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Joint inversion of geophysicaldata for site characterisation

Deliverable D1.4

Jochen Kamm, Mehrdad Bastani, Laust B. Pedersen,Daniele Boiero, Corrado Calzoni, Valentina Socco,Sebastiano Foti, Alberto Godio, Alessandro Arato, EstherBloem and Helen French


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Deliverable D1.4

Soil Contamination: Advanced integrated characterisation and time-lapse Monitoring

Title Joint inversion of geophysical data for site characterisation

Author Jochen Kamm, Mehrdad Bastani, Laust B. Pedersen, Daniele Boiero,Corrado Calzoni, Valentina Socco, Sebastiano Foti, Alberto Godio,Alessandro Arato, Esther Bloem and Helen French

Report No. Deliverable 1.4

ISBN

Organisation name of lead

contractor for th is deli verable

UUUppsala University

No. of pages 48

Due date of deliverable: May, 2011

Actual date of deli very July, 2011

Disseminati on level PU

Key words Joint inversion, modelling, RadioMagnetoTelluric (RMT), ElectricalResistivity Tomography (ERT)

Title of project: Soil Contamination: Advanced integrated characterisation and time

lapse Monitoring (SoilCAM)

Instrument: 6.3 Environmental technologies, Topic ENV.2007.3.1.2.2, Development oftechnologies and tools for soil contamination assessment and site

Contract number: 212663

Start date of project: June 2008, Duration: 48 months

Project co-funded by the European Commission within the Seventh Framework Programme (2007-2013)

Disclaimer

The information provided and the opinions given in this publication are not necessarily those of the authors or the

EC. The authors and publisher assume no liability for any loss resulting from the use of this report.


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Deliverable D1.4

Summary

Joint inversion of RadioMagnetoTelluric (RMT) and Electrical Resistivity Tomography (ERT) data werecarried out for the Trecate test site with a view to get an enhanced image of the variability in sedimentarystructure above the ground water table as well the geometry of the underlying aquifer. Whereas ERT dataare very sensitive to changes in the upper few meters the RMT data is more sensitive to deepervariations, especially variations in the geometry of conductive zones like aquifers. Joint inversion along

profiles of the two datasets thus gives a more complete image than each individual inversion image.However, it is unclear as to whether the many small details observed in the images for the upper 10meters can be trusted. Only observations in boreholes can shed light on this question.

In addition to the joint inversion along profiles, for the first time 3D inversion of the RMT data wereattempted. Compared with the 2D models the 3D models show a more consistent and distinct image of

the presumed water table at about 12 meters depth, but less details in the upper few meters than the jointinversion result. The inversion results were validated by comparing with cross-borehole results at asingle pair of boreholes in the area. A remarkable coincidence between the resistivity in the 3D modeldistribution between the boreholes and the 3D image were found in the whole depth interval from thesurface down to a depth of 16 meters. This might indicate that the 3D model is a better image than themore scattered image obtained from the joint inversion of RMT and VES data, because a 2D modelapproach is too simplified even for the relatively undisturbed sedimentary sequence found at Trecate.

Joint inversion of resistivity and radar travel time data have been performed at the Moreppen test site,bordering Gardermoen airport with a view to get an enhanced image of the time evolution of tracerconcentration in the vadose zone upon snowmelt. The cross-gradient approach was used to search formodels that possibly share a common geometry. The common time-evolution of the models can most

easily be seen when visualising the change of the estimated parameters. The resistivity models showmany small scale anomalies, probably inversion artefacts due to bad data quality. However, on a coursescale a simultaneous and collocated decrease of the respective model parameters can be observed, andthe approximate location of the melt-water front can be located as a function of time in both images. Thestrongest dynamic changes occurred over a short time interval from April 3 to April 12 in the depthinterval from 0 to 4 meters.

Due to the conspicuous data quality of the ERT data at the Moreppen site it was concluded that moreadvanced attempts to estimate hydrological parameters such as water saturation as a function of timewould not be feasible. Instead synthetic data were constructed to test our new algorithm for estimatingfrom time lapse measurements both static fields (parameters) such as porosity and dynamic fields

(parameters) such as water saturation. The result was that even with a joint inversion approach and timelapse measurements it is not possible to determine those fields while at the same time not knowing theempirical relations like those found in Archies law. The parameters in the empirical laws need to beestimated from sample measurements in situ or in the laboratory.

In addition to the experiments mentioned above this report also describe additional very high qualityradar measurements conducted between two boreholes at the Trecate site, where also ERT measurementshad been performed. Because of the electrode array used (pole-dipole) for the ERT measurements wewere not able yet to conduct a joint inversion with present version of the program.

Finally a new approach is presented for the joint inversion of seismic P-wave velocities and surface wavedispersion curves for improved reliability of site characterisation and for estimating the porosity in the

saturated zone in shallow aquifers.


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Deliverable D1.4

Content

1. INTRODUCTION ............................................................................................................. 12. JOINT INVERSION OF GEOPHYSICAL DATA AT TRECATE SITE ...................... 32.1 2D joint inversion of surface ERT and RMT data ....................................................................... 32.2 3D individual inversion of RMT data ............................................................................................ 62.3 Comparison between 2D joint inversion and 3D individual inversion results ........................... 62.4 3D imaging of the aquifer using 3D RMT resistivity model ...................................................... 122.5 Discussion on the joint 2D and individual 3D inversion results ................................................ 143. JOINT INVERSION OF CROSSHOLE DATA AT MOREPPEN & TRECATE ...... 163.1 Implementation .............................................................................................................................. 163.2 Moreppen, site and measurement ................................................................................................ 173.3 Data acquisition, inspection and processing ............................................................................... 17

3.3.1 Ground penetrating radar (GPR) .............................................................................................. 173.3.2 Electrical resistivity tomography (ERT) .................................................................................. 18

3.4 Joint inversion ............................................................................................................................... 213.4.1 Joint inversion with Cross-gradient constraints ....................................................................... 213.4.1.1 Numerical experiments on synthetic models ......................................................................... 223.4.1.2 Results of joint inversion of a field data set .......................................................................... 253.4.2. Joint inversion based on physical laws .................................................................................... 32

3.5 Inversion of Cross-hole GPR data in Trecate ............................................................................. 354. JOINT INVERSION OF COMPRESSIONAL AND SURFACE WAVES FOR SOIL

POROSITY ESTIMATE ....................................................................................................... 394.1Background ..................................................................................................................................... 39

4.2 Inversion Scheme ........................................................................................................................... 404.3 Synthetic Example ......................................................................................................................... 414.4 Example on real data .................................................................................................................... 434.5 Porosity estimate............................................................................................................................ 434.6 Conclusions of joint inversion of p-wave and surface wave ...................................................... 443. REFERENCES ................................................................................................................... 45


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Deliverable D1.4

APPENDIX AJOINT INVERSION RESULTS, MOREPPEN, NORTH WALL OF

LYSIMETER TRENCH ........................................................................................................ 49


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1. Introduction

In the frame of the SoilCAM project multiple geophysical datasets have been acquired with

the aim of generating models of corresponding physical parameters e.g. electrical resistivityor radar velocity to elucidate more details for example about geological structures, geometryof pollution plume, time variation of contamination front, etc. The geophysical data werecollocated to enable us to either make a joint interpretation or a joint inverse modelling, a so-called joint inversion. The collected data can be modelled in two- or three-dimensional (2Dand 3D) spaces depending on the data acquisition geometry and parameters. In the last fewdecades a vast amount of work has been carried out to develop algorithms for the 2D jointinversion of multiple geophysical datasets. Vozoff and Jupp (1977) were among the first to

jointly invert two geophysical datasets. They generated synthetic electrical resistivity andmagnetotelluric (MT) datasets from a three-layer model and showed that resolution at certain

depths would be slightly enhanced by including the higher frequency MT data, while byadditional DC data at larger spacing no improvements could be observed. The joint inversionwas then applied to the electrical resistivity and magnetotelluric data collected in a shallowsedimentary basin. Lines et al. (1988) describe the concepts of cooperative interpretation ofmultiple geophysical data by referring to the paper by Golizdra (1980) where three types ofapproaches for the inversion of gravity and seismic datasets were explained. The approachesintroduced are separate inversions (no coupling), unified inversions (coupling between

physical properties, e.g. density and seismic velocity) and mixed inversions (assuming arelationship between the models or structures in the models). Liners et al. (1988) call the latter

joint inversion. Haber and Oldenburg (1997) assume that two models resulting from a jointinversion should have the same structure and accordingly define a structure operator thatmakes use of local spatial gradients or curvature of the estimated model. The operator selected

by Haber and Oldenburg (1997) for the joint inversion is a measure defined as the sum ofsquares of differences between the curvatures of the two models in a predefineddomain/space. They then minimize the measure subject to fitting both datasets to a

predetermined level. More recently Gallardo and Meju (2003) introduced the cross-gradientmethod to jointly invert Direct Current (DC) electrical resistivity and seismic velocity data in2D. Later they used the same approach (see Gallardo and Meju, 2007) to make 2D joint

inversion of MT and seismic travel-time data to classify petrophysical and structuralvariations based on the resulting electrical resistivity and seismic velocity models. Linde et al.(2006) used the cross-gradient method combined with a stochastic regularization operator toinvert cross-hole electrical resistivity and ground-penetrating radar data to improve the hydro-geophysical characterization of Sherwood Sandstone in an area close to Eggborough, NorthYorkshire, UK. Later in another paper Linde et al. (2008) applied the same concept to bothsynthetic and field cross-hole GPR and seismic traveltime datasets. They also compare theresults of joint inversions with individual inversions of each dataset by presenting scatter plotsof estimated electrical resistivities and seismic velocities and show that the joint inversionresults are much less scattered than the individual ones.


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In this deliverable the results of joint inversion of collocated geophysical data from the

Trecate and Gardermoen test sites are presented. First the resistivity models from the 2D jointinversion of ERT and RMT data acquired at the Trecate site are shown. The results ofindividual 2D inversions of each dataset are presented in Deliverable 1.1. We also show theresistivity models from the individual 3D inversion of RMT data and compare them with the2D resistivity models from the 2D joint inversion. Both sets of models are also compared withthe resistivity models from inversion of cross-hole resistivity data prepared by POLITO (seeDeliverables 1.2 and 1.3). Finally the 3D resistivity model is compared to the lithologicalinformation collected in the boreholes located in the study area to make a semi 3D lithologicalmodel over the study area. Then the results from joint inversion of cross-hole resistivity andIP data carried out by POLITO at Trecate site are presented. Uppsala University and POLITOconducted cross-hole ERT and GPR data in mid November 2010. Both datasets were jointlyinverted by Uppsala University to study the spatial distribution of smearing zone by analysingthe resulting electrical resistivity and GPR velocity models. Finally, the electrical resistivityand radar velocity models as a results of the joint inversion of cross-hole time-lapse electricalresistivity (collected by Bioforsk) and GPR data (acquired by Uppsala University) inMoreppen area are presented.


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2. Joint inversion of geophysical data atTrecate site

In this section we present the results of joint inversion of collocated geophysical datasets thatwere collected with different methods to image details of the geometry of the aquifer as wellas the smearing zone at the Trecate site. Resistivity models from the 2D joint inversion ofsurface ERT and RMT data are presented first and then compared to the resistivity modelsfrom individual 3D inversion of the RMT data.

2.1 2D joint inversion of surface ERT and RMT data

As mentioned in the introduction, joint inversion of MT and electrical resistivity data date

back to 1977 (see Vozoff and Jupp, 1977) with the aim to increase the resolution of theresulting resistivity models by benefiting from the fact that each method has differentresolution and depth penetration. Recently Candansayar and Tezkan (2008) introduced analgorithm for the joint inversion of ERT and RMT data using smoothness constraints. In their

paper they mention that the RMT data are not very sensitive to the near surface resistivityvariations due to the limited frequency range used, while at the same time ERT data withsmall off-sets have relatively high resolution to detect near surface resistivity variations. Onthe other hand RMT data carry more information about the deeper structures than the ERTdata. Therefore when the two datasets are jointly inverted more details over a wider depth

range can be extracted from the resistivity models. We have used program EMILIA (Electro-Magnetic Inversion with Least Intricate Algorithms) for the joint inversion of RMT and ERTdata. The program is developed by Thomas Kalscheuer (see Kalscheuer et al., 2010) in whichthe code for the RMT forward and sensitivity calculations is based on the REBOCC programdeveloped by Siripunvaraporn and Egbert (2000). DCR and RMT data were inverted in 2Dwith a smoothness-constrained scheme. Details of the joint inversion can be found inKalscheuer et al. (2010). Here we describe some of the parameters used in the joint 2Dinversion of ERT and RMT data.

Location of the ERT and RMT profiles are respectively shown in Figures 2.6 and 2.7 of

Deliverable 1.2. We have made a single figure of the ERT and RMT profile locations (see,Figure 2.1) to facilitate the referencing made in this deliverable. The ERT and RMT data wereacquired by POLITO and UU, respectively. Profiles E1 and E2 are 108 m long and the ERTdata were collected using 72 electrodes arranged in a Wenner configuration with a minimumelectrode separation of 1.5 m. Profiles E3 to E10 are 96 m long and the ERT data werecollected using the same array configuration and a minimum electrode spacing of 2 m. TheRMT data were collected along the same profiles, marked as R2 to R11, with a stationspacing of 10 m and 11 stations per profile. There were also some extra RMT profiles thatwill not be shown in this report.


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The results and details of individual inversions of each dataset have been reported indeliverable D1.1. The joint inversion parameters were adjusted after running a few inversionsand checking the overall datafits including both the ERT and the RMT data. Figure 2.2 is aschematic that shows the model discretization which is based on the geometry of the fieldsetup. The lateral cell size within the measuring area is half the minimum electrode separation(in this case 2 m) used in the ERT measurements to allow for more accurate calculation of theERT sensitivity matrix. The cell thickness is increased with depth logarithmically. In theexample shown in Figure 2.2 the rate of increase is by a factor of 101/20meaning that the cellsin row 20 are 10 times thicker than those in row 1. The width of the cells outside themeasuring area (both to the left and to the right) also increases logarithmically.

The regularization used for the inversion is of Occam type where a smooth model is soughtsubject to fitting the data to a predefined level. We have used a smoothing anisotropy of 3

where the lateral smoothing is three times the vertical one. The ERT and RMT data wereweighted equally when minimizing the objective function (see Kalscheuer et al., 2010) toestimate the model parameters, namely resistivity of the cells. The target overall RMS whichis a standard measure of datafit was taken to be 1.

Figure 2.1: Location of ERT (blue text) and RMT (red text) profiles measured in the Trecate site.


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Figure 2.2: Schematic of model discritization for the 2D joint inversion of ERT and RMT.

Due to the electrode separation along profiles E1 and E2 (minimum a of 1.5 m) andalso the

RMT station spacing it was not possible to carry out a joint inversion of ERT and RMT dataalong the two profiles. It can be easily seen in Figure 2.2 that the measuring points should becollocated with cell nodes. The ERT and RMT setup used do not allow this collocation.

Prior to presenting the results of the 2D joint inversion and individual 3D inversion of theRMT data we summarize some of the key concepts in RMT data acquisition. As explained insection 2.1.4 of deliverable D1.1, three components of the magnetic field and two horizontalcomponents of the electric field from the electromagnetic signal in the frequency range 10-250 kHz are measured in the RMT method. The two horizontal electric field components andthe vertical magnetic field component are related to the horizontal magnetic field components

via the earth electromagnetic transfer functions, namely impedance tensor (Z) and tippervector (T), respectively. In the frequency domain the relationship between is very simple andis given as

)()(

)()()(,

)(

)()(

)(

)(

fZfZ

fZfZf

fH

fHf

fE

fE

yyyx

xyxx

y

x

y

xZZ (1)

wherex,yand fdenote the north-south direction, the east-west direction and the signalfrequency, respectively.


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Zis a complex tensor and contains information about the electrical resistivity variations in3D. In the 2D joint inversions we have used the determinant of Zat each frequency to reducethe 3D effects (see, Pedersen and Engels, 2005).

2.2 3D individual inversion of RMT data

Three-dimensional inversion codes for magnetotelluric (MT) and controlled source MT datahave become more and more powerful, even though they remain computationally expensive.Recent processing codes have been developed and applied by e.g. Farquharson (2002),

Newman (2003), and Zhdanov (2010) on a variety of targets.

Available for the academic community is a single processor version of the algorithm

WSINV3DMT by Siripunvaraporn 2005, which is a data-space based Occam-like minimum-structure inversion scheme. Using this code all elements of the impedance tensor shown in (1)can be included. When applying 3D inversion it is no longer necessary to meet anydimensionality assumptions. Additionally, any local inhomogeneities can be accounted for inthe modeling/inversion process, whereas in the 2D case only those along the profile directioncould be taken into account, and furthermore the resulting 2D models can be heavily biased

by the distortions caused by off-profile conductivity variations (3D structures).

For convenience, prior to the 3D inversion all the tensor data were rotated by -15 into a localcoordinate system in which the y-axis is along the measured profiles. The model space was

divided into 313128 cells; horizontal cell sizes were designed to allow for two cellsbetween neighboring sites (5 m long profiles and 4 m across them). The vertical discritizationwas the same as in the 2D case (see Figure 2.2), starting with a cell size of 0.5 m. Weincluded the data of 104 sites on lines R2-R11 (excluding five noisy sites on line 7) and allnine measured frequencies (10-250 kHz). An error floor of 5% (on off-diagonal elements,ZxyandZyx) was applied to the main off-diagonal impedance elements and the same absolute errorfloor was applied to the corresponding additional diagonal impedance elements, Zxxand Zyy,respectively. The starting model was a homogeneous half space of 1000 Ohm-m. After fiveiterations (that each took ca. 8 h on a 2.6 GHz desktop computer) the final model with an

overall RMS of 1.5 was achieved.

2.3 Comparison between 2D joint inversion and 3D individualinversion results

In order to compare the results from the 2D joint inversion of ERT and RMT data with thosefrom the individual 3D inversion of RMT data we have chosen the resistivity sections fromthe 3D resistivity model where they coincide with the ERT+RMT profiles. Figures 2.3a and2.3b depict eight resistivity sections extracted from the 3D model along profiles R4-R11 (or


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E3-E10, shown in the right columns) and their corresponding resistivity models from the 2Djoint inversion (models in the left column).


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Figure 2.3: A comparison between the results of individual 3D inversion of RMT data (models to theright) and joint 2D inversion of ERT and RMT (models to the left) at Trecate site. a) Models alongprofiles R4-R7 (E3-E6) and b) models along profiles R8-R11 (E7-E10).


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The resistivity models show similar trends and have clear differences in details. We firstdiscuss the similarities and then address the differences.

Figure 2.4: Location and Lithological logs of three boreholes B-I, B-J, and B-N located closest to thearea covered by ERT and RMT measurements. The information is extracted from deliverable D3.1.

For the comparison we have also used information from deliverable D3.1 mainly shown inFigures 2.1-1, 2.1-2a, 2.1-2b, and 2.1-3. We have compiled the information in Figure 2.4 tocompare the resistivity models with the existing information from the boreholes in the area.

Figure 2.5 shows the lithological logs from boreholes B-I, B-J and B-N that were closest tothe study area. According to the information given in deliverables D1.1 and D3.1 at theTrecate site subsurface consists mainly of low-consolidated sediments (sand, silt and gravel).Boreholes marked in Figure 2.4 with a B belong to the network that focuses on the plumecore. Stratigraphical descriptions show alternation of gravel in silty sandy matrix andsilty/sandy layers, down to a depth of more than 10 m (deliverable D3.1). Gravel is seenalmost in all boreholes, whereas silty sandy layers are indicated as rather thin layers in the

boreholes. Gravel is found even in the top layer in point B-J, the top layers consist mainly ofsandy silt, sand, silty sand in points B-N, B-O, B-P, with some gravel. The stratigraphic data

in boreholes B-I and B-J show similarly that gravel in a silty, sandy matrix is dominant from


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near soil surface down to a depth of more than 6 m. Dominance of gravel layers in sandy siltmatrix results in high permeability in most part of the Trecate site. Top soil and silty sandlayers have lower permeability.

In deliverable D6.1 it is concluded that in the Trecate site there are three main layers that canbe taken into consideration:

- Layer 1: top soil (siltysandy organic composition)

- Layer 2: silty sand layer with gravel

- Layer 3: gravel in siltysandy matrix.

Relatively coarse material (layer 3) is dominant at the site, and layer 2 is not present under the

top soil in some areas.

Both models show a thin low resistivity layer at the topmost part that thins out towards south.This layer might be interpreted as the siltysandy top soil layer with organic composition (seeFigure 2.4). Models from the 2D joint inversions show more detailed variations compared tothe 3D models and the reason is due to the following facts: a) The ERT data with a 2 melectrode separation contain more information and have higher resolution power at the surfacethan the RMT data and b) the cell width used in the 3D inversion (5 m) is five times coarserthan the width of cells (1 m) in the 2D inversion. This means that the lateral resistivityvariation seen in the 3D model is an average value of 5 cells in the 2D model. A considerably

more resistive layer underlies the first layer that is thicker towards WSW and has a changingresistivity and thickness. This layer might represent coarser sediments, namely the gravelsshown in Figure 2.4. The resistivity of this layer is generally estimated higher with the 2D

joint inversion than those with the 3D inversion. The second layer appears thicker in the 2Djoint inversion models than the 3D which might be due to the less sensitivity of ERT data tothe lower resistivity below (see synthetic examples shown in Kalscheuer et al., 2010). Thethird layer that most probably is the saturated aquifer has a significantly lower resistivity thanthe overlying layers. The change in the resistivity might be also related to the changes in thelithology where, according to lithological logs, below a depth of 10 m more silty sands have

been detected. The third layer is located deeper in the 2D joint inversion models than inresistivity models from the 3D inversion. In order to validate the estimated depth to the thirdlayer we have used some of the materials presented in the previous deliverables. Indeliverable D1.3 page 96 the estimated water content from 2D modeling of MRS datacollected by IRIS and POLITO were shown in Figure 6.8. We show the same figure here (seeFigure 2.5) to find qualitative correlations between the water content the resistivity models.


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Figure 2.5: 2D distribution of water content (1 = 100%) from 2D inversion of MRS data collectedalong a profile in the Trecate site. For the exact location of profile see Figure 5.14 in deliverable D1.3.This figure is a copy of Figure 6.8 shown in deliverable D1.3.

As seen from Figure 2.5 below a depth of 14-20 m (marked by the first black line) the watercontent increases considerably and this depth can be considered as the top of the saturatedzone, named as the first aquifer in deliverable D1.3. The second black line marks the bottomof high water content zone (Figure 6.8) and below this line the estimated water content dropsclearly. Comparing the water content model with the resistivity models shown in Figure 2.3aand 2.3b one perceives that the low resistivity zone, namely layer 3, coincides very well with

the highest water content zone, especially in the models extracted from the 3D resistivitymodels.

We have also used the results shown in deliverable D1.1 (page 33) where the resistivity modelfrom the inversion of cross-hole ERT data was compared to the part of resistivity model fromthe 2D inversion of surface ERT data. In Figure 2.6 we show these two models together withthe resistivity cross-section extracted from the 3D resistivity model of RMT data along profileR2 (E1), and also the location of boreholes B-S3 and B-S4 (see Figure 3.4 in D1.1 for moredetails). The part of 3D model that coincides with the area within the two boreholes isenlarged to be compared with the other two cross-sections. As it is clearly seen the resistivity

models from the cross-hole ERT and 3D RMT inversion show more correlation than theresistivity model from 2D inversion of the surface ERT data. Two distinct features to point atare the more resistive feature at 4-10 m depth and the underlying lower resistivity zone. Itseems that the depth and resistivity estimates in the cross-hole model and 3D RMT model areclearly more similar. However, there are obviously some differences between the two modelsthat are mainly because of the fact that a higher resolution exists in the cross-hole ERT dataand also the coarser block size used in the 3D inversion (partly because of reducing thecalculation time and partly RMT station separation that controls the resolution).


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Figure 2.6: A comparison between the resistivity models from the 3D inversion of RMT data, 2Dinversion of cross-hole ERT data and the 2D inversion of surface ERT data in the Trecate site. Modelsmarked by B and C are taken from Figure 3.3 in deliverable D1.1. Note that the colour scales are thelog10 of the estimated resistivities.

2.4 3D imaging of the aquifer using 3D RMT resistivity model

As discussed above the 3D resistivity model can be used as a reasonably accurate tool toimage the geometry of unsaturated and saturated zones, namely the vadose zone and theaquifer, in three dimensions. However, effects of regularization used in the inversion processmay always propagate into the final model and therefore this model has to be interpreted withcare. The model, when combined with the other information for example water conductivity,can provide useful information about variation of porosity in 3D. Since the clay content is

small (see borehole logs shown in Figure 2.4) Archies law provides a good approximation toconstruct 3D porosity models from the 3D electrical resistivity model.

The resistivity model from 3D inversion of RMT data is shown in three dimensions in Figure2.7. The dominating layers mentioned above are clearly seen in this figure. The moreconductive layer at depths of below 10 meters is interpreted to represent the water saturatedlayer and the zone above to be the vadose zone. To facilitate the understanding of thegeometry of these layers we have emphasized those cells which lie in a given resistivityrange, which in turn may be related to a change either in lithology or in water saturation.

Figure 2.8a depicts the 3D model where the cells with an electrical resistivity < 300 m are

clipped. In Figure 2.8b we show two iso-resistivity surfaces of 300 and 1000 m


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Figure 2.7: Resistivity model from 3D inversion of RMT data in the Trecate site.

Figure 2.8: a) Clipped resistivity cell for values below 300 m. b) Iso-Resistivity surfaces (red 1000

and orange 300 m) to show the approximate morphology of the unsaturated layer in 3D.


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to image the morphology of the unsaturated zone in 3D. Ascan be seen, this zone is thicker inthe central part and deepens towards the south. This is in good agreement with the boreholeinformation shown in Figure 2.4 indicating that the coarse grain gravel layer is thicker in

boreholes B-J and B-N than in borehole B-I. The morphology of saturated layer isapproximated by making iso-resistivity surface of 100m that is shown in Figure 2.9 in threedifferent views. To see more details the surface with 50 m is also shown. The thickness ofthe water saturated layer reaches its maximum at the centre of the study area.

Figure 2.9: Iso-resistivity surface for 50 and 100 Wm to image the morphology of the saturated

layer/zone in 3D using 3D RMT resistivity model. Three views are presented to show more details.

2.5 Discussion on the joint 2D and individual 3D inversion results

From the results shown in this section we can conclude that

- The joint 2D inversion of ERT and RMT data provided more details of the variation ofelectrical resistivity at the surface.

- 3D inversion of the RMT data has a better resolution of deeper conductors andprovides resistivity values that are closer to those measured in the cross-hole

measurements.


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- 3D inversion of RMT data is less affected by the near surface inhomogeneities.

In Figure 2.9 we have shown both unsaturated and saturated zones imaged from the 3Dresistivity model and we also refer to the discussion made in Deliverable 1.1 where POLITO

provides some facts about the available data in that area. They write:

The water conductivity is in the range between 450-620 S/cm, with a decrease from the topof the saturated zone to the deeper level; these values are equivalent to water resistivity of 16-22 Ohm-m. This leads to an estimate in the saturated zone of the bulk conductivity in therange of 75-100 S/cm or 100-135 Ohm-m (soil porosity = 0.4; Archie parameters: a=1,m=2). From page 41 D1.1.

Considering this information and the structure shown in Figure 2.9it is possible to make anestimate of the porosity variation in 3D to be used for transport modelling.

Figure 2.9: Morphology of saturated and unsaturated zones estimated from 3D RMT resistivity modelin the Trecate area.


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3. Joint inversion of Crosshole Data atMoreppen & Trecate

Time-lapse Crosshole Radar Tomography (GPR) data (by Uppsala University) and CrossholeElectrical Tomography (ERT) data (by Bioforsk) have been collected. The overlapping periodwas the critical phase of a tracer measurement, the snow melt period between 2010-03-29 and2010-04-21.

We have implemented two different inversion methods for the inversion of time-lapse GPRand ERT datasets. The first step was to apply the methods to synthetic data (section 3.4.2) totest their functionality and optimise the algorithm. The first approach is the well-known jointinversion with cross gradient constraints to enforce structural similarity. It was applied to oneofthe field data sets (section 3.4.3).

In the second approach (section 3.4.4), we try to study the feasibility of a joint inversion interms of estimation of hydrological subsurface parameters and water saturation state based onempirical laws with unknown but bounded coefficients, that are determined during theinversion as well. However, the results were not suitable to apply the method to the field data,especially when taking the poor data quality of the Moreppen data set into account.

Details of the geological characterisation of the Moreppen test site, design and evaluation ofthe tracer experiment, GPR and ERT measurement set-up, GPR data treatment can be foundin Deliverable 1.3, section 4 along with results of the individual interpretations of those datasets. Only the relevant information will be restated here.

The resistivity and radar velocity models from the individual inversion of cross-hole ERT andradar data that were acquired in a field campaign by POLITO and UU are delivered.

3.1 Implementation

The implementation of the inversions was carried out in Matlab. The forward routine for ERThas been developed in FORTRAN by Naser Meqbel in the context of Kalscheuer et al. 2010with modifications for the borehole configuration following (A. Pidleysecky) and was

interfaced with Matlab as a MEX routine. For travel time data the C-routine PSTOMO_eq6-62 (Tryggvason 1998) has been interfaced as a MEX routine as well.

The inversion is a general package that allows exchange and combinations of various forward,transformation, filtering and regularization operators. Using this package, all kinds of jointand single inversions can be derived as long as the necessary operators have already beenimplemented. The core routines of the package currently consist of 11287 code lines in 153files, not counting the interfaced C and FORTRAN routines. It is fully object-oriented andtherefore easily extendible. The corresponding data processing is tied into this core.


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The inversions shown here are all carried out using the Quasi-Gauss-Newton algorithm(Tarantola 1982). The regularization parameter was adjusted by an automatic OCCAMscheme (Constable, 1987).

3.2 Moreppen, site and measurement

Data were acquired at Moreppen, near Oslos Gardermoen airport in Norway (Figure 3.1) byUppsala University (UU) with assistance from the National Research-Development Institutefor Environmental Protection (ICIM) with an instrument rented from NGI (see Farmani,2008).

Moreppen is a test site which is geologically similar to the airport site. Hence, in order toconclude on possible processes at the airport, measurements at Moreppen are valid for

extrapolation.The geological environment consists of glacial sediments, mostly sands and gravel. Thegroundwater table lies at approx. 4 m depth. Two pairs of boreholes (EF and CD) were drilledthrough the vadose zone (Figure 3.2) down to the groundwater table. Borehole separation was2.2 m for the GPR measurements. The borehole pairs for the ERT encompass them with aseparation of 3.2 m. The boreholes are located next to a trench that has been dug before inorder to characterise the site.

On 26.03.2010, before the start of the snow melt period, a highly conductive bromide tracerwas brought upon the snow together with de-icing chemicals (potassium formate and

propylene glycol). Then ERT and GPR measurements were carried out in order to monitor thetransport of the de-icing chemical with the melt water. It is expected that the conductivityanomaly from the melting water is augmented by the tracer, thus enhancing its visibility in theresistivity data.

In this chapter, we present results from the processing and inversion of data acquired in theborehole pair CD (i.e. the South wall of the trench). The results from borehole pair EF (i.e. theNorth wall) are shown in appendix A. They are significantly less useful due to worse qualityof the ERT data.

3.3 Data acquisition, inspection and processing

3.3.1 Ground penetrating radar (GPR)

The transmitter and receiver positions were varied to all possible combinations between theboreholes with a vertical spacing of 20 cm down to 4.2 m depth. During 19 measurementdays, a total number of 9614 radar traces has been collected in borehole pair CD.


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We refer the reader to Deliverables D1.1 and D1.2 where we address some of the peculiarproblems with the GPR data inherited and the way we treated them. There, we also describethe data processing tool and workflow, as it was developed for the GPR data.

A total number of 6619 out of 9614 radar traces could be used.A data error of 0.8 ns wasassumed in order to account for all the reported problems and to minimize the artefacts causedby previously described systematic errors. The traces recorded with a source or receiver above1 m depth was mostly contaminated by air waves travelling over instead of through the soiland could not be used. Below the ground water table (ca.4 m), almost no traces could be picked dueto the strong damping of the water.

Figure 3.1: Moreppen site location (French et al., 2002).

3.3.2Electrical resistivity tomography (ERT)

Electrodes were installed at vertical positions in the boreholes every 15 cm down to 5 depth.The recording device used was a Multi-Channel Syscal Pro device by IRIS instruments. TheERT data were collected with a dipole-dipole electrode configuration and a dipole separation

of 3electrode distances. A total number of 31444 independent (non-reciprocal) apparent


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resistivities were measured in the bore hole pair CD. Receiver dipoles were located within thesame borehole as well as the opposite borehole. In order to obtain a reliable estimate of thedata quality, reciprocal measurements have been performed, i. e. transmitter (currentinjection) and receiver (potential) dipoles were exchanged. Theory predicts equal values for ameasurement and its reciprocal repetition. Therefore deviations can be considered as anestimate for the measurement error.

Additionally, the device also reports a data quality factor Q obtained during the stackingprocess. This estimate does not account for any systematic (reproducible) error and is usuallyorders of magnitude lower than the reciprocal error. Consequently, it gives underestimation ofthe data error and is not used in the inversion.

Previously, we reported on the development of a data inspection, visualization, and processingtool for Crosshole GPR data (see D1.3). A similar tool for the inspection of ERT data has

been developed (Figure 3.3). It allows data visualization in various spatial or spatio-temporalpseudo section arrangements and interactive or semi-automatic removal of data points (singledata points, electrode related data points or thresholding different quantities like Q, reciprocalerror, apparent resistivity or geometry factor).

The steps of data rejection were: 1) Removal of data points with bad reproducibility Q. 2)Visual inspection of data. As the errors related to the electrodes may be found due to theirlinear signatures in the pseudo sections, the inspection tool was used to spot and remove thosevalues. Individual outlier values were also removed. 3) Inspection of the reciprocal errordistribution (Figure 3.4). As the distribution is very wide, no Gaussian error model could be

fit to it. As Gaussian statistics, i.e. normally distributed errors, are a prerequisite to ourinversion algorithm, values with reciprocal errors above 10 % were truncated. After that, aGaussian distribution could be fit to it. It is clear, though, that the data does in reality notfollow Gaussian statistics. Consequently, even though the fitted distribution has a much lowerstandard deviation, we assume a data error of 10% Gaussian noise to account for theinaccuracy of the error model. Moreover, the 2D assumption of the model may not be valid,since, in addition to other possible 3D effects, the lysimeter trench is located only ca. 2 maway from the boreholes. The remaining data set consisted of 9675 apparent resistivities.


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Figure 3.2: Borehole positioning in Moreppen in relation to the trench. In boreholes (dots) crossholeGPR measurements (Uppsala University, inner boreholes, with letters) and crosshole ERTmeasurements (Bioforsk, out boreholes) have been carried out.

Figure 3.3: Screenshot of Geoelectrics Visualization and Processing tool.


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Figure 3.4: Histograms of relative reciprocal errors. The data is made fit to a Gaussian error model by

a sufficiently high threshold.

3.4 Joint inversion

Geophysical data sets obtained from a common area at the same time are probably inherentlyrelated by the target. Joint inversion is the task of incorporating this similarity assumption intoa combined inversion of several data sets. This might improve the usage of measured data by

reducing the number of models that can explain the individual data sets to those that explainall data sets while fulfilling a specified assumption of connection between the resultingmodels. As the true nature of these connections is generally unknown, the constraints andtheir respective levels enforcement have to be appropriate. Otherwise, possible choices can bestructural constraints, such as the Cross-Gradient constraint (Gallardo and Meju, 2007, Lindeet al., 2006) or known or unknown physical relationships between the quantities or empiricalapproximations thereof.

3.4.1 Joint inversion with Cross-gradient constraints

Generally, when performing least-squares inversion, a subsurface model is sought that is thenconsidered to be a particular solution to the inverse problem, when the difference between itsresponse and the measured data (in a L2-norm sense) falls within the confidence limits of themeasurement. That is equivalent to the condition of achieving an RMS error in the vicinity of1. Additional constraints may (or have to) be applied to limit the number of possible models,to stabilise the inversion, i. e. ensuring convergence, and in order to obtain geologically soundmodels that match our a priori information.

In the joint inversions of two collocated physical quantities m1 and m2 that have been

examined by two individual data sets d1 and d2, we perform a least-squares inversion


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simultaneously on both data sets and connect them by applying constraints involving bothmodels and/or both data sets at the same time.

The cross gradient constraint

t(m1,m2)= 21 mm

was described by Gallardo and Meju, 2007. Minimizing its magnitude along with the datamisfits means to minimize cross-product between the gradient fields of the spatiallycollocated models along with the data fit. It is optimal (its square norm is zero) when bothmodel gradients are either parallel or anti-parallel, or when one of the models has no gradient.

Therefore, as soon as one model exhibits a structure in terms of gradients, the other modelwill be driven to adapt the same structural features.

3.4.1.1 Numerical experiments on synthetic models

The synthetic experiment is identically set up as the Moreppen experiments, i.e. we used thesame configuration of measurements and boreholes as in Moreppen. The error levels assumedwere 0.8 ns Gaussian noise on the travel-times and 10% error on the apparent resistivities,

which is identical to the assumptions made for the Moreppen data. However, in the syntheticcase, the true error model as well as the true velocity and resistivity model is known (Figure3.5, right column). Additionally, we omit the time lapse aspect of the data in order to saveinversion run time. We invert the synthetic data sets individually (no cross gradients or otherconnection between the data sets or models) and afterwards jointly with cross-gradientconstraints added. Then we compare the resulting models with the synthetic input model.

First, the weighting factors of different constraints were optimised for the single inversions.For the ERT data, we observed that a strong damping constraint and only a very weaksmoothness constraint (first difference flatness operator) are adequate. These constraints are

described in detail in Menke (1989).We conclude that the ERT sensitivity imposes strongsmoothness by itself. The electric fields are spatially smooth and so are the ERT sensitivities.On the other hand, the ERT inversion convergences slower than the GPR inversion, whichalong with a strong need for damping constraint indicates stronger non-linearity. Additionally,the sensitivities are concentrated around the electrodes. The data can mostly be explained bystructures around the boreholes. In order to image the volume in between, it was helpful to re-weight the damping according to the sensitivities, thus punishing changes close to theelectrodes and encouraging them farther away.

The GPR tomography problem behaves very different. It does not need a lot of damping as

the problem is less non-linear. The sensitivities, as they follow from the tracing of rays are not


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smooth in model space. Therefore, strong smoothing is necessary to ensure non-scatteredmodels.

The results of single 2D inversions of each dataset are shown in (Figure 3.5, left column) and

will be discussed later on.The joint inversion combines the two algorithms as they are, without any changes to theoptimised single inversion algorithms. The only extension is the cross-gradient constraint. Theweight for the cross-gradient term was determined by trial and error.

The results of the joint inversion are shown in (Figure 3.5, middle column).

The thin layer in the central part of the model is highly conductive and of low radar velocity.Therefore, it is not easy to detect it with both methods. It can be seen that even though bothmethods see the layer individually, it is more distinct in the joint inversion image.Additionally, the geometry of the layer in the individual velocity model is not as accurate asin the joint inversion model.

The overall joint inversion shows a clearer image of the underlying structure below thisconductive layer, but there are some artefacts introduced that appear in the process ofmatching the cross-gradient constraint. For example, at a depth of ca. 1.8 m near the left

boreholes, a high resistivity structure can be seen that is not present in the synthetic model atall. This structure is more pronounced in the joint inversion model.

The different detailed behaviour of the model can also be analysed more closely by examiningcross-plots of collocated model parameters (Figure 3.6). For a more fair and precise

comparison, only the part of overlapping data coverage is displayed, i.e. between the GPRboreholes (0 m < x < 2.20 m) and down to ~4m depth. The overall scatter of the points inparameter space is less for the joint inversion, indicating the systematic connection betweenthe model parameters as it has been enforced by the cross-gradient constraint. Especially the

part in the centre of the model at intermediate depths is better resolved as the parametervalues obtained come closer to the true values.


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Figure 3.5: Results of inversion of synthetic data calculated from the resistivity (top) and velocity(bottom) models shown in the right column. Left: Results of individual inversions. Centre: Cross-gradient constrained joint-inversion result. Dots: Electrodes in ERT-boreholes (outer boreholes), GPRsource-receiver locations (inner boreholes).The structure of the synthetic model is overlain with black

and white lines.

Figure 3.6: Cross-plots of resistivity versus velocity for the results shown in Figure 5 . Left panel:individual inversion results. Right panel: Joint inversion. The circles' centres denote the zonescomposing of the synthetic input model. Depths are colour-coded.


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3.4.1.2 Results of joint inversion of a field data set

Having this high comparability with the Moreppen data enabled us to directly apply theoptimised algorithm to the field data set (see below).

Even though the data set is problematic, a joint inversion could be carried out and the datacould be explained with the error model that we assume to hold for this data set (see section3.3).

Resistivity models

The resistivity models are displayed in Figure 3.7. They exhibit very high resistivities

between 103

and 105.5

m.These very high values can only be explained by an extremely dryor frozen subsurface.

The models can coarsely be partitioned into 4 zones:

1. A very resistive, but possibly not well constrained layer near the surface (4 m, ~103 m) is still to be considered highly resistive, but moreconductive than the overburden. We observe a slight increase in the resistivity

As expected, the highest resistivities were found above the ground water table at times beforethe start of the snow melt. We expect that the water influx as well as the bromide tracer giverise to lower bulk resistivity. Near the surface, the resistivities start to decrease first. There is aclear decrease of resistivity in zone 1 between 30.03.10 and 12.04.10. It can also be seen thatthe resistivity of zone 2 decreases gradually over course of time, starting at the top andmoving downwards. Before the snow melt, zone 2 is nearly homogeneously resistive but ahorizontally layered structures at ~1.8 m appears after 04.04.10 which becomes graduallymore conductive over time.

Velocity models

The same 4 zones can be identified in the velocity model:

1. A low velocity near surface later (4 m, ~0.1 m/ns).


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The velocity in the zones 1 and 4 is probably not as well constrained as in zone 2 and 3, asalmost no travel times were picked from sources or receivers at those depths. Moreover, low

velocities are generally resolved poorer than high velocities in radar, since the rays travelaccording to Fermat's principle, choosing the path that takes least travel time.

As in the resistivity sections, a gradual decrease of velocities can be observed over time.However, the low velocities enter the model more like a well defined front that propagatesdownwards until the high velocity zone is affected almost completely by the decrease ofvelocity. The radar velocity decrease is a clear evidence for the change of water content andthus can be related to water flow and thus used for hydrological modelling.

Between 14.04.10 and 16.04.10 an abrupt increase of the velocity can be observed over all themodel space. It is not clear whether it is a real physical change that has not been recorded as

there is no measurement on the 15.04.10, or whether it is caused by a replacement of theoperators and thus detail changes in the acquisition procedure. As we observe an almost staticshift in the velocities (see D1.3) and as there is no visible change in the resistivity sections ofthe corresponding days, the latter is probable argument is more probable.


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Figure 3.7: Results of joint inversion with cross-gradient constraints for boreholes C(left) and D(right).RMS 1.00, RMSERT 1.01, RMSGPR 0.98. The first and third columns are the consecutive resistivity

models, the second and fourth are velocities.


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Combined models

As previously outlined, the coarse structure of the models is a similar 3 layer case for

resistivity and velocity. Although some of the detailed structures observed in the resistivitiesare not reflected in the much smoother velocity models, there are some details that coincidevery well. For example the structures directly above the ground water table a high resistivethin layer of higher velocity is located that seems to remain stable through most of the snowmelt, suggesting that it is of lower permeability and thus is circumvented by the water.Moreover, the thin layer structure at ~3.5 m is present in both models as a conductor and alayer of slightly lowered velocity, suggesting higher permeability and porosity.

The common time-evolution of the models becomes more obvious when visualising thechange of the estimated values (Figure 3.8). The resistivity models show many small scale

anomalies, which we mostly regard as inversion artefacts. However, on a coarser scale, wecan see a simultaneous and collocated decrease of the respective model parameters in bothmodels. The interpreted water front is indicated by a horizontal black line in Figure 3.8.

It can also be seen, that the time with strongest dynamics lies between around 03.04.10 and12.04.10. We learn from this that daily measurements are imperative to resolve this kind oftransport process.

3.4.1.3 Comparison with single inversion results

Even though the data sets have already been inverted separately in a traditional manner (see D1.3, chapter 4), we repeated the inversions using the algorithm obtained in section 3.4.1.1. inorder to get as comparable images as possible, that is not obscured by a different choice ofconstraints or the optimisation algorithm.

The results of the individual inversions are displayed in Figure 3.9. The representation isidentical to the one of the joint inversion results Figure 3.7 to allow for easy comparison ofthe detail features.

It is especially obvious that the joint radar image is much more detailed and the transitionsbetween the zones are sharper. The resistivity models also show slightly more pronouncedstructures, but as far as the single inversions contain artefacts, those are more pronounced inthe joint inversions as well.

The correlations of the model parameters can be analysed in cross plots (Figure 3.10) as hasbeen done in the synthetic example as well. We especially emphasise, that in the jointinversion the near surface (4 m). Both areas are not well constrained

by GPR data, so we conclude that the similarity is introduced by the cross gradient constraint.


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Figure 3.8: Results of joint inversion, same models and representation as in Figure 3.7, but the changewith regard to the model from 30.03.10 is displayed. Red is a decrease of resistivity or velocity, bluean increase. Red colours indicate an increase in soil moisture. In every model, a horizontal line is

drawn that outlines the interpreted lower limit of the water plume.


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Figure 3.9: Results of individual inversions for comparison with the joint inversion results in Figure3.7. The first and third columns are the consecutive resistivity models, the second and fourth are

velocities.


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Figure 3.10: Cross-plots of resistivity versus radar velocity for selected days from joint inversionresults for borehole pair CD. The depth position of the model parameters is colour-coded.

Figure 3.11: Cross-plots of resistivity versus radar velocity for selected days from individual inversion

results for comparison with joint inversion results in Figure 3.10.


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3.4.2. Joint inversion based on physical laws

3.4.2.1 Formulation

By assuming that resistivity and radar velocity can both be described by empiricalrelationships to partially common hydrological and petrological parameters, theserelationships can be incorporated into the inverse problem by a model parameter transform.The relationships considered are Archie's law (Archie, 1942)

bulk= a -mSwater

-n water (1)

and the CRIM model (Tinge, 1973)

vbulkr= vsoilt(1) +vwaterSwater+vair(1-Swater) (2)

a, m, n: Archie's coefficients, : porosity, Swater : water saturation, water: water resistivity(inverse of water conductivity), vair, vwater, vsoil : radar velocities in pure air, water or soilmaterial. The model parameters to invert for are

m = [, a, m, n, Swater, water, soil]T

here, soil= (vlight/vsoil)2, and all parameters but waterare taken to be spatially variable, and

Swater is considered to vary over time as well. Therefore, we achieve a decomposition of theproblem into a static and dynamic part.

The velocities vlight = 3x108m/s , vair v light and vwater v light/9 are known constants from the

literature. When formulating the inverse problem in this way, it is straightforward to applysmoothness constraints to those parameters instead of resistivity and velocity. Moreover,relatively tight bounds on most of these parameters can be given:

Porosity and water saturation values must lie between 0 and 1.Porosity can often be even

further constrained. For the synthetic experiment, we assume 0.05 < < 0.4. For the soilpermittivity, we assume 3 < soil< 20. Bounds on the Archie parameters are given from theliterature: 0.5 < a < 1, 1 < m < 4, 1.5 < n < 2.5. Physically, waterhas to be > 0.

The bounds are enforced by transforming the bounded parameters via a non-linear logarithmictransform to an unbounded domain. The transform has the form:

mtrans= log(mlb) - log(ubm) (3)


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Inverting for mtrans instead of mand using the inverse of equation (3) afterwards will alwaysmake sure that lb < m < ub.

For this approach, time-lapse data is required. As we assume all spatially variable parametersto lie on the same 2D grid with M cells, we get 2M model parameters per day when invertingfor resistivity and velocity, while in this approach only M model parameters (Swater) for theeach day are added. However, there are 5M (, a, m, n,soil ) + 1 (water) static model

parameters to be considered regardless of the number of measurement days. When using atleast 6 or more days, the number of model parameters in this approach is be smaller than inthe conventional scheme:

2MD > (5+D)M + 1 if D > 5, D measurement days

This indicates that for a longer measurement, the model parameters will be increasingly welldetermined. Because of the non-linear nature of the empirical laws a separate resolution of themodel parameters can be expected to a certain degree. However, from a mere repetition of themeasurement on a static water saturation model the non-uniqueness of the model parameterscannot be resolved. It is necessary that every day independent information is addedi.e. thatthe water saturation state has to change in order to obtain different states of the empiricallaws. Only then, it may be possible to determine the static variables. Similarly, welldetermined static parameters are necessary to obtain a stable estimation of the dynamic watersaturation. Additionally, temporal and spatial smoothness helps to make the problem betterdetermined.

3.4.2.2 Numerical results

The synthetic input model is displayed in the first row of Figure 3.12 (static parameters) andFigure 3.13 (temporally variable water saturation). A snow melt scenario similar to theMoreppen situation has been modelled to examine the capability of the approach. Thesynthetic data sets are identically configured as the Moreppen data sets regarding the numberof measurements and the measurement configuration for a particular day. We modelled 15days with variable water saturation degrees. The synthetic data set consists of 21429 apparentresistivities and 5165 radar travel times. Note that while the number of radar travel times is inthe same order as it was in the Moreppen experiment, the apparent resistivities considereduseful are approximately two times more numerous. We applied very favourable noiseconditions: 0.3 ns for the travel times, and 5% relative error and 0.01 m absolute error onthe apparent resisitivities. The noise is normal distributed, with the given errors being onestandard deviation. The conditions were chosen favourable in order to explore the limits of theapproach.

The water saturation was chosen to mimic a water plume from the snow melt passing throughthe lithology. The model parameters vary on a 2D grid with 28x38 cells. In total, 21281 model

parameters have been used to create the synthetic data and had to be estimated in the inverseproblem.


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Figure 3.12:Static parameters. Top: synthetic input model and bottom: recovered model.

Figure 3.13: Water saturation over time. Top: synthetic input model and bottom: recoveredmodel.

The results are displayed in the lower rows of Figure 3.12 (static parameters) and Figure 3.13(temporally variable water saturation). The obtained models were obtained after 25 iterationsexplain the data to an RMS of 1.14. Later models do fit the data completely (RMS ~1) but50+ iterations were needed and the models obtained do not visibly differ from the displayedones.

Even though the models explain the data down to the very low noise level and tight bounds onthe model parameter ranges, we see that the static Archie parameters and the porosity couldnot be very well separated. They are not sufficiently independent. The time-variant watersaturation can be inferred well. The soil permittivity is also clearly resolved, which can beattributed to the high resolution of the radar data. Figure 3.14 shows the water contentdistribution of the synthetic model and the result by calculation of = Swater for everywater saturation state. We observe that the water content can be recovered robustly, eventhough it was not directly inverted for.


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3.4.2.3 Conclusions in Moreppen

It could be shown on synthetic examples that joint inversions with structural constraints doimprove structural investigation when it is known that the structures do coincide. By applyingthe developed algorithm to a suitable field data set, a reinforced interpretation of the watertransport could be made based on two data sets. Comparisons with the individual inversionresults show the usefulness of joint inversion and validates to a certain extend the structuralsimilarity assumption for the Moreppen situation.

We conclude that even under very favourable conditions, the parameters necessary toconstruct the empirical laws are not determined from GPR and ERT data without addingadditional constraints. These constraints could come from drillings and laboratorymeasurements on soil samples.

Since the quality of the field data collected in Moreppen are problematic, we do not considertesting this approach on them. We also see that an easier model formulated in terms of thewater content could be more promising, as it appears to be a very well determined parameter.

Figure 3.14: Water content over time, derived from results displayed in Figure 3.10 and 3.11.Top: synthetic input model and bottom: recovered model.

3.5 Inversion of Cross-hole GPR data in Trecate

On 19.10.2010, POLITO and Uppsala collaborated to collect a radar tomography data set atthe Trecate field site. The instrument was provided by POLITO. The two boreholes B-S3 andB-S4 are separated by 6 m. The source and receiver antennae were moved downwards in theirrespective borehole in 25 cm steps between 3 m an 17 m depth. This gives 57 antenna

positions. All possible permutations of source and receiver were measured, resulting in 3249radar traces. From those 2306 travel times could be picked. The data quality was very good

(see Figure 3.15). The main reason for not being able to pick certain radar traces was the


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diminished amplitude with higher offset. The boreholes are relatively close compared to theirdepths. The range of the offsets is very large (between 6 m and 15.2 m) and affects theamplitude of the radar wave strongly. Therefore, large offset traces can be picked with less

precision as short offset traces. We counteract by weighting the obtained travel time by theirrespective offset. The error for the i-th picked travel times is assumed to be

i= Offseti/ Offsetmin* 1[ns]

where the Offsetmin = 6 m.

Figure 3.15: Sample of a source gather of radar-gram together with the corresponding picked first-

arrivals (red circles).

The results of the inversion are displayed in Figure 3.16. A clear 4 layer case can beidentified. All the layer boundaries are slightly inclined. The velocity is decreasingdownwards. We interpret:

1. An unconstrained surface layer of intermediate velocity (~0.1 m/ns, down to


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Figure 3.16: Inversion result.

Figure 3.17: A comparison between the results of inversion of a) crosshole ERT and b) cross GPRdata. The zones with low, intermediate and high values are shown in c. Note that the measurementswere not conducted at the same time.


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In Figure 3.17 we compare the results of the single inversions of crosshole GPR and cross-hole ERT (shown in Figure 2.6). Note that the data were not collected at the same time andthe main idea behind the comparison is mainly to show the zones that might partly reflect themorphologies especially in the vadose zone (depths above 10 m). In the vadose zone we see

three zones marked by LH, LI and HI where the first letter indicates the resistivity and thesecond the radar velocity. Letter L, I, and H represent Low, Intermediate and High,respectively. If we compare this zonation with the lithological log collected in borehole B-Ithat is very close to B-S4 the HI zone corresponds to the sandy layer and LH and LI zonescorrespond to gravel layer that might partly have silt inside.


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4. Joint inversion of compressional and surfacewaves for soil porosity estimate

We discuss an approach for joint inversion of seismic velocity (compressional and surfacewaves) that can be useful both for improving the reliability of site characterisation and for anestimate of the porosity in the saturated zone in shallow aquifer.

The approach is based on the data acquisition using the same geophone array and sources ofcompressional and surface waves, and then joint inversion of P-wave and surface wave datasets and finally reconstruction of 2D images of Poisson coefficient distribution and 2D imageof porosity distribution.

We shortly describe the approach by the discussion on a real dataset, that has been recently

acquired in a test site where the methodology has been checked. Were working on new dataset at Trecate site where the suggested methods has to be calibrated with the results given byother methods, such as interpretation of radar data for estimating the soil moisture (see

previous deliverable).

The joint inversion of seismic data can produce a significant improvement of the resultreliability, mitigating the effect of illness and interpretation ambiguities which are typical ofthe two methods. In a joint inversion approach the coupling of different data set can be

performed in several ways accounting for only the geometry or also introducing physical lawsto link unknown parameters (Dal Moro, 2008).

We present here a joint inversion method for refracted P and surface-wave data specificallydesigned to deal with laterally varying layered environments which can present strongvelocity contrasts with depth. In point of fact, as seen from other authors (Auken andChristiansen, 2004), a smooth minimum-structure inversion produces smooth models even forgeological models, which are intrinsically layered. Hence, with respect to other similarmethods (Re et al., 2010), it differs in the parameterization, Frchet derivatives, and inversealgorithm and it is able to incorporate a-priori information available over the site and any

physical law to link model parameters.

After describing the inversion algorithm we apply the joint inversion to synthetic and field

data sets and finally we discuss the approach for converting P/S wave velocity into porosity.

4.1Background

Seismic surface-wave analysis (SWA) and body-wave refraction tomography (BWT) are bothwidely applied to build P- and S-wave (VPand VS) near surface velocity models, providinguseful information for near-surface geological modelling. SWA is mainly used for VSestimation in the field of geotechnics, seismic hazard, vibration propagation and geological

studies (Socco et al., 2010). BWT is widely used to estimate depth and morphology of the


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bedrock and to define geological boundaries and property distribution in sedimentaryenvironments (Zhang and Toksoz, 1998). SWA and BWT present several synergies. The datacan be jointly gathered with a significant optimization of fieldwork, since the acquisitionschemes are very often similar. Furthermore, since they have different depths of investigation,different intrinsic limitations and different resolution and sensitivity with depth, the couplingof the two methods helps in building consistent VPand VS models (Piatti and Socco, 2010).Moreover the gathered analysis of VP and VS dataset leads to the imaging of mechanical

parameter (e.g. Poisson coefficient) or in sedimentary environment the results can beconverted into distribution of soil porositySWA inverse problem is ill-posed, mix determined and strongly non-linear and suffers from asolution non uniqueness. Thus, different models may supply equally good fitting withreference to the experimental dispersion curve (Socco and Boiero, 2008). BWT is an ill-posedand ill-conditioned inverse problem as well. The presence of low velocity layers embedded in

stiffer layers or gradual velocity increases over interfaces with strong velocity contrastproduce hidden layers and significant errors in the final velocity profile regardless of the

algorithms that are used to invert the traveltimes (Palmer, 2010).

4.2 Inversion Scheme

To couple SWA and BWT data we discretise Earths surface (x,y)using a (regular) grid of Kpoints, and at each point k we consider a layered 1D model mkcharacterised by VP(z), VS(z)and, the density (z). In the following we illustrate the scheme for the 2D case but theextension to 3D case is straightforward. We use a (damped) weighted least square algorithm

to jointly invert dispersion curves and first breaks. This involves minimization of the misfitfunction Q:

-1 -1

-1 -1

- - - -

- - - -

TT

obs obs obs prior prior prior

TT

R pr

Q fw fw

pr pr

d m C d m m m C m m

Rm C Rm pr m C pr m

(1)

where the forward responsefw(m) evaluates for each model mkthe correspondent dispersioncurve using the Haskell (1953) and Thomson (1950) method and the P-wave first breaks

through the finite difference solution of the eikonal equation as proposed by Podvin andLecomte (1991), which can compute traveltimes in very contrasted velocity models as alayered one could be. For P-wave traveltime computation a 2D/3D grid mesh is built bynearest-neighbour interpolation of all the mk. The covariance Cobs (inferred from themeasurements) weights the observed data dobs. The vector mprior represents the a-prioriinformation with the associated covariance Cpriorwhereaspr represents the expected value fora physical propertypr(m) of the model parameters (i.e. the Poissons ratio which links V PandVS)with its covariance Cpr. In eq. (1), the regularization consists in the inclusion of Rwhichhelps to minimize the differences between model mkand the models at surrounding points. Itseffectiveness depends on the covariance matrix CR (Auken and Christiansen, 2004). To

minimise Q, the model solution at the nthiteration can be expressed as


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-1-1 -1 -1 -1

1-1 -1 -1 -1

- - - -

T T T

obs prior R pr obs pr

n nT T T

obs obs n prior prior n R n pr pr nfw pr

G C G C R C R G C G Im m

G C d m C m m R C Rm G C pr m

(2)

where the Jacobian Grepresents the sensitivity matrix of the dispersion curves and the firstbreaks respect to the model parameters (VP(z), VS(z)) and where is the damping parameter,which is used to stabilize the solution. It is worth noticing that the sensitivity of the first

breaks respect to VP(z) is evaluated without performing ray-tracing, following the approachproposed by Ammon and Vidale (1993). This presents several advantages, for instance, theinversion is less biased by the reference model and could consider finite frequency effects.Because this method explicitly constructs the sensitivity matrix (Frchet derivatives) byrepeating the forward calculation for each perturbed parameter, this approach is limited to a

small number of model parameters as in the case of a quasi-layered model. The matrix Gprcontains the partial derivatives of the considered physical property respect to model

parameters.

4.3 Synthetic Example

The synthetic datasets, representing a multi-fold seismic survey, has been generated through aFEM code (Comsol Multiphysics, stress-strain module), using an axial symmetric schemeand a Ricker source type (dominant frequency 10 Hz). The model is linear elastic isotropic(its seismic properties are reported in Table 4.1) and presents topographic unevenness and a

dipping half-space (Figure 4.1a).

Figure 4.1: The synthetic model: a) the model geometry; the sensor positions (red triangles) and thecentres of the window used to estimate the dispersion curves (blue points) are shown; b) the estimateddispersion curves; c) the picked first breaks and the source positions (black stars).


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The finite difference solution of the eikonal equation is computed using FDTIMES by PascalPodvin (Gophysique - Centre de Gosciences - ARMINES/cole des Mines de Paris).Concerning the SWA processing, all seismograms have been windowed in the spatial domainusing a sliding window according to the method proposed by Socco et al. (2009). The f-kspectra from different shots with the same spatial windowing have been stacked, and the

picking of maxima has been performed on each stacked f-k spectrum, yielding a set of 12dispersion curves (Figure 4.1b) evenly-spaced (25m) along the survey line (Figure 4.1a). First

breaks for each channel and for different sources have been also picked (Figure 4.1c).Dispersion curves and first breaks are inverted together using the scheme described by the eq.2. The adopted reference model is the layered 1D VP(z) and VS(z) profile described in Table4.2.

Table 4.1 The seismic properties of the

ModelLayer

VP(m/s)

VS(m/s)

(kg/m3)

1 240 120 18002 340 170 21003 500 270 2400

Table 4.2 The seismic properties of the Reference

ModelLayer

Thk(m/s)

VP(m/s)

VS(m/s)

(kg/m3)

1 10 200 100 18002 10 400 200 21003 - 600 300 2400

The results of the inversion are shown in Figure 4.2, where each sounding corresponds to anmk . The maximum differences with respect the true model are about 10% for thickness and7.5% for the velocity values (for the second layer which is hidden for P-wave BWT for themajority of the profiles).

Figure 4.2: The inversion result: a) the estimated S-wave velocity model; b) the estimated P-wavevelocity model. The true model geometry is superimposed.


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4.4 Example on real data

A dataset has been acquired with a receiver array made up of 48 vertical 4.5 Hz geophoneswith 1.5 m spacing. The acquisition parameters in time domain were 2 s acquisition length

and 0.5 ms sampling rate (Figure 4.3a). Ten (10) evenly spaced dispersion curves (Figure4.3b) have been extracted using the same scheme adopted for synthetic data using 10 differentGaussian windows according to the method proposed by Bergamo et al. (2010). The P-wavefirst breaks have been visually picked for all the seismograms (Figure 4.3c). The VS and VPmodels retrieved through the inversion are shown in Figure 3d and 3e, respectively. Both ofthem point out a shallower interface at 2-4 m depth gently sloping to the left and a deeperinterface at 8-12 m depth leaning to the right. The results are in agreement with previousstudies.

Figure 4.3 Field case: a) an example of raw data; b) the dispersion curves estimated along the line; c)the P-wave first breaks; d) the estimated S-wave velocity model; e) the estimated P-wave velocitymodel.

4.5 Porosity estimate

Several approaches are available for estimating the soil porosity from seismic waves dataset; aexhaustive review on this topic is given by Berryman (1995). We suggest using the approachgiven by Foti and al. (2002) based on the development of Biots theory. The approachassumes the following:

1. in the low frequency range no motion between the fluid and the solid phase isobserved:

2. the soil grains are incompressible.


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The soil porosity can be estimated according to the following relationship (Foti et al. 2002):

where n is the value of soil porosity, sis the soil density, fis the fluid density, Kfis the bulkmodulus of the fluid, vskis the Poisson coefficient of the solid skeleton, varying between 0.1and 0.4; it can be shown that the Poisson coefficient of the solid skeleton has a negligibleinfluence on the final estimate (Foti et al., 2002).

4.6 Conclusions of joint inversion of p-wave and surface wave

The proposed algorithm of joint inversion of surface-wave dispersion curves and P-wave firstbreaks has proved to be effective in the case of laterally varying layered sites (models) tobuild VP and VSmodels. It presents advantages with respect to individual SWA and BWTsince it imposes internal consistency for all the model parameters reducing the required a-

priori assumptions (i.e. Poissons ratio in SWA) and the illness of the two methods (i.e.hidden layers for BWT).Were also exploring the reliability and sensitivity of the approach in assessing the soil

porosity in the saturated zone at Trecate site; particularly were dealing with the sensitivity of

the method in improving the data interpretation of georadar data in detecting the gradualchanges of the soil porosity in the most contaminated area as due to the combination ofseveral effects such as mineralisation and mineral precipitation, pore clogging.


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3. References

Ammon C. J. and Vidale J. E. 1993. Tomography without rays. Bulletin of Seismological Society ofAmerica 83, 509-528.

Archie, G. E., 1942.The electrical resistivity log as an aid in determining some reservoircharacteristics. T. Am. Inst. Mineral. Metall. Petrol. Eng., 146, 54-62.

Auken E. and Christiansen A. V. 2004. Layered and laterally constrained 2D inversion of resistivitydata. Geophysics69, 752-761.

Bergamo P., Boiero D. and Socco L.V. 2010. Retrieving 2D Structures from Surface Wave Data byMeans of a Space-varying Spatial Windowing. 16th EAGE Near Surface 2010. Zurich,

Switzerland.Extended Abstracts.

Berryman, J. G. 1995. Mixture theories for rock properties. In Rock physics and phase relations: Ahandbook of physical constants (ed. T. J. Ahrens), pp. 000000. Washington: AmericanGeophysical Union.

Boiero D., Calzoni C. Socco L.V. 2011, Joint Inversion of Surface-wave Dispersion and PwaveRefraction Data for Laterally Varying Layered Models, Proc. 73rdEAGE Conference & Exhibition2011, Vienne, Austria.

Candansayar, M.E. & Tezkan, B., 2008. Two-dimensional joint inversion of radiomagnetotelluric and

direct current resistivity data, Geophys. Prospect., 56(5), 737749.

Constable, S. C., Parker, R. L. and Constable, C. G., 1987. Occam's inversion: A practical algorithmfor generating smooth models from electromagnetic sounding data. Geophysics, 52(3), 289-300.

Dal Moro G. 2008. VS and VP vertical profiling via joint inversion of Rayleigh waves and refractiontravel times by means of bi-objective evolutionary algorithm. Journal of Applied Geophysi

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