalculate the center of mass and the variances of the plumes(see Table 2). The individual GPR inversion model over-estimates the injected mass of 2100 l (+39% (+57%) forday 3 (day 5)), whereas it is more reasonable for the jointinversions (+4% (+37%) for day 3 (day 5)). The individual(50% (53%) for day 3 (day 5)) and joint ERT (56%(54%) for day 3 (day 5)) models both underestimate themass with no improvement for the joint inversions.

[22] The differences in the center of mass between the

individual GPR and ERT models (0.35 m (0.34 m) for day 3(day 5)) are improved for the joint inversion at both time steps (0.05 m (0.18 m) for day 3 (day 5)). The varianceestimates of the GPR and ERT plumes are smaller in thehorizontal direction for the joint compared to the individualinversions (24% on average) indicating that the jointinversion improves resolution.

4. Concluding Remarks

[23] A synthetic experiment based on flow simulationstogether with field data from a water injection experiment inunsaturated sandstone show clearly that crossgradients

joint inversions of crosshole timelapse ERT and GPR tra-veltime data decrease horizontal smearing of plumes, thatthey increase the similarity between models, and improvethe estimated center of mass of plumes compared to indi-vidual timelapse inversions. The examples also illustratethat higher resolution 2D traveltime GPR data might ben-

efit from lower resolution 3

D ERT data. We emphasize thatthe inversion methodology presented here is only validwhen one state variable varies at each location of the modeldomain. For example, if the fluid salinity of the tracer wasdifferent to the native pore water salinity then the resistivitywould be a function of moisture content and salinity, whichwould violate the assumptions of structural similarityunderlying the crossgradient constraints.

Figure 3. Scatter plots of moisture content change (D) inferred from (a) individual and (b) joint timelapse inversions ofthe Hatfield data at day 3, (c, d) same as Figures 3a and 3b but for day 5. The black triangles indicate model cells with GPRray coverage.

Table 2. Statistics of the GeophysicallyDefined Plumes for the

Hatfield Water Injection Experimenta

HatfieldMass[m3]

Center of Mass [m] Variance [m2]

x y z sxx2

syy2

szz2

Day 3GPR individual 2.92 2.87 4.05 4.77 1.78 1.21 0.85ERT individual 1.06 3.07 4.24 4.99 0.79 0.96 0.86GPR joint 2.18 3.13 4.21 4.84 1.02 0.92 0.81ERT joint 0.93 3.10 4.17 4.85 0.87 0.76 0.70

Day 5GPR individual 3.30 3.49 4.08 5.55 2.35 1.44 1.85ERT individual 0.85 3.28 4.34 5.47 1.42 1.29 1.19GPR joint 2.80 3.47 4.31 5.57 1.63 1.36 1.69ERT joint 0.92 3.34 4.23 5.66 1.11 0.84 1.19

aThe ERT data and model weight is 1.8 times that of the GPR in the jointinversions to assure that both models converge to the same target datamisfit.


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[24] Acknowledgments. The field data were acquired by P. Winshipand R. Middleton, formerly of Lancaster University. We thank T. Gnther,A. Tryggvason and collaborators for providing the ERT and traveltimeforward solvers. A. Binley is grateful to A. Green at ETHZ for generoussupport that enabled his contribution during a sabbatical leave. We thankFrederick DayLewis and Luis Gallardo for their reviews. Funding was

provided by the Swiss National Science Foundation.

References

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A. Binley, Lancaster Environment Centre, Lancaster University,Lancaster LA1 4YQ, UK.

J. Doetsch, Institute of Geophysics, ETH Zurich, Sonneggstr. 5, CH8092Zurich, Switzerland. ([email protected])

N. Linde, Institut e of Geophysics , Universit y of Lausanne, CH1015Lausanne, Switzerland.


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