Diploma Thesis

The eﬀect of the topography on Magnetic Resonance Tomography (MRT)

Diplomarbeit

Ines Rommel

Technical University Berlin Institute of Applied Geoscience Department of Applied Geophysics

October 2006

Hiermit versichere ich, die vor- liegende Arbeit selbstst¨andig, ohne unerlaubte fremde Hilfe und nur mit den angegebenen Hilfsmitteln angefertigt zu haben.

Berlin, den 6. Oktober 2006

Ines Rommel

I II Zusammenfassung

Der Einﬂuss der Topographie auf die Magnetische Resonanz Tomographie (MRT)

Die Magnetische Resonanz Sondierung (MRS) ist eine geophysikalische Methode um Grund- wasser direkt von der Oberfläche aus zu detektieren und zu untersuchen. Mittlerweile ist es möglich in der Anwendung der MRS mit getrennter Sende- und Emp- fangsspule zu messen. Diese Messungen eignen sich fur¨ hochauflösende 2D-Untersuchungen der oberflächennahen Wassergehaltsverteilung, da sie eine erhöhte räumliche Auflösung und zuverlässige Signalamplituden liefern. In bisherigen Modellierungen fur¨ gleiche und getrennte Sende- und Empfangsspulen wurde die Topographie vernachlässigt. Diese Arbeit untersucht und bewertet den Einfluss der Topographie auf MRS Messungen und stellt den Vergleich zur flach modellierten Erde mit horizontaler Oberfläche her. Zusätzlich dazu wurden die Spulenseparation, die Profil- richtung und der Neigungswinkel der Spulen mit in Betracht gezogen. Die 2D Modellierung einer Dunenstruktur¨ mit Hilfe der Finiten Elementen Methode zeigt durch Veränderungen in der Neigung der Dune¨ und die damit einhergehende relative Veränderung der Spulen- lage, Einflusse¨ des zusätzlichen oder reduzierten Untersuchungsvolumens und des Winkels zwischen den Spulen. Die Abweichung der Amplituden zur flach modellierten Erde steigt mit zunehmender Spulenseparation und steigender Neigung der Dune.¨ Zusätzlich zu den 2D Modellierungen wird eine 2D Inversion fur¨ Messungen mit getrennten Spulen unter Berucksichtigung¨ der Topographie durchgefuhrt.¨ Die Inversion spiegelt das Modell wieder und gibt zuverlässige Ergebnisse. Die Inversion einfacher Wassermod- elle mit vernachlässigter Topographie liefert offensichtlich falsche Ergebnisse. Trotzdem der invertierte Wassergehalt vertikal zur wahren Topographie hin verschoben wird, bleiben große Abweichungen zum Modell erhalten. Um die Verbesserung durch die Einbeziehung der Topographie in der Dateninterpretation zu zeigen, wurden MRS Messungen uber¨ eine Dune¨ im Nordosten Deutschlands durchgefuhrt.¨ Ein detailliertes und plausibles Modell einer 2D Wassergehaltsverteilung wurde er- mittelt, das durch gleichstromgeoelektrische und induzierte Polarisationsmessungen bestärkt werden kann. Die Betrachtung der Topographie verbessert die Inversionsergebnisse und eine bessere Anpassung an die Daten wird erzielt. Als Ergebnis dieser Arbeit ist zu sagen, dass es nicht vertretbar ist, bei Untersuchungen mit getrennten Spulen die Topographie in der Modellierung und der Inversion zu vernach- lässigen.

III IV Abstract

The eﬀect of the topography on Magnetic Resonance Tomography (MRT)

Magnetic Resonance Sounding (MRS) is a geophysical method to detect and explore groundwater directly from the surface. Recently, in the application of the MRS, it is possible to measure with separated transmitter and receiver loops. This measurement provides enhanced spatial resolution and sufficient signal amplitudes to perform high resolution 2D surveys to obtain the water content distribution of the shallow subsurface. So far, the topography is neglected in conventional modeling of coincident and separated loop measurements. This study investigates and assesses the effect of the topography on the MRS measurements in comparison to the flat modeled earth with a horizontal surface. In addition, the loop separation, the profile direction and the tilt angle of the loop are taken into account. The 2D modeling of a dune structure, using the finite element method, shows through the changes of the slope of the dune and the associated relative changes in the orientation of the loops, the effects of the additional or reduced investigated volume and of the angle between the loops. The deviation of the amplitude to the flat modeled earth increases with increasing separation of the loops and increasing slope of the dune. In addition to the 2D modeling, a 2D inversion for separated loop surveys is performed, which takes the topography into account. The inversion reflects the model and gives reliable results. The inversion of water models neglecting the topography provides obviously wrong inversion results. Despite shifting the inverted water content vertically to the real topography, high deviations to the model remain. To show the improvements of incorporating the topography into the data interpretation a MRS survey across a dune in the northeast of Germany has been realized. A detailed and reasonable model of a 2D water content distribution, which can be encouraged by the measured geoelectrical and IP data was determined using the inversion scheme. The consideration of the topography improves the inversion results and thus a better adaptation to the data is achieved. As the result of this thesis, it is not justifiable to neglect the topography in the modeling and the inversion for separated loop surveys.

V VI Contents

1 Introduction 1

2 Theory 3 2.1 The basics of Magnetic Resonance Sounding (MRS) ...... 3 2.2 The Finite Element Method (FEM) ...... 13

3 Synthetic Data 17 3.1 Comparison of a modeling with a square and a circular loop and different separations ...... 17 3.2 Modeling with coincident and separated loop configuration ...... 21 3.2.1 Modeling with COMSOL ...... 22 3.3 Homogeneous half space ...... 24 3.3.1 Coincident soundings ...... 29 3.3.2 Edge-to-edge measurements ...... 34 3.4 Three layer model ...... 38 3.5 Comparison between north-south and east-west profile ...... 41 3.6 Inversion of synthetic data ...... 45

4 Field data set 53 4.1 Geophysical measurements ...... 55 4.2 Data processing ...... 57 4.3 Modeling of the dune ...... 61 4.4 2D inversion ...... 66

5 Conclusions 71

Bibliography 73

A Appendix 75 A.1 Appendix to Chapter 2 ...... 77 A.2 Appendix to Chapter 3 ...... 80

VII Contents

VIII 1 Introduction

The Magnetic Resonance Sounding (MRS), also known as Surface Nuclear Magnetic Reso- nance (SNMR), is a relatively new geophysical method to assess groundwater. It is the only geophysical method that allows the direct detection of water and a reliable estimation of the water content from the surface. The connection of the measured signal and the amount of the water results in a very high potential of this method in environmental, engineering and geophysical questions. The measured transient signal also contains information about the pore structure and the hydraulic conductivity. In 1946, the Nuclear Magnetic Resonance was discovered by Bloch [1946] and Purcell et al. [1946], and became an established tool in physics, medicine and chemistry. The initial idea of using this technique for the exploration of groundwater was developed in the early 1960s. Only in the 1980s the equipment was designed by Russian scientists. The applications and results of the surveys for water prospecting was first published by Semenov [1987]. In 2000, new and improved equipment was designed by Iris Instruments with increasing power and the possibility for measuring with separated transmitter and receiver loops. The extended formulation of the theory for different transmitter and receiver loop was developed by Weichman et al. [1999; 2000]. In 2003, first measurements and the first assessment of the properties of separated loop measurements were published by Hertrich and Yaramanci [2003]. A detailed physical description of separated loop measurements and their contribution to high resolution two dimensional water content distribution mapping is studied in the PhD thesis of Hertrich [2005]. The high potential of the measurement with separated transmitter and receiver lies in the investigation of 2D structures. For the interpretation of the MRS data, the loop separation, separation direction and separation orientation have to be taken into account for each individual sounding of the survey. In conventional modeling of 2D separated loop measurements, flat and horizontally strat- ified earth is assumed, but for the extension to arbitrary field applications, the effect of topography has to be considered. If the loops are located on the same plane, the apparent inclination angle can simply be changed as a sum of the true inclination of the Earth’s magnetic field and the tilt angle of the loop. As soon as the loops are not on one plane this simplification is not valid. The numerical computation of the magnetic field is demonstrated with the help of the Finite Element Method (FEM) in three dimensions. It is possible to model and calculate any arbitrary topography and internal subsurface conductivity, as well as every wanted 2D or 3D water content distribution. The advantage of this method is the use of the unstructured tetrahedron mesh with local mesh refinement and the flexible description of arbitrary geometry.

1 1 Introduction

Chapter 2 gives a general introduction into the theory of the Magnetic Resonance Sound- ing including the properties of the magnetic field of the loop in the Earth’s magnetic field and a detailed description of the MRS response signal. An overview of soundings with different separations and profile directions is shown, as well as the spatial distribution of their sensitivity. For the numerical solving of a partial differential equation (PDE), the Fi- nite Element Method is described and compared to the Finite Difference Method, another numerical method for solving PDEs. The modeling of a synthetic data set for a dune structure with different slopes of the dune is shown in chapter 3. To clarify the influence of the topography, the profile direction and the resistivity to the soundings, all these properties are varied. On the basis of two models with a 1D and a 2D water content distribution, a new 2D inversion scheme shows the influence of the non-consideration of the topography in the inversion. Chapter 4 shows the application of the MRS technique on the field over a dune with a slope of 15° and 20° in the northeast of Germany. Additionally, a modeling of the 2D water content distribution based on the data from the geoelectrical and IP measurements as well as a 2D inversion of the whole survey with and without the consideration of the topography are performed. Finally, chapter 5 gives a conclusion of the results and a short outlook.

2 2 Theory

In section 2.1, the principle of the Magnetic Resonance Sounding (MRS) is described concentrating on the exciting magnetic field and on the three parts of the MRS response signal. Here, mainly the derivation according to Weichman et al. [2000] is applied. Section 2.2 gives an overview of the numerical method of the finite elements and therefore the solution of partial differential equations.

2.1 The basics of Magnetic Resonance Sounding (MRS)

The protons of a hydrogen atom are positive charged particles, they have a speciﬁc mass, a microscopic magnetization µ and a spin (angular momentum). The magnetization µ is coupled with the angular momentum over the Planck’s constant ~ and the gyromagnetic ratio of the protons γp µ = γp~I (2.1) with nT γ = 0.267518 . (2.2) p Hz

In the presence of an external ﬁeld, the local Earth’s magnetic ﬁeld B0, a resulting macro- scopic magnetization establishes, called the net magnetization,

2 (0) γp ~B0 N 0 MN = (Curie s law), (2.3) 3kBT V

N where V is the number of Spins per volume, kB is the Boltzmann constant and T is the temperature in kelvin [Yaramanci and Hertrich, 2006]. When an additional magnetic field BT oscillating with the Larmor frequency is generated by a large transmitter loop at the surface, the axis of the protons are deflected from their (0) equilibrium. MN is deflected into the direction of the field BT and precesses clockwise with the angular Larmor frequency

ωL = −γpB0 = 2πfL (2.4) around the field [Levitt, 2002]. The excitation or resonance frequency ωL is determined by measuring the Earth’s magnetic field at the test site using a magnetometer. The Larmor frequency fL varies worldwide between 800Hz and 3000Hz. The magnetic field BT is generated by an alternating current I(t) = I0 cos(ωLt) for a limited time τ and with an amplitude of I0, that is passed through the transmitter loop.

3 2 Theory

The penetration depth depends on the excitation intensity, called pulse moment q = I0 · τ, the choice of the loop size, the loop layout as well as the electrical conductivity. After the field BT is removed, the perpendicular component of the net magnetization decays ∗ exponentially back to the equilibrium with the decay time constant T2 and generates an additional magnetic field which can be measured on the surface [Yaramanci, 2000]. But only the component of the transmitter field BT perpendicular to the Earth’s mag- ⊥ netic field BT has a physical effect on the Spin system ⊥ ˆ ˆ BT = BT − b0 · BT b0, (2.5)

ˆ ⊥ where b0 is the unit vector of the Earth’s field. In general, BT is elliptically polarized and can be described by two circularly polarized fields with different amplitudes and opposite rotation sense.

⊥ + − BT = B T + BT (2.6) h ˆ ˆ ˆ i = I0 αT cos (ωLt − ζT ) bT + βT sin (ωLt − ζT ) b0 × bT (2.7)

with

I h i B± = 0 (α ∓ β ) cos (ω t − ζ ) bˆ ∓ sin (ω t − ζ ) bˆ × bˆ (2.8) T 2 T T L T T L T 0 T ± I0 B = (αT ∓ βT ) . (2.9) T 2 + BT (co-rotating part) rotates clockwise and thus in the same spinning sense as the pro- − tons, and BT (counter-rotating part) rotates counterclockwise around the Earth’s magnetic ˆ field B0. The major axis of the ellipse αT is in the direction of bT , the unit direction vector of the exciting magnetic field BT . The minor axis of the ellipse βT is in the direction of ˆ ˆ b0 × bT . The phase ζT determines the orientation of the vector at t = 0 and is chosen in a way that αT and βT are real. The elliptically polarized field lies in the plane, that is ˆ ˆ ˆ spanned by the vectors b0 × bT and bT , perpendicular to the Earth’s magnetic field. Figure 2.1 shows the magnetic field lines produced by the loop in the east-west (a) and ⊥ north-south (b) direction, respectively, the magnetic field BT and its decomposition. In the east-west direction, the exciting field BT is radial symmetric in respect to the loop center. The co- and counter-rotating parts have same amplitudes but contrary rotation sense in the east and the west of the loop. In the north-south direction, the resulting exciting field is linearly polarized, so the minor axis of the ellipse βT is zero. The co- and counter-rotating parts have equal amplitudes but different rotation sense. The angle between the Earth’s magnetic field B0 and the exciting field is larger in the south of the loop center in the northern hemisphere, as a consequence of the inclination of the Earth’s magnetic field. So the amplitudes of the co-and counter-rotating parts and of the linear polarized field are larger in the south than in the north of the loop [Braun, 2002], [Hertrich, 2005].

4 2.1 The basics of Magnetic Resonance Sounding (MRS)

(a)

(b)

⊥ Figure 2.1: The mathematical decomposition of the exciting magnetic ﬁeld BT perpendicular to the Earth’s magnetic ﬁeld (blue) into the co-rotating (red) and counter-rotating (green) part in the east-west (a) and north-south (b) direction [Hertrich, 2005].

5 2 Theory

The MRS response signal for the induced voltage in the receiver loop is given according to Weichman et al. [2000] by the volume integral

Z 3 (0) q + VR(q) = ωL d r MN (r) sin −γp BT (r) I0 2 − i|ζT (r,ωL)+ζR(r,ωL)| × BR(r) · e I0 h i ˆ⊥ ˆ⊥ ˆ ˆ⊥ ˆ⊥ × bR(r, ωL) · bT (r, ωL) + ib0 · bR(r, ωL) × bT (r, ωL) , (2.10)

(0) where MN (r) is the magnetization vector and the indent T and R stands for the transmitter and the receiver, respectively. This volume integral depending on the puls moment q, is the envelope of the transient oscillating MRS signal at t=0 and consists of three parts:

The first line of the equation gives the signal amplitude of the Spin system. The q + excitation or deflection angle θT = −γp B (r) is composed of the magnitude of I0 T the co-rotating part of the magnetic field of the transmitter and depends on the pulse moment q. Only the amplitude of the co-rotating part of the magnetic field of the transmitter causes the deflection of the protons. The magnetization, which is influenced by the number of the protons and the static magnetic field B0, as well as the deflection angle contributes to the amplitude of the MRS signal.

The second part of the equation shows the sensitivity of the receiver to a signal in the subsurface. It is inﬂuenced by the counter-rotating part of the magnetic ﬁeld of the receiver but independent of the excitation pulse q. The signal undergoes a phase lag (ζT (r, ωL) + ζR(r, ωL)) from the transmitter to the sample and back to the receiver as a consequence of the electromagnetic attenuation in a conductive media.

The third and most important part for separated loop measurements demonstrates the vectorial character of the MRS signal. It is called the geometric term and describes the dependency of the signal on the mutual orientation of the transmitter and receiver loop and the orientation to the Earth’s magnetic field. The first part of the sum is a scalar vector multiplication and the second part, the imaginary part, is a vector cross product of unit direction vectors of both fields and the scalar product with the Earth’s magnetic field. For coincident measurements, the expression in the brackets becomes real and equal to one. The imaginary part vanishes and the real part becomes unity, because BR = BT and ˆ ˆ bR k bT (collinear). For separated loop measurements, the MRS signal is always complex valued, so the resulting phase is the geometrical phase [Hertrich, 2005]. The third part depends

6 2.1 The basics of Magnetic Resonance Sounding (MRS)

only on the distance between the transmitter Tx and the receiver Rx and the profile direction but not on their mutual orientation, due to the unit direction vectors of the fields. Therefore one loop configuration has the same third part in equation 2.10.

(0) The magnetization vector MN (r) can be replaced by the magnitude of the nuclear magnetization vector M (0) for protons in water at thermal equilibrium, which is assumed to be constant for one test site multiplied with the water content f(r) at the speciﬁc point r

(0) (0) MN (r) = M · f(r). (2.11)

The water content f(r) in the equation is closely related to free mobile water. The re- ∗ laxation constant T 2 for water stored in ﬁne pores or for bounded water is very small, as a consequence of the fast loss of energy during the relaxation. Such short relaxation constants are not detectable with the present technique since the equipment has a dead time depending on ωL of approximately 30ms. Due to the dead time, the transient MRS signal has to be extrapolated to the time t = 0, where the pulse is cut oﬀ. This study concentrates on the amplitude of the complex MRS signal. The MRS kernel function in the case of a 3D water content distribution in Cartesian coordinates is given by

(0) q + K3D(q, x, y, z) = ωLM sin −γp BT (x, y, z) I0 2 − i|ζT (x,y,z,ωL)+ζR(x,y,zωL)| × BR(x, y, z) · e (2.12) I0 h i ˆ⊥ ˆ⊥ ˆ ˆ⊥ ˆ⊥ × bR(x, y, z, ωL) · bT (x, y, z, ωL) + ib0 · bR(x, y, z, ωL) × bT (x, y, z, ωL)

and the induced voltage in the receiver loop is the volume integral of the 3D kernel function multiplied with the 3D water content distribution

Z ∞ Z ∞ Z ∞ VR(q) = f(x, y, z) K3D(q, x, y, z)dx dy dz. (2.13) 0 −∞ −∞ The integral in equation 2.13 for one measurement at one loop configuration is only dependent on the spatial distribution of water. The 3D kernel function in equation 2.12 can be reduced to 2D or 1D by integrating the kernel in one or two directions, respectively. A 2D kernel function in dependency of depth and profile direction can be obtained by integrating over one dimension perpendicular to the profile direction for a water content distribution along the profile direction and variable with depth

Z ∞ Z ∞ VR(q) = f(x, z) K2D(q, x, z)dx dz (2.14) 0 −∞

7 2 Theory

with Z ∞ K2D(q, x, z) = K3D(q, x, y, z) dy −∞ or in dependency of two horizontal directions by integrating over the depth for a water content distribution variable in the horizontal directions Z ∞ Z ∞ VR(q) = f(x, y) K2D(q, x, y)dx dy (2.15) −∞ −∞ with Z ∞ K2D(q, x, y) = K3D(q, x, y, z) dz. 0 Integrating the kernel function along the two horizontal directions, a 1D kernel function is obtained that only allows a variation of water with depth

Z ∞ VR(q) = f(z) K1D(q, z) dz (2.16) 0 with Z ∞ Z ∞ K1D(q, z) = K3D(q, x, y, z) dx dy. −∞ −∞ The calibration sounding curves with a hypothetical and assumed water content of 100% show the maximal amplitudes and are used in the modeling to show the characteristics of measurements with arbitrary separation. Figure 2.2 shows the initial amplitude and the phase for every pulse moment. The sounding curves were modeled with an Earth’s magnetic field of 48000nT, an inclination of 60°N and a homogeneous half space of 100Ωm. The square loops with a side length of 45m lie coincident, half-overlapping and edge-to- edge in the east-west and the north-south direction, respectively. This corresponds to a separation of 0m, 22.5m and 45m. For coincident configuration, the sounding curve has a maximum at small pulse moments and becomes constant with increasing excitation intensities. The phase increases linearly to nearly 15°. For a profile direction in east-west the amplitudes are more or less the same for one configuration, but they slightly differ with increasing pulse moments. The phase is nearly mirror symmetric with a small shift to positive values due to electromagnetic phase effects. The phase effects due to the vectorial character of the MRS signal and due to the EM attenuation yield the resulting modeled phase. The second effect becomes more prominent with decreasing resistivity. In the insulating case only the geometric phase has an effect and the phase becomes symmetric in the case of an east-west separation [Hertrich, 2005]. In the case of separated loops in the north-south direction, the amplitudes differ consid- erably. If the transmitter lies half-overlapping in the north of the receiver, the amplitudes

8 2.1 The basics of Magnetic Resonance Sounding (MRS) 180 2000 90 0 1000 phase [ ° ] Tx south of Rx amplitude [nV] edge−to−edge −90 0 0 2 4 6 8

0 2 4 6 8

10 12 14 16

10 12 14 16 −180 pulse moment [As] moment pulse pulse moment [As] moment pulse 180 2000 90 Tx north of Rx 0 1000 overlap phase [ ° ] amplitude [nV] −90 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16 −180

pulse moment [As] moment pulse [As] moment pulse 180 2000 90 0 1000 phase [ ° ] Tx east of Rx amplitude [nV] edge−to−edge −90 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16 −180

pulse moment [As] moment pulse pulse moment [As] moment pulse 180 2000 90 Tx west of Rx 0 1000 overlap phase [ ° ] amplitude [nV] −90 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16 −180

pulse moment [As] moment pulse [As] moment pulse 180 2000 90 0 coincident 1000 coincident phase [ ° ] amplitude [nV] −90 0 0 2 4 6 8

0 2 4 6 8

10 12 14 16

10 12 14 16 −180 pulse moment [As] moment pulse pulse moment [As] moment pulse

Figure 2.2: Calibration sounding curves with diﬀerent separation (coincident, half- overlapping, edge-to-edge). Modeled with square loops with a side length of 45m, a homogeneous half space with a resistivity of 100Ωm, an Earth’s magnetic ﬁeld of 48000nT at ° the inclination of 60 N. 9 2 Theory

Figure 2.3: The 2D kernel function for four selected pulse moments with the same modeling conditions as in Figure 2.2. The transmitter lies coincident, half-overlapping and edge-to- edge in the west of the receiver. The midpoints of the loops are marked with arrows.

10 2.1 The basics of Magnetic Resonance Sounding (MRS)

Figure 2.4: The 2D kernel function for four selected pulse moments with the same modeling conditions as in Figure 2.2. The transmitter lies coincident, half-overlapping and edge-to- edge in the north of the receiver. The midpoints of the loops are marked with arrows, so the loops are arranged symmetrically around zero.

11 2 Theory

increase with increasing q. In the opposite layout, there is a small maximum at small pulse moments followed by a minimum. The configurations with a separation of 22.5m (half-overlapping) have nearly the same phase behavior as the data modeled for a coincident loop. With a higher separation of 45m (edge-to-edge) the sounding curve possesses a minimum at small pulse moments where the amplitudes drop off to zero, followed by an increase of the amplitudes with increasing q. The characteristic in the phase behavior is a phase shift from 180° to nearly zero in the area where the amplitudes are minimal. It is caused by a change in the sign of the real part of the MRS signal. The 2D kernel function, as it is described in equation 2.14, is the integration of the 3D kernel function over one dimension. The resulting function depends on the pulse moment q and two directions, e.g. the profile and the vertical direction or two horizontal directions. It contains information about the spatial distribution of the signal and therefore about the spatial sensitivity of the sounding to the water content distribution at this location. An infinite extension in the direction of integration is assumed. The spatial sensitivity for the soundings of Figure 2.2 for four selected pulse moments are shown in Figure 2.3 and 2.4 as well as in Figure A.1 and A.2 in the Appendix. The arrows illustrate the midpoints of the loops. For coincident loops the area of the main signal increases with increasing pulse moments. The signal spreads radially around the loop center and penetrates higher depth with increasing q. For an east-west separation, the signal is symmetric in respect to the loop center. For a north-south separation, the signal shows a displacement to the south as result of the inclination of the Earth’s magnetic field. In the case of separated loops, the main signal is shifted into the direction of the receiver. This is more visible with increasing pulse moments and higher separations. The main part of the signal for coincident loops covers a bigger area while for separated loops the signal is concentrated on shallow depth under the receiver, so the penetration depth is smaller for separated loop configuration than for coincident soundings. On the other hand separated loop measurements are more sensitive to shallower depth and investigate different areas for a reverse configuration of transmitter and receiver. With such a 2D MRS survey with separated loops and different layouts along a profile, a high 2D sensitivity is given and the 2D water content distribution is derivable. This kind of measurement is a tomographical application and it is called, like in other geophysical applications, the Magnetic Resonance Tomography (MRT).

12 2.2 The Finite Element Method (FEM)

2.2 The Finite Element Method (FEM)

The Finite Element Method is a numerical method for approximate solving of partial differential equations (PDE), especially elliptical partial differential equations. The basic elliptic partial differential equation is

− 5 · (c 5 u) + au = f (2.17) in the domain Ω. c, a, f and the unknown solution u are complex functions deﬁned on Ω. There are three main steps to solve such a problem: First, the geometry of the domain Ω and the conditions on the boundary ∂Ω have to be described. Possible conditions are:

Dirichlet: The searched function assume speciﬁc values on the boundary and is harmonic within the domain. hu = r (2.18)

Generalized Neumann: The normal derivation of the searched function assumes spe- ciﬁc values on the boundary and is harmonic within the domain.

−→n · (c 5 u) + qu = g (2.19)

Mixed condition: Is a combination of the Dirichlet and the generalized Neumann condition [Bronstein et al., 2001].

−→n is the outward unit vector and g, q, h and r are functions defined on the boundary, c is a complex function defined in the domain and u is the unknown solution of the problem. Second, the domain Ω has to be divided into smaller parts, e.g. using triangles in 2D and tetrahedrons in 3D. The triangles or tetrahedrons form the mesh, that can be refined on special points. Finally, the PDE and the boundary conditions on the mesh have to be discretised to obtain the linear system KU = F. The weak form of the PDE has to be determined to the given problem of the boundary value. Therefore a smooth test function v, which vanishes on the boundary, is defined. Multiplying it with the PDE and integrating on Ω, leads to Z Z −(5 · (c 5 u))v + auv dx = fv dx. (2.20) Ω Ω Integrating by parts and using the Green formula results in Z Z Z (c 5 u) · 5v + auv dx − −→n · (c 5 u)v ds = fv dx. (2.21) Ω ∂Ω Ω Assuming generalized Neumann conditions on the boundary, the boundary integral can be replaced by the boundary condition from equation 2.19:

13 2 Theory

Z Z Z (c 5 u) · 5v + auv dx − (−qu + g)v ds = fv dx. (2.22) Ω ∂Ω Ω By rearranging the equation in the following way, Z Z (c 5 u) · 5v + auv − fv dx − (−qu + g)v ds = 0 (2.23) Ω ∂Ω the variational or weak form of the partial diﬀerential equation is obtained. Any solution of the PDE is also a solution of the weak form of the PDE [MathWorks, 2005]. For the searched function u(x), a polynom for each small domain (subdomain) has to be prepared. The polynom can be linear, quadratic or any higher order. The polynomial function has to be continuous at the crossing from one element to the next, so that a continuous solution is developed [Bronstein et al., 2001].

N X uh(x) = Uiφi(x) (2.24) i=1 is called the piecewise linear basis function, φi(x) is linear on each triangle and takes the value 0 at all nodes xj except for xi, φi(xi) = 1, so that uh(xi) = Ui. N is the number of the edge elements and Ui are scalar coefficients and the values of uh at the nodes. N unknown coefficients has to be determined, so the test function v = φj, j = 1, 2, ..., N is defined. For x inside a triangle uh(x) is found by linear interpolation from the nodal values. Putting equation 2.24 and the test function for each element into the variational form 2.23

  Z Z Z Z PN i=1  (c 5 φi) · 5φj + aφiφj dx + qφiφj ds Ui = fφj dx + gφj ds Ω ∂Ω Ω ∂Ω | {z } | {z } KU = F (2.25) the linear system KU = F is obtained, which is easy to solve. The matrices K and F contain integrals in terms of the test functions and the coefficients defining the problem. The FEM approach is to approximate the PDE solution u by a piecewise linear function uh. Using the FEM technique, the solving of more general problems is possible, e.g. time- dependent and eigenvalue problems. The Finite Difference Method (FDM) is another numerical method and an alternative way for solving partial differential equations (PDE). The FDM requires the use of rectan- gular shapes and regular, orthogonal grids, which limits the choice of the geometry. The refinement on a regular grid leads to an immoderate increase of the number of elements and therefore to a higher numerical effort and more computing resources. The FEM mesh is more flexible, unstructured and irregular. In contrast to the FDM, it is possible to refine

14 2.2 The Finite Element Method (FEM) the mesh at specific points and to define any arbitrary geometry and boundary. The refinement is only needed in those parts, where the error is large, so the usage of an adaptive mesh is practicable. In this case, the linear system is solved for an initial mesh, the error is estimated and the refinement is only done on the triangles, where the error is large. This is a good tool to save computational time and work. Problems in the calculation with FEM occurs in an open not locked domain. The boundary conditions must be defined [Kost, 1994].

15 2 Theory

16 3 Synthetic Data

The modeling of different separated loop measurements, using the method of finite elements with the program COMSOL MULTIPHYSICS, should demonstrate the influence of the topography. The modeled loops are square loops. The advantage of a square loop is that the loops are easier to lay out if there is a slope of over 10° or an area covered with plants or trees. Nevertheless, it is not possible to model a circular loop profile with COMSOL over topographical structures with the necessary refinement around the loop, because at the point where the loops touch each other, the area is too small to generate the mesh. First of all, the differences between a sounding with a circular loop and a square loop are shown. After that, the modeling of a dune structure, the calculation of the synthetic sounding curve and the illustration of the 2D kernel function are explained to show the effect of the topography. In this chapter all models were performed for a square loop with a side length of 45m, one turn and an Earth’s magnetic field of 48000nT at an inclination of 60°N. These settings are typical for northern Europe.

3.1 Comparison of a modeling with a square and a circular loop and diﬀerent separations

In order to show the topographical effects in the modeling, square loops with a side length of 45m were used, but for many applications circular loops are preferred. Therefore, a comparison of the different layouts with different separations has to be shown, to verify the comparability. Figure 3.1 shows the calibration curves, assuming a water content of 100%, of the modeled circular and square loop data over a homogeneous half space with a resistivity of 100Ωm and a separation in the north-south direction. The circular loop has a diameter of 50.8m that corresponds to the same area as a square loop with a side length of 45m covers. The differences in the calibration curves between the two loop geometries increase with increasing separation of the loops. Table 3.1 illustrates the proportional deviation of the calibration curves between a square loop and a circular loop measurement.

17 3 Synthetic Data

deviation [%] Tx coincident with Rx 1.05 Tx half-overlapping and south of Rx 11.47 Tx half-overlapping and north of Rx 9.00 Tx edge-to-edge and south of Rx 27.00 Tx edge-to-edge and north of Rx 146.88

Table 3.1: The deviations of the calibration curves between a square loop and a circular loop modeling.

The 2D kernel function is a function of the pulse moment q and depth, profile direction or the two horizontal directions. The equation 2.12 describes the 3D kernel function. The integration along the horizontal direction perpendicular to the profile direction or along the vertical direction gives the 2D kernel function. Figure 3.2 shows sections of the three kernel parts and the full kernel function of the edge-to-edge configuration for one pulse moment (q = 2.7As), thereby transmitter (Tx) lies in the north of receiver (Rx). For this pulse moment, the maximal deviation of the two measurements can be found. The 2D kernel function is useful to show the 2D sensitivity of the measurement. The first part of the kernel function in the first column of the figure, shown as a map and a section view, depends on the pulse moment q. It shows the spatial distribution of the MRS signal amplitude. It is influenced only by the co-rotating part of the transmitter loop. In the map view, differences in a small area around the transmitter loop can be seen due to the geometry of the loops. In the second column, the second part of the 2D kernel function is illustrated. The amplitudes increase exponentially with the distance from the loop. The deflection of the spatial distribution of the first two parts of the kernel is caused due to the higher amplitudes of the co- and counter-rotating part in the south of the loop center (see also Figure 2.1 in the previous chapter). The third part of the kernel function is the geometric or projection term of the equation which scales the NMR effect. For coincident measurement, the third part of the kernel function becomes unity and has no scaling effect. In the section view of the figure, a shadow zone can be seen under the loop in the north (in this case under the receiver) of the configuration with a small deflection to the north. But it is not the characteristic of the receiver because for the third part of the kernel function it is the same for reverse transmitter and receiver configuration. In the map view, the figure shows one minimum under the point where the loops touch each other and a second minimum under the northern part of the northern loop, with a spatial displacement of the minima for the square loop sounding. The spatial differences between a sounding measured with a square loop and a circular loop are very small. The spatial distribution of the amplitudes of the kernel differ due to the different geometry of the loops only close to the loop. The differences vanish with increasing distance from the loop.

18 3.1 Comparison of a modeling with a square and a circular loop and diﬀerent separations Tx north − circular Tx south − circular Tx north − square Tx south − square 180 1600 1200 90 800 phase [ ° ] amplitude [nV] edge−to−edge 400 0 0 0 2 4 6 8

0 2 4 6 8

10 12 14 16

10 12 14 16 pulse moment [As] moment pulse pulse moment [As] moment pulse 180 1600 1200 90 800 overlap phase [ ° ] amplitude [nV] 400 0 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16

pulse moment [As] moment pulse [As] moment pulse 180 1600 1200 90 800 phase [ ° ] coincident amplitude [nV] 400 0 0 0 2 4 6 8

0 2 4 6 8

10 12 14 16

10 12 14 16 pulse moment [As] moment pulse pulse moment [As] moment pulse coincident − circular coincident − square

Figure 3.1: The amplitude and the phase of the calibration sounding curves with a separation in the north-south direction. The transmitter (Tx) lies coincident (black), half-overlapping and edge-to-edge in the north (blue) and in the south (red) of the receiver (Rx), respectively. The modeling is realized by a square loop with a side length of 45m and a circular loop with a diameter of 50.8m, a homogeneous half space of 100Ωm, an Earth’s magnetic ﬁeld of 48000nT and an inclination of 60°N.

19 3 Synthetic Data

Figure 3.2: The parts (1st-3rd column) and the full 2D kernel function (4th column) of the modeled edge-to-edge configuration for one selected pulse moment at q = 2.7As. The transmitter (Tx) lies in the north of the receiver (Rx). The first two lines show the kernels of the circular loops and the square loops in a map view with the orientation of the two loops (black line). The following lines illustrate the kernels in the section view. The arrows indicate the midpoints of the loops. Modeled for a homogeneous half space of 100Ωm, an Earth’s magnetic field of 48000nT and an inclination of 60°N. 20 3.2 Modeling with coincident and separated loop configuration

3.2 Modeling with coincident and separated loop conﬁguration

As a demonstration of the topographical effect, a dune structure with a homogeneous half space with a resistivity of 100Ωm is used. The slope of the dune is varied between 1° and 30° (1°, 2°, 5°, 10° and 30°). The slopes and the top of the dune have a length of 45m. This is equivalent to the side length of one loop. Figure 3.3 shows the arrangements of nine square loops as a top (top) and a side view (bottom) of the dune with a maximal elevation of 9m that is equal to a slope angle φ of 10°. The numbers in the figure show the midpoints of the loops. Loops with odd numbers describe loops, which lie in one plane (e.g., loop 5 is located at the top of the dune). Whereas loops with even numbers lie in two planes, for example one half of loop 4 lies on the top and the other half on the slope of the dune. The measurements along the dune are calculated in the north-south direction and for one case in the east-west direction (slope of 10°). Except for one model, a survey of 39 measurements along the dune with separations of 0m, 22.5m and 45m is simulated. Thus, the loops lie coincident, half-overlapping and edge-to-edge to each other. In one case where the dune is extended in the north-south direction and has an slope of 10°, all 81 possible configurations are calculated to show the influence of the increasing separation and volume of the dune in comparison to the flat earth. Beside the coincident sounding, there are eight possible separations of the loops, from half-overlapping to 3 loops separation.

0 distance [m] −20 100 80 60 40 20 0 −20 −40 −60 −80 −100 distance [m] N S 40

20 4 5 6 3 7 elevation [m] 1 2 φ 8 9 0 100 80 60 40 20 0 −20 −40 −60 −80 −100 distance [m]

Figure 3.3: Modeling of a dune with square loops with a side length of 45m, a separation of 0m, 22.5m and 45m, 100Ωm half space and an Earth’s magnetic ﬁeld of 48000nT with an inclination of 60°N. The ﬁgure shows a sketch of the arrangement of the loops in the top (top) and the side view (bottom). The dune has a maximum height of 9m and a slope angle φ of 10°. The loops lie half-overlapping. The numbers indicate the midpoints of the loops.

21 3 Synthetic Data

3.2.1 Modeling with COMSOL For the modeling of the dune, the electromagnetics module from COMSOL MULTI- PHYSICS was used. This program is based on the method of the finite elements and is used to compute the magnetic field of the loops in any arbitrary field. Choosing a 3D model, there are three categories of the application modes, the statics, quasi-statics and high-frequency waves mode. By using the quasi-static mode and defining two application modes for one model, it is possible to solve the magnetic fields of two loops independently and simultaneously. The physical assumption of the quasi-static mode, which is used in the multiphysics model, is that the currents generating the electromagnetic (EM) fields vary so slowly in time, that the EM fields are the same at every instant as if they had been generated by stationary sources. In the application of MRS, only the calculation of the magnetic and not of the electrical potentials is of interest. First of all, the geometry of the dune and the position of the loops have to be constructed. The volume around the loops was defined sufficiently large to minimize the influence of the −16 A boundary conditions. A minimal magnetic field of H0 = 2.2 ∗ 10 m is committed on the external boundaries. An unstructured tetrahedron mesh was generated with local mesh refinement using three factors: the element growth rate, the maximum element size of the whole volume and the maximum element size of the edges, in this case of the loops. The refinement of the mesh has only been done on the loops where the magnetic field has to be calculated. A maximum element size of 1.2m on the edges of the loops were defined. The mesh cell size grows with increasing distance from the loops with the selected element growth rate of 1.2 to the maximum element size of 40m on the boundaries. The results of a coincident sounding were compared with the analytically calculated magnetic field of a circular loop with a diameter of 50.8m to find the best refinement around the loop and the maximal needed size of the model. The mesh has nearly 550000 elements in a volume of 1.2 · 109m3. An overview of the statistics of such an unstructured tetrahedron mesh is given in Table 3.2. Figure 3.4 illustrates the 3D volume of the COMSOL model and the orientation of the loops. The upper half space is the atmosphere, the lower half space is the homogeneous underground with a defined resistivity and the loops on the surface.

number of elements 550000 boundary elements 25000 edge elements 1000 degree of freedom 1.5 · 106

Table 3.2: Statistics of the unstructured tetrahedron mesh.

22 3.2 Modeling with coincident and separated loop conﬁguration

Calculating the magnetic fields of the loops, GMRES was used as iterative solver, because direct solvers run out of memory. But the chosen solver does often not converge, so the magnetic field of the resistive (insulating) case was calculated, where the elliptical partial differential equation (PDE) is simplified. This field has been given as a start value for the magnetic field of the half space with a resistivity of 100Ωm. The magnetic flux density of the loop at the top of the dune structure with a slope 10° is shown in Figure 3.5. The upper half space has insulating electrical properties whereas the lower half space has a resistivity of 100Ωm. The magnetic field is defeated by a higher attenuation as a result of the lower resistivity. The slope and the orientation of the dune (north-south and east-west direction) was varied, and in one case a layer in 10m depth with a resistivity of 20Ωm was added. The body of the dune and the lower half space under the layer have a resistivity of 1000Ωm and 5Ωm, respectively.

Figure 3.4: The 3D COMSOL model with nine loops lie in the north-south direction over a dune structure. The upper half space is the atmosphere and the lower one is the underground .

23 3 Synthetic Data

Figure 3.5: The calculated magnetic ﬂux density of the loop at the top of the dune (Tx 5). The subsurface has a resistivity of 100Ωm and the atmosphere has insulating electrical properties.

3.3 Homogeneous half space

A homogeneous half space is modeled with a resistivity of 100Ωm and with different slopes of the dune. The calibration curves calculated for a homogeneous hypothetical water content of 100% show the influence of the topography in comparison to the flat earth on a synthetic data set with 39 configurations (Figure 3.6). The logarithmic spaced pulse moment ranges from 0.29As to 17As and the Earth’s magnetic field is 48000nT at an inclination of 60°N. The calculated calibration curves (red) of the synthetic data set over the dune are dis- played in Figure 3.6 for a slope of 10° and in the Appendix in Figure A.3 and A.4 for 5° and 30°. In addition Figure A.5 shows the whole synthetic data set of the dune with a slope of 10° with all 81 possible configurations. The sounding curves are compared to the flat earth (black line). The main diagonal of the figure shows the coincident soundings. In the first secondary diagonal, the half-overlapping and in the second secondary diagonal, the edge-to-edge soundings can be seen. For the upper half of the plot the transmitter lies in the south and for the lower half the transmitter lies in the north of the receiver, respectively. The subplot in the right corner presents the geometry of the model and the

24 3.3 Homogeneous half space

numbers and midpoints of the loops with a side length of 45m. For coincident soundings, the influence of the slope of 10° is very small. The deviations are larger for small pulse moments and decreases with increasing excitation intensities. Thus, the influence of the topography is stronger near the surface and decreases with increasing penetration depth. The amplitudes of the half-overlapping measurements also have higher variations for smaller pulse moments, but with increasing pulse moments they converge. The largest differences can be found in the edge-to-edge measurements. The configurations Tx3Rx5, Tx4Rx6, Tx5Rx7 and the reverse configurations over the top of the dune show increased amplitudes in comparison to the flat earth. Configurations are almost unaffected where one half of the loop lies on the slope of the dune and the other half lies horizontal (Tx2Rx4, Tx4Rx2, Tx6Rx8 and Tx8Rx6). For the other four configurations at the edges of the dune, the amplitudes are clearly smaller. For an improved clarification and for the comparison to a slope of 5° (a) and 30° (c), Figure 3.7 (b) illustrates the deviation between the topographical case and the flat earth. Here, the deviation was defined as the mean ratio of the amplitudes. P |Etopo|  |E | dev = log √ hori , (3.1) 2  n 

where n is the number of pulse moments, |Etopo| and |Ehori| are the amplitudes of the sounding for the topographical case and the flat earth, respectively. The logarithm was used to map values of 50% and 200% to values corresponding to -1 and 1. Ratios four times bigger have the same mapped absolute value as four times smaller ratios, but different signs. A deviation of 1 and -1 means that the amplitude of the sounding over the dune is twice as big as and the half of the amplitude for the flat earth, respectively. Figure 3.7 (b) shows, for example for the configuration Tx1Rx3 that the amplitude is about three quarters of the amplitude of the sounding without any topography. The mean deviation considering coincident soundings is very small (±0.03). A comparison of all three subplots in Figure 3.7 shows a typical pattern. The amplitudes of the coincident soundings south of loop 3 are always smaller than for the flat earth, but nearly uninfluenced, if the loop lies horizontal in the north or in the south of the dune structure (loop position 1 and 9). The deviations are higher for separated loop measurements, but always higher, if the transmitter lies in the north of the receiver. The amplitudes of all configurations with Tx3, Tx4 and Tx5, whereas Tx is situated in the north of the Rx, are raised due to the topographical effect. For the reverse configuration, this is only true for the edge-to-edge measurements with the transmitter Tx5, Tx6 and Tx7. The influence of the topography increases with increasing separation and slope of the dune.

25 3 Synthetic Data

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 5 5 5 −40N S Rx 1 10 10 10 −20 3 4 5 6 7 15 15 15 0 1 2 8 9 0 3K 0 3K 0 3K 20

0 0 0 0 elevation [m] 40 5 5 5 5 100 50 0 −50 −100 distance [m] Rx 2 10 10 10 10 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 3 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 4 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 5 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 6 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 7 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 5 5 5 5 Rx 8 10 10 10 10 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 0 0 5 5 5 Rx 9 10 10 10 15 15 15 0 3K 0 3K 0 3K

Figure 3.6: The calibration curves (red) of a survey over the dune with a slope of 10°. The black line shows the corresponding sounding for the ﬂat earth.

26 3.3 Homogeneous half space

Tx Tx Tx 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5 Rx Rx Rx

6 6 6

7 7 7

8 8 8

9 9 9

−0.4 −0.2 0 0.2 0.4 −0.5 0 0.5 −1 −0.5 0 0.5 1 deviation deviation deviation (a) (b) (c)

Figure 3.7: The deviation in the amplitude between the soundings over the dune structure with a slope of 5° (a), 10° (b), 30° (c) and the ﬂat earth.

An overview of the spatial distribution of the sensitivity of the sounding for all configurations can be seen in Figure 3.8 (sounding sensitivity). The measurements are arranged in the same order as in Figure 3.6. The arrows show the position of the transmitter (blue) and the receiver (red). The 2D kernel function is the integral of the 3D kernel function over the direction perpendicular to the profile direction as it is described in equation 2.14. The 2D kernel function is a function of the pulse moment q. The sensitivity of the whole sounding, the sounding sensitivity, can be obtained by the sum over all pulse moments. The area of high sensitivity decreases with increasing separation and is confined for the separated loop measurements to a zone under the receiver. So, changing transmitter and receiver position for one configuration, the sounding differs and reveals different information about the underground. The sensitive area is elevated with the topography and shifted over the dune structure. The 2D profile sensitivity in Figure 3.9, defined as the sum of all 2D kernel functions over all pulse moments and all configurations, shows the spatial distribution of the sensitivity of the entire profile. The area with the highest sensitivity is situated in the body of the dune and decreases with distance from the loops with a small displacement to the south. The profile sensitivity contains information about the penetration depth and about of the horizontal sphere of influence for the whole synthetic data set.

27 3 Synthetic Data

Figure 3.8: The 2D sounding sensitivity of the synthetic data set with an homogeneous half space of 100Ωm and a slope of the dune of 10°. The arrows reveal the position of the transmitter (blue) and the receiver (red).

28 3.3 Homogeneous half space

Figure 3.9: 2D proﬁle sensitivity of a dune with a slope of 10°, a homogeneous half space of 100Ωm and an Earth’s magnetic ﬁeld of 48000nT at 60°N using nine square loops with a side length of 45m and one turn.

3.3.1 Coincident soundings A direct comparison of all coincident soundings is given in Figure 3.10. It shows the dependency of the amplitude on the location of the loop and the slope of the dune. When the slope is 30° significant changes in the coincident soundings can be seen. Soundings over the dune with angles smaller than 30° are mainly influenced for smaller pulse moments, i.e. for q < 2As. For soundings north of the dune structure, the amplitude increases with increasing slope, whereas the amplitude for soundings in the south of the dune decreases with increasing slope of the dune. That is the result of the inclination of the Earth’s magnetic field. Figure 3.11 illustrates the resulting angle between the loop and the Earth’s magnetic field. For an inclination of 60°N and a slope of 30°, the resulting angle is 90° in the south at loop position 7 and 30° in the north of the dune at loop position 3. For the horizontal loops at position 1, 5 and 9, the resulting angles remain constant.

29 3 Synthetic Data 2K Tx9Rx9 1K 0 2 4 6 8 10 12 14 16 2K Tx8Rx8 1K 0 2 4 6 8 10 12 14 16 2K Tx7Rx7 30° 1K 0 2 4 6 8 10 12 14 16 2K 10° Tx6Rx6 1K 0 2 4 6 8 10 12 14 16 5° 2K Tx5Rx5 2° amplitude [nV] 1K 0 2 4 6 8 10 12 14 16 2K 1° Tx4Rx4 1K 0 2 4 6 8 10 12 14 16 0° 2K Tx3Rx3 1K 0 2 4 6 8 10 12 14 16 2K Tx2Rx2 1K 0 2 4 6 8 10 12 14 16 2K Tx1Rx1 1K

0 2 4 6 8

10 12 14 16 pulse moment [As] moment pulse

Figure 3.10: The ﬁgure shows the amplitudes of all coincident soundings over the dune (1-9) at diﬀerent slopes (0°-30°).

30 3.3 Homogeneous half space

To compare the sounding curves to corresponding soundings of the flat earth and show the effect of the resulting angle, a modeling was performed with a homogeneous half space and a constant magnetic field of 48000nT but with different inclinations. The calculated sounding curves in the three subplots of Figure 3.12 (red) illustrate the dependency of the amplitude on the inclination of the Earth’s magnetic field in the horizontal case for an inclination of 30°N (left), 60°N (center) and 90°N (right). The amplitudes decrease with increasing inclination of the Earth’s magnetic field. In the left and right subplot the sounding curves of the coincident soundings on the slope with a slope angle of 30° are added (Tx3Rx3 and Tx7Rx7). For both configurations, the deviation from the flat earth increases with increasing pulse moments. But the deviations are small. For the configuration Tx7Rx7 on the southern side of the dune, the amplitudes are smaller than for a flat earth with an inclination of 90°N. Despite the effect of the angle, there is the effect of the volume, that increases with increasing depth and pulse moment, respectively. In Figure 3.13, the spatial distribution of the sensitivity shown as kernel parts (1st-3rd column) and the full 2D kernel function (pulse sensitivity, 4th column) for one selected pulse moment over all calculated slopes of the dune (0°-30°) is given for the configuration Tx7Rx7. The 2D kernel function has a deflection to the south, due to the inclination of the Earth’s magnetic field. An area of decreased amplitude increases on the southern side of the loop 7 with increasing slope and consequently increasing volume of investigation. This leads to smaller amplitudes of the sounding with increasing slope. The subplot in the center of Figure 3.12 shows the amplitudes of the horizontally lying loops, i.e. for the soundings of Tx1Rx1, Tx5Rx5 and Tx9Rx9. The deviation between the flat earth and these coincident soundings are relatively large, although the loops are lying horizontal and there cannot be any effect of the angle. Thus the volume affects the soundings, caused by the lifting of the dune. The amplitudes of the sounding Tx5Rx5 are clearly smaller, as a consequence of the reduced volume of investigation, whereas the amplitudes of the sounding Tx1Rx1 and Tx9Rx9 are larger due to the additional volume of investigation. The effect of the volume can be seen explicitly in Figure 3.14. The figure shows spatial distribution of the MRS signal amplitude (first column), of the sensitivity of the receiver (second column) and the full 2D kernel function (fourth column) for the configuration Tx1Rx1 and different slopes of the dune (0°-30°). The third part of the 2D kernel function is unity for coincident soundings. In the southern side of loop 1, the volume of investigation increases due to the lifting of the dune. This results in higher amplitudes for sounding with higher slopes.

31 3 Synthetic Data

Figure 3.11: The ﬁgure shows the resulting angle between the loop and the inclination of the Earth’s magnetic ﬁeld of 60°N (black arrows). The dune has a slope of 30°. The resulting angle increases in the south of the dune at loop position 7 and decreases in the north of the dune at loop position 3.

angle 30° angle 60° angle 90° 0 0 0

2 2 2

4 4 4

6 6 6

8 8 8

10 10 10 pulse moment [As] pulse moment [As] pulse moment [As]

12 12 12

14 14 14

16 16 16

18 18 18 1000 1200 1400 1600 1800 2000 1000 1200 1400 1600 1800 2000 1000 1200 1400 1600 1800 2000 amplitude [nV] amplitude [nV] amplitude [nV]

horizontal Tx3Rx3 horizontal Tx1Rx1 Tx5Rx5 Tx9Rx9 horizontal Tx7Rx7

Figure 3.12: Dependency of the amplitudes on the inclination of the Earth’s magnetic field (30°N, 60°N, 90°N) for the flat earth (red lines). The figure shows different coincident soundings, whereas the loops lie only in one plane with an resulting angle of 30°(left), 60°(horizontal lying loops, center) and 90°(right).

32 3.3 Homogeneous half space

Figure 3.13: A comparison of the parts (column 1st-3rd) and the full 2D kernel function (4th column) of the coincident sounding Tx7Rx7 for diﬀerent slopes of the dune and one selected pulse moment at q = 8.79As.

Figure 3.14: A comparison of the parts (column 1st-3rd) and the full 2D kernel function (4th column) of the coincident sounding Tx1Rx1 in the north of the dune for diﬀerent slopes of the dune and one selected pulse moment at q = 8.79As.

33 3 Synthetic Data

3.3.2 Edge-to-edge measurements For separated loop measurements, the effect of the volume and the effect of the slope cannot easily be distinguished, because both loops contributes to the sounding and are in a certain angle to each other. Figure 3.15 shows the sounding curves of all configurations and different slopes of the dune. In the first and second row of the figure the transmitter lies in the north and south of the receiver, respectively. This reveals that the amplitude of the configuration Tx1Rx3, Tx7Rx9 and the reverse configuration decreases with increasing slope of the dune, whereas the configuration Tx2Rx4, Tx6Rx8 and the reverse configuration are nearly uninfluenced by the topography. For all other configurations over the top of the dune, the amplitudes increase with increasing slope. For the spatial distribution of the sensitivity for different slopes of the dune, the 2D kernel function for three selected configurations is shown in Figures 3.16, 3.17 and 3.18. Figure 3.16 shows the single integrated parts and the full 2D kernel function for the configuration Tx4Rx6 for one selected pulse moment (q=8.79As) and different slopes. This configuration is representative for all convex edge-to-edge measurements. The transmitter lies in the north of the receiver. One half of the loops is situated edge-to-edge on the ridge and the other half is on the slope of the dune. As a consequence of the convex configuration, the sensitive area of the receiver coincides with the area of the maximum amplitudes as defined by the transmitter. The third part of the kernel function shows a shadow zone under the transmitter and in the south of the configuration. This part is independent of the orientation of the loops and therefore is the same for the reverse configuration. With increasing slope of the dune, this zone is lifted up and increases. But the influence of the shadow zone with the decreasing amplitudes in the third part is very small, due to the fact, that the zone is shifted out for the convex configuration Tx3Rx5. This can be seen in Figure A.6 in the Appendix. Hence, the effect of the coincide volume is higher. In Figure 3.17, the 2D kernel function of the concave configuration Tx1Rx3 is illustrated for different slopes of the dune. The transmitter lies in the north of the receiver and remains constant, whereby the receiver is lifted up. The loops lie in an obtuse angle to each other, i.e. an angle between 90° and 180°. For a slope of the dune of 10°, the angle between the two loops is 170°. By lifting the dune, the amplitude of the MRS signal covers an increasing area and the volume of investigation increases, but the sensitive area of the receiver is turned away from the transmitter. The common area of the receiver and the transmitter decreases. The shadow zone of the third part of the 2D kernel function increases and is shifted into the body of the dune with increasing angle. All these facts result in the decrease of the amplitude of the sounding curves and a clearly reduced sensitive area in the 2D kernel function with increasing slope (fourth column).

34 3.3 Homogeneous half space 3K 3K 2K 2K 1K 1K Tx7Rx9 Tx9Rx7 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 3K 3K 2K 2K 30° 1K 1K Tx6Rx8 Tx8Rx6 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 3K 3K 10° 2K 2K 1K 1K Tx5Rx7 Tx7Rx5 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 5° 3K 3K 2K 2K 1K 1K 2° Tx4Rx6 Tx6Rx4 0 0 amplitude [nV] amplitude [nV] 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 3K 3K 1° 2K 2K 1K 1K Tx3Rx5 Tx5Rx3 0 0 0 2 4 6 8 0 2 4 6 8 0° 10 12 14 16 10 12 14 16 3K 3K 2K 2K 1K 1K Tx2Rx4 Tx4Rx2 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 3K 3K 2K 2K 1K 1K Tx1Rx3 Tx3Rx1 0 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16

pulse moment [As] moment pulse [As] moment pulse Tx north of Rx Tx south of Rx

Figure 3.15: The figure shows the amplitudes of all edge-to-edge soundings over the dune at different slopes (0°-30°) of the dune. The transmitter lies in the north (first row) and in the south (second row) of the receiver, respectively.

35 3 Synthetic Data

For the third kind of loop layout for the edge-to-edge measurement, the 2D kernel function of the configuration Tx6Rx8 is presented in Figure 3.18. All loops with even numbers lie in two planes. One part of the loops lies on the slope of the dune and the other half of the loops lies horizontally. For loop 6 and 8, these parts move towards and away from each other, respectively. In the first row of the figure, the MRS signal amplitude, which depends on the pulse moment q, can be seen. The total volume of investigation for the first part of the kernel function is reduced due to the lifting and the convex bended loop (6). The spatial distribution of the sensitivity of the receiver increases, due to the concave bended loop (8). The third part changes slightly and only outside of the sensitive area of the receiver. The amplitudes of the sounding curves for such a kind of loop layout are more or less identical to the flat earth. Thus, the effect of the volume, the effect of the slope and the influence of the projection term is balanced. An overview of all half-overlapping measurements but with different slopes of the dune is shown in Figure A.7 in the Appendix. All calculated 2D kernel functions of the homogeneous model with a resistivity of 100Ωm can be found in an extra appendix to this thesis.

Figure 3.16: Presentation of the 2D kernel functions in the north-south direction. The transmitter (Tx4) is situated in the north of the receiver (Rx6). The rows show the 2D kernels for diﬀerent slopes of the dune (0°-30°) for the 13th pulse moment at q=8.79As. The columns show the three parts of the kernels and the full kernel function.

36 3.3 Homogeneous half space

Figure 3.17: Presentation of the 2D kernel functions in the north-south direction. The transmitter (Tx1) is situated in the north of the receiver (Rx3). The rows show the 2D kernels for diﬀerent slopes of the dune (0°-30°) for the 13th pulse moment at q=8.79As. The columns show the three parts of the kernels and the full kernel function.

Figure 3.18: Presentation of the 2D kernel functions in the north-south direction. The transmitter (Tx6) is situated in the north of the receiver (Rx8). The rows show the 2D kernels for diﬀerent slopes of the dune (0°-30°) for the 13th pulse moment at q=8.79As. The columns show the three parts of the kernels and the full kernel function.

37 3 Synthetic Data

3.4 Three layer model

A three layer model is constructed to show the effect of the layered model with a high resistive layer and a high conductive half space. For the modeling a water content distribution of 100% is assumed. Thus, only the influence of the pure electrical conductivity is demonstrated. A layer (10m) with a resistivity of 20Ωm was added under the dune with a slope of 10°. The body of the dune has a resistivity of 1000Ωm and the half space under the layer has 5Ωm. The synthetic sounding curves are compared to a one dimensional two layer model, neglecting the dune body. The first layer with a thickness of 10m at 0m has a resistivity of 20Ωm followed by a 5Ωm homogeneous half space. Figure 3.19 presents the calibration curves of the three layer model (red) and the layered model without the body of the dune (black). The amplitudes of the coincident soundings as well as of the half-overlapping measurements increases with decreasing distance from the dune center. This is caused by the high resistive body of the dune. The strongest differences occur in the edge-to-edge measurements. Figure 3.20 shows the deviation between these two models. The deviation is calculated as described in equation 3.1. Almost all amplitudes are increased in comparison to a horizontal surface of the earth’s. Here, the resistive body of the dune plays the major role. There are very high differences between these two models, but the amplitudes differ more if the transmitter lies in the north of the dune. It is the same effect as it is for the homogeneous half space. In comparison to the homogeneous half space with a resistivity of 100Ωm, the penetration depth is smaller due to the stronger attenuation as a result of the high conductive layer in the underground. The amplitudes of the soundings are smaller, too. The 2D profile sensitivity of the model in Figure 3.21 is much smaller and there is no clear displacement into the south as it is for the homogeneous half space with a resistivity of 100Ωm (see Figure 3.9). There is no 1D conductive model for such a three layer model with a high resistive dune. By the neglect of the topography, a 2D model must be assumed. But the effect of the volume and of the angle between the loop and Earth’s magnetic field, as it is shown in section 3.3, influences the amplitude as well. The simplification of a topographical layered model to a 1D conductive layered model is not sufficient.

38 3.4 Three layer model

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 5 5 5 −40N S Rx 1 10 10 10 −20 3 4 5 6 7 15 15 15 0 1 2 8 9 0 2000 0 2000 0 2000 20

0 0 0 0 elevation [m] 40 5 5 5 5 100 50 0 −50 −100 distance [m] Rx 2 10 10 10 10 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 0 0 0 0 5 5 5 5 5 Rx 3 10 10 10 10 10 15 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 2000 0 0 0 0 0 5 5 5 5 5 Rx 4 10 10 10 10 10 15 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 2000 0 0 0 0 0 5 5 5 5 5 Rx 5 10 10 10 10 10 15 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 2000 0 0 0 0 0 5 5 5 5 5 Rx 6 10 10 10 10 10 15 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 2000 0 0 0 0 0 5 5 5 5 5 Rx 7 10 10 10 10 10 15 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 2000 0 0 0 0 5 5 5 5 Rx 8 10 10 10 10 15 15 15 15 0 2000 0 2000 0 2000 0 2000 0 0 0 5 5 5 Rx 9 10 10 10 15 15 15 0 2000 0 2000 0 2000

Figure 3.19: The calibration curves (red) of the three layer model with a resistivity of 1000Ωm in the body of the dune, 20Ωm in the 10m thick layer and 5Ωm in the half space. It was modeled for a dune with a slope of 10°, an Earth’s magnetic ﬁeld of 48000nT at 60°N. The black lines shows the corresponding sounding curves for a two layer model without the high resistive dune body. 39 3 Synthetic Data

Tx 1 2 3 4 5 6 7 8 9

5 Rx

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 deviation

Figure 3.20: Deviation between the ﬂat layered Earth and the three layer model with topography.

Figure 3.21: 2D proﬁle sensitivity of the three layer model. The body of the dune has a resistivity of 1000Ωm, the 10m thick layer under the dune has a resistivity of 20Ωm and the homogeneous half space has 5Ωm. 40 3.5 Comparison between north-south and east-west proﬁle

3.5 Comparison between north-south and east-west proﬁle

The dune with a homogeneous half space of 100Ωm and a slope of 10° strikes in the north- south direction. The loops lie perpendicular to it. The calibration sounding curves of the synthetic data set with a profile in the east-west direction can be seen in Figure 3.22. The differences between the horizontal surface and the dune model are small. This is mainly caused by the properties of the Earth’s magnetic field, which is a vector field with an intensity and a direction. In the east-west direction, the magnetic field is symmetric to the loops center and the resulting angle between the loop and the magnetic field is 90° for the flat earth. Thus, the direction vector is zero along the east-west direction. For the modeling of a coincident sounding neglecting the topography, the component in the east-west direction is unequal to zero, i.e. in the case of a dune with a slope of 10°, the resulting angle between the Earth’s magnetic field and the slope is 80° to the west and to the east for loop 3 and loop 7, respectively. The resulting angle along the north-south direction remains constant with an angle of 60° to the north. A comparison between the deviation with a separation in the east-west (a) and the north-south (b) direction in Figure 3.23 shows that the influence of the topography for a survey with a profile in the east-west direction is clearly more slightly. These two subplots are scaled differently. The maximum of the deviation in the east-west direction is ±0.16, whereas the maximum in the north-south direction is ±0.57. The influence of the slope of the dune is nearly the same in both profile directions. All edge-to-edge measurements with a convex loop configuration have higher and with a concave loop configuration have smaller amplitudes compared to the flat earth. The highest deviation in the east-west direction for the coincident soundings can be found for Tx5Rx5, which is the same for the north-south direction (-0.03). The spatial distribution of the parts and the full 2D kernel function for the sounding Tx5Rx5 are demonstrated in Figure 3.24. The first two rows and the last two rows show the kernel with a profile in the north-south and east-west direction, respectively. The first column illustrates the spatial distribution of the MRS signal amplitude depending on the pulse moment q. For a profile in the east-west direction and for the flat earth a small deflection to the west can be seen, due to the higher amplitudes of the co-rotating part in the west of the loop center (Figure 2.1), whereas the second part of the kernel function shows a deflection to the east, due to the higher amplitudes of the counter-rotating part of the magnetic field of the receiver. The full 2D kernel function is symmetric to the loop center and shows the resulting contribution of the recorded MRS signal. Loop 5 is only lifted up as a consequence of the topography of the dune. This results in a reduction of the volume of investigation and a decrease of the amplitude. Figure 3.25 shows the spatial distribution of the sensitivity for the edge-to-edge measurement Tx4Rx6. The third part of the 2D kernel function illustrates an area of decreased amplitudes under the loops. This zone is symmetric to the midpoint of the two loops for the profile in the east-west direction, but the changes due to the lifting of the dune are

41 3 Synthetic Data very small. Here, the effects for the convex configuration Tx4Rx6 with a north-south and east-west profile are the same, e.g. the effect of the coincide volume (see section 3.3). Only the resulting angle between the loop and the inclination of the Earth’s magnetic field differs. It is in the east-west direction 80° to the west for loop 3 and 90° for loop 5. This leads to a smaller amplitudes compared to the flat earth.

42 3.5 Comparison between north-south and east-west proﬁle

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 W E 5 5 5 −40 Rx 1 10 10 10 −20 3 4 5 6 7 15 15 15 0 1 2 8 9 0 3K 0 3K 0 3K 20

Figure 3.22: The calibration curves (red) of a survey over the dune with a slope of 10°and a proﬁle in the east-west direction. The black line shows the corresponding sounding for the ﬂat earth. 43 3 Synthetic Data

Tx Tx 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

1 1

2 2

3 3

4 4

5 5 Rx Rx

6 6

7 7

8 8

9 9

−0.15 −0.1 −0.05 0 0.05 0.1 0.15 −0.5 0 0.5 deviation deviation

Figure 3.23: The deviation in the amplitude between the soundings over the dune structure with proﬁle direction east-west (left), north-south (right) and the ﬂat earth.

Figure 3.24: The parts (1st-3rd column) and the full 2D kernel function (4th column) of the modeled coincident sounding at q = 2.7As. The first two rows and the last two rows show the kernels of the sounding Tx5Rx5 with a profile direction north-south and east-west, respectively. Modeling was performed for a homogeneous half space of 100Ωm, an Earth’s magnetic field of 48000nT at 60°N.

44 3.6 Inversion of synthetic data

Figure 3.25: The parts (1st-3rd column) and the full 2D kernel function (4th column) of the modeled edge-to-edge measurement at q = 2.7As. The first two rows and the last two rows show the kernels of the sounding Tx3Rx5 with a profile direction north-south and east-west, respectively. Tx lies in the north and in the west, respectively. Modeling was performed for a homogeneous half space of 100Ωm, an Earth’s magnetic field of 48000nT at 60°N.

3.6 Inversion of synthetic data

A new inversion code has been developed by M. Hertrich (unpublished). It allows the inversion of the 2D water content distribution for separated loop surveys taking the topography into account. First of all, a mesh using distmesh for the program MATLAB is generated. This mesh program gives the position of the nodes and the indices of the triangles (details given by Persson and Strang [2004]). The two dimensional mesh can be seen in Figure 3.26 (black). The third dimension consists of equidistant slices in the direction perpendicular to the profile direction. The 3D kernel function was interpolated on the refined 3D mesh and was integrated over one direction to get the 2D kernel function. For the refinement, each triangle of the mesh calculated with distmesh was divided into four equally sized triangles (green). The two dimensional and coarser (black) mesh is used for the inversion. The triangles have a side length of 2m at the surface under the loops, which increases with increasing distance from the loops.

45 3 Synthetic Data

−20

60 elevation [m]

100 −200 −150 −100 −50 0 50 100 150 200 distance [m]

Figure 3.26: 2D mesh of the dune structure in dependency of the proﬁle direction and the depth generated with the program distmesh. The black irregular mesh was reﬁned (green) for the interpolation of the 3D kernel function.

To show the eﬀect of the topography for the inversion, the sounding curves for two kinds of models were calculated. Therefore a 1D layered aquifer and 2D water bearing lens with water content of 30% surrounded by 5% were modeled on the reﬁned grid. The electrical properties of the models are homogeneous with a resistivity of 100Ωm. Figure 3.27 shows the models with the corresponding water content distributions. The layered aquifer lies in 1m depth and has a thickness of 10m. The lens is located 4m below the surface and is therefore inside the body of the dune. It has an extension of 50m and a thickness of 10m.

Figure 3.27: The model with a homogeneous resistivity of 100Ωm and the water content distribution of a layered aquifer in 1m depth with a thickness of 10m and a water bearing lens with a minor axis of 10m and a major axis of 50m can be seen. The lens is situated 4m below the top of the dune. The dune has a slope of 10°.

The sounding curves (red) of all 39 configurations based on the layered aquifer, the forward calculated data as well as the inversion results can be seen in Figure 3.29. The inversion result at the bottom left in the figure illustrates a nearly 10m water bearing aquifer with an average water content of 30%. The differences between the inversion

46 3.6 Inversion of synthetic data

and the model at the top of this figure demonstrate, that there is a good agreement in the area where the 2D profile sensitivity is high. The profile sensitivity in Figure 3.28 gives a good approximation of the sensitive area for the inversion. The water content is underestimated towards the borders, whereas the water content under the dune around the layer is slightly overestimated. The water content for the layered model is in average 0.6vol.% underestimated. The model error of the 1D model1 is 54.2%. The relatively big error is caused by the large area, which is considered in the inversion and which is larger than the sensitive area of the profile. The area of the 2D kernel function considered for the inversion was 4 · 104m2 and had a thickness of 100m. The inversion results and the differences between the model and the inversion of the 2D water bearing lens are presented in the Figure 3.30. It shows the lens in the depth of 4m below the surface. As it can be seen in the figure at the top right, the water content in the south under the lens is easily overrated. The error of the 2D model is 24.0% and small in comparison to the 1D model. This is caused by the geometry of the lens. The lens is situated in an area, where the sensitivity is high. Outside of the sensitive area, the error of estimated water content is smaller, due to the homogeneous water content distribution of the half space. The whole water content for the 2D model is in average 0.1vol.% slightly underestimated.

Figure 3.28: 2D proﬁle sensitivity of the model with a slope of 10°.

1 The root mean square of the model: norm fmes−fcal rms = √ fcal · 100 [%], with N is the number of nodes, f and f are the calculated water N cal mes content and the water content of the model [MathWorks, 2005].

47 3 Synthetic Data

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 −20 3 4 5 6 7 0 1 2 8 9 Rx 1 10 10 10 20 40 20 20 20 60

0 300 0 300 0 300 elevation [m] 80 0 0 0 0 150 100 50 0 −50 −100 −150 distance [m] Rx 2 10 10 10 10

20 20 20 20 0 300 0 300 0 300 0 300 0 0 0 0 0 −0.2 −0.1 0 0.1 0.2 difference Rx 3 10 10 10 10 10

20 20 20 20 20 0 300 0 300 0 300 0 300 0 300 0 0 0 0 0

Rx 4 10 10 10 10 10

20 20 20 20 20 0 300 0 300 0 300 0 300 0 300 0 0 0 0 0

Rx 5 10 10 10 10 10

20 20 20 20 20 0 300 0 300 0 300 0 300 0 300 0 0 0 0 0

Rx 6 10 10 10 10 10

20 20 20 20 20 0 300 0 300 0 300 0 300 0 300 0 0 0 0 0

Rx 7 10 10 10 10 10

20 20 20 20 20 0 300 0 300 0 300 0 300 0 300 0 0 0 0 −20 4 5 6 1 2 3 7 8 9 Rx 8 0 10 10 10 10 20 40 20 20 20 20 60 0 300 0 300 0 300 0 300 80 0 0 0 150 100 50 0 −50 −100 −150 Rx 9 distance [m] 10 10 10

20 20 20 0 0.1 0.2 0.3 0.4 0.5 0 300 0 300 0 300 water content

Figure 3.29: The ﬁgure shows the sounding curves of the layered aquifer models (red), the forward calculated data (blue), the inversion results (at the bottom left) and the diﬀerence of the model and the inversion result (on the top right). 48 3.6 Inversion of synthetic data

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 −20 3 4 5 6 7 0 1 2 8 9 Rx 1 10 10 10 20 40 20 20 20 60

0 250 0 250 0 250 elevation [m] 80 0 0 0 0 150 100 50 0 −50 −100 −150 distance [m] Rx 2 10 10 10 10

20 20 20 20 0 250 0 250 0 250 0 250 0 0 0 0 0 −0.2 −0.1 0 0.1 0.2 difference Rx 3 10 10 10 10 10

20 20 20 20 20 0 250 0 250 0 250 0 250 0 250 0 0 0 0 0

Rx 4 10 10 10 10 10

20 20 20 20 20 0 250 0 250 0 250 0 250 0 250 0 0 0 0 0

Rx 5 10 10 10 10 10

20 20 20 20 20 0 250 0 250 0 250 0 250 0 250 0 0 0 0 0

Rx 6 10 10 10 10 10

20 20 20 20 20 0 250 0 250 0 250 0 250 0 250 0 0 0 0 0

Rx 7 10 10 10 10 10

20 20 20 20 20 0 250 0 250 0 250 0 250 0 250 0 0 0 0 −20 4 5 6 1 2 3 7 8 9 Rx 8 0 10 10 10 10 20 40 20 20 20 20 60 0 250 0 250 0 250 0 250 80 0 0 0 150 100 50 0 −50 −100 −150 Rx 9 distance [m] 10 10 10

20 20 20 0 0.1 0.2 0.3 0.4 0.5 0 250 0 250 0 250 water content

Figure 3.30: The ﬁgure shows the sounding curves of the 2D water bearing lens (red), the forward calculated data (blue), the inversion results (at the bottom left) and the diﬀerence of the model and the inversion result (on the top right). 49 3 Synthetic Data

Now, the topography is neglected in the inversion to show the importance of the consideration of the topography. The 2D kernel function of a 100Ωm homogeneous half space without any topography was calculated and used for the inversion of the synthetic data set. The models (top) and the different inversion results with (middle) and without (bottom) the consideration of the topography are illustrated in Figure 3.31. The root mean square error (rms) of the adaptation is calculated as described in the matlab manual2 [Math- Works, 2005] and is presented as the average rms of all soundings in Table 3.3. When the topography is neglected in the inversion, the adaptation error increases. The effect of the angle between the slope and the Earth’s magnetic field, which vary in dependency of the loop position, and the effect of the volume, as it is presented in section 3.3, influences the sounding curves. This is not considered in the inversion neglecting the topography and makes a correct adaptation impossible. The shifting of the 2D water content distribution to the true altitude is shown in Figure 3.32. The neglect of the topography in the inversion is a simple approach known from other geophysical methods. The estimated water content is shifted in dependency of the size of the grid for the inversion at the surface. The 2D model of the water content distribution is interpolated on a regular grid with a size of 2m and is shifted only vertically to the true altitude of the topography. The regular grid is not finer than the irregular inversion grid to get no additional smoothing into the data and also not coarser to avoid additional deformation in the data. For the 1D layered model, the water content in the north and south of the dune is raised and situated directly under the surface. After shifting, the layer is indicated but has more internal structures. The bottom line of the layer is more or less identical with the given model. On the northern side of the dune body, the water content is increased to nearly 30%, due to the higher amplitudes, e.g. of the coincident sounding Tx3Rx3 in comparison to the flat earth. This results in a higher water content at this location. The shifted inversion results of the 2D model shows that the lens is hardly recognizable and clearly distorted to the south. In comparison to the water content from the inversion including the topography, the water content is raised in the entire model. The raised water content at nearly ±100m at the surface is explainable by the effect of the volume e.g. of a coincident measurement as it is described in section 3.3. The amplitudes increase due to the slope of the dune and the associated higher volume of investigation. Thus, the water content for the flat earth is clearly overestimated. The homogeneous half space under the structure of both models has higher differentiations. Due to the fact, that the water content in the inversion neglecting the topography is raised directly under the surface and not only directly under the topographical structure, it can be concluded that, for a MRT measurement along the profile, the influence of the topography cannot be neglected.

|E |−|E | norm mes cal |E | 2 √ cal rms = n · 100 [%], with n is the number of pulse moments, |Ecal|and |Emes| are the amplitudes of the sounding for the forward calculated data and measured data.

50 3.6 Inversion of synthetic data

model rms [%] 1D layer 21.6 1D layer without topography 38.3 2D water bearing lens 13.5 2D lens without topography 43.9

Table 3.3: Average root mean square error (rms) of the inversion.

Figure 3.31: At the top, the model with the corresponding water content distribution and below, the inversion results with and without the consideration of the topography is illustrated.

51 3 Synthetic Data

Figure 3.32: The shift of the water content to the true altitude after the inversion without topography.

52 4 Field data set

To validate the modeled data, an exemplary survey has been realized across a dune on the Darss peninsula, at the Baltic sea in the northeast of Germany. The Darss borders in the southeast on the peninsula Fischland and in the northeast near Prerow on Zingst (Figure 4.1). The peninsula Darss is divided into three parts: Altdarss in the southeast, Neudarss in the north and Vordarss in the southwest. The test site is situated on the Vordarss (Figure 4.2). It is bordered in the south by the Schifferberg, in the north by the Rehberge and in the east by the Altdarss. In the northwest of the Vordarss is the Baltic sea and in the south the Saaler Bodden. The geology of the area mainly consists of fine and medium sands and till. The first aquifer consists of glacio-fluvial holocene sands and occurs close to the surface at mainly the whole Darss. The aquifer has a thickness of 2-5m and is followed by an aquiclude composed of pleistocene till [Braatz, 2005]. The conductivity S of the seawater is 1.49 m , this is equal to 0.67Ωm. The Baltic sea water is characterized as a brackish water with an average salinity of 12 (Leibniz-Institut fur¨ Ostseeforschung Warnemunde,¨ www.io-warnemuende.de).

Figure 4.1: The peninsulas Fischland, Darss and Zingst, at the Baltic sea, in the northeast of Germany. The test site is situated in the southwest of the Darss. A detailed image of the test site can be seen in Figure 4.2.

53 4 Field data set

Figure 4.2: A detail of the southern part of the Darss peninsula with the area under investigation (rectangle). The antenna (circle) in the southwest of the test site was nearly 1.8km away from the proﬁle.

NNW SSE 10 3 5 2 4 1 5 0 elevation [m] −5 0 10 20 30 40 50 60 70 80 distance [m]

Figure 4.3: The topography of the dune on the Darss peninsula. The Baltic sea is situated in the north northwest of the proﬁle. The numbers shows the midpoints of the square loops with a side length of 25m.

Figure 4.3 shows the topography of the dune. It has a slope of nearly 15° on the northern side and nearly 20° on the southern side. The Baltic sea is situated in the north northwest of the proﬁle. The relative elevation is based on the water level. The dune has been surveyed using Geoelectric, Induced Polarisation (IP) and Magnetic Resonance Tomography (MRT).

54 4.1 Geophysical measurements

4.1 Geophysical measurements

The electrical model was derived from geoelectrical and Induced Polarisation (IP) measurements along the dune. The geoelectric data with a spacing of 1m and a profile of 124m were measured with a wenner configuration. Figure 4.4 shows high resistivities in the body of the dune with about 3000Ωm and a layer with a resistivity of about 20Ωm. The IP data were measured with a schlumberger configuration to show the differences between the till with a high IP effect and thus higher chargeability and the water bearing layer in lower depths. It has a profile length of 65m with a spacing of 1m at the same profile. Within the first 5m the chargeability is very low indicating a water layer. It increases with increasing depth. An aquiclude is assumed at approximately 10m. However, for a reliable determination of the resistivity and the differentiation between the aquifer and the aquiclude, the penetration depth is not sufficient. For the modeling, a very dry dune and a 10m water bearing layer followed by an aquiclude is assumed.

Figure 4.4: Section of the resistivity (top) measured with a wenner conﬁguration and of the Induced Polarisation (bottom) measured with schlumberger conﬁguration.

55 4 Field data set

For the measurement at the test site, five square loops with a side length of 25m and two turns were designed at the surface. The data were measured with the NUMIS Lite device from Iris Instruments for separations of 0m, 12.5m and 25m. Thus, 19 measurements are possible, with coincident, half-overlapping and edge-to-edge configuration. The numbers in Figure 4.3 indicate the midpoints of the loops. The local Earth’s magnetic field at the test site was 49600nT, which is equal to a Larmor frequency fL = 2111.8Hz, with an inclination of 69°N. The profile direction runs from north-northwest to south-southeast. The excitation intensities ranged from 0.1 to 4 As. Both the single stacks and the noise are very spiky (Figure 4.5 (a) and (c)). This was possibly influenced by an antenna in the southwest of the test site, which was about 1.8km away from the profile (Figure 4.2). The spikes range from 2000nV to 6000nV whereas the signal amplitudes are around 600nV. In the stacked signal in Figure 4.5 (d) the spike from subplot (c) can be seen at around 100ms.

56 4.2 Data processing

stacked noise noise (single) 250 5000 real part real part imag. part imag. part 200 abs. part 4000 abs. part 150

3000 100

2000 50 Voltage [nV]

Voltage [nV] 0 1000 −50

0 −100

−1000 −150 0 50 100 150 200 250 0 50 100 150 200 250 time [ms] time [ms] (a) (b)

stacked data single stack 300 5000 real part real part imag. part 4000 imag. part 200 abs. part abs. part 3000

2000 100 1000

0 0 Voltage [nV]

Voltage [nV] −1000 −100 −2000

−3000 −200 −4000

−5000 −300 0 50 100 150 200 250 0 50 100 150 200 250 time [ms] time [ms] (c) (d)

Figure 4.5: Field data set. Noise of one single stack (a), stacked noise (b), one single stack (c) and stacked signal (d).

4.2 Data processing

In the MRS technique, the signal in the range of a few hundred nV is measured using the synchronous detection. As a result, two components of the MRS signal are observed, the in-phase and out-of-phase signal [Legchenko and Valla, 2002]. The complex MRS signal is composed of the wanted signal and the noise. The noise can be distinguished between the spiky noise, the electromagnetic noise produced by electrical power lines and the stochastic or natural noise. The ﬁrst two kinds of noise are man-made. To improve the quality of the data, i.e. the signal-to-noise ratio, the data should be stacked with a rate of 30 to 120 single stacks. In√ the case of Gaussian or stochastic noise, the signal-to-noise ratio increases with the factor n, with n numbers of single stacks. The measurements at the test site are done with 24 and 32 single stacks which is quite little.

57 4 Field data set

The initial attempt of improving the signal-to-noise ratio by post-processing the data after the registration was the elimination of the bad single stacks with spiky events, which have an amplitude 5 times higher than the mean maximal signal amplitude. By the removal of the spiky single stacks, from 20% to 40% of the single stacks were deleted. The signal- to-noise ratio decreases as a consequence of the relatively small number of single stacks, and smoothness of the sounding curves was not improved. A post-processing scheme based on the master thesis of Strehl [2006] achieves the best possible improvement. First of all, the spiky noise was eliminated using the discrete wavelet transform. After transforming the signal into the wavelet domain, large wavelet coefficients are associated with the high amplitudes in the signal. The interfering part of the signal with large wavelet coefficients can be reconstructed using the inverse wavelet transform and is subtracted from the MRS signal in the time domain. By applying this algorithm to all single stacks of the survey, the remaining noise is reduced to the range of the natural noise. Figure 4.6 (a) shows the signal of one single stack as the amplitude, the in-phase and the out-of-phase plot with a spike at 140ms (black), which is filtered out using the wavelet transform (red). The wavelet transform leads to only small changes in the signal just around the spike. Figure 4.6 (b) presents the time signal of the stacked data before (black) and after (red) the application of the spike-filter. It reveals that the spike at 140ms affects the stacked signal, too. For the further post-processing the part of the harmonic noise produced by power lines must be reduced. Therefore, the application of a notch-filter is possible (see also Legchenko and Valla [2003]). Here, only frequencies of the harmonic noise are filtered out, which are contained in the MRS signal. These harmonic frequencies of the noise are known from the registered noise before every sounding. The frequency spectrum of the processed stacked data from Figure 4.6 (b) is given in Figure 4.7 (a). The spectrum shows the MRS signal peak at ∼0Hz as well as the power line harmonics, which are eliminated after filtering. The corresponding MRS signal in the time domain can be seen in Figure 4.7 (b). The red signal is the resulting MRS signal after applying the spike- and notch-filter. Subsequently, a nonlinear least square fitting algorithm to the in-phase and out-of-phase signal was used to determine the initial amplitude, the decay time, the phase and the frequency [Legchenko and Valla, 1998]. This post-processing scheme improves the adaptation to the initial amplitude and consequently the smoothness of the sounding curve. Figure 4.8 shows the amplitudes of the stacked data from the NUMIS Lite device with a low pass filter (black), the data stacked without any filter (green) and the post-processed data (red). A standard low pass filter with a cutoff frequency of 100Hz was applied to the stacked data [Legchenko and Valla, 2002], but the single stacks are saved without any filter. The single stacks for the configuration Tx1Rx2 were not registered, so the data was not processed. The figure demonstrates a good improvement of the smoothness of the sounding curves after executing the post-processing algorithm (red). The stacked data with an applied standard filter and the stacked unfiltered data shows more spikes in the sounding curves. Consequently, the adaptation to such a noise-influenced data set is difficult and inaccurate.

58 4.2 Data processing

signal amplitude (single record) signal amplitude (stacked signal) 5000 300 after spike filtering before spike filtering after spike filtering 4000 before spike filtering 200 3000

2000 100 Voltage [nV] Voltage [nV] 1000

0 0 0 50 100 150 200 250 0 50 100 150 200 250 time [ms] time [ms] in−phase out−of−phase in−phase out−of−phase 2000 1000 200 200

100 100 0 0 0 0 −2000 −1000 −100 −100 Voltage [nV] Voltage [nV] Voltage [nV] −4000 Voltage [nV] −2000 −200 −200

−6000 −3000 −300 −300 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 time [ms] time [ms] time [ms] time [ms] (a) (b)

Figure 4.6: (a) Application of the spike-filter (red) on one single stack (black). (b) Effect of the post-processing on the stacked MRS signal. The figure shows the signal amplitude (top) as well as the in-phase and out-of-phase signal (bottom).

signal amplitude amplitude spectrum 300 70 before notch filtering after notch filtering after notch filtering before notch filtering 200 60

100 50 Voltage [nV]

0 40 0 50 100 150 200 250 time [ms] in−phase out−of−phase 30 100 200 Voltage [nV]

50 100 20 0 0

−50 −100 10 Voltage [nV] Voltage [nV] −100 −200

0 −150 −300 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0 50 100 150 200 250 0 50 100 150 200 250 frequency [kHz] time [ms] time [ms] (a) (b)

Figure 4.7: Application of the notch ﬁlter (red) on the MRS signal with harmonic frequencies. (a) Frequency spectrum of the MRS signal. (b) The time domain of the notch- and unﬁltered signal.

59 4 Field data set

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5

0 0 0

1 1 1

Rx 1 2 2 2

3 3 3 pulse moment [As] 4 4 4 0 150 300 0 150 300 0 150 300

0 0 0 0

1 1 1 1

Rx 2 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300

0 0 0 0 0

1 1 1 1 1

Rx 3 2 2 2 2 2

3 3 3 3 3 pulse moment [As] 4 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300 0 150 300 amplitudes [nV]

0 0 0 0

1 1 1 1

Rx 4 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300 amplitudes [nV]

0 0 0

1 1 1 Rx 5 2 2 2

3 3 3 pulse moment [As] 4 4 4 0 150 300 0 150 300 0 150 300 amplitudes [nV] amplitudes [nV] amplitudes [nV]

Figure 4.8: Comparison of the amplitudes of the stacked data with (black) and without (green) a low pass ﬁlter and the post-processed data (red).

60 4.3 Modeling of the dune

4.3 Modeling of the dune

Using the finite element method, a 2D model based on the geoelectrical and IP data, as shown in Figure 4.9, was created with an infinite extension perpendicular to the profile direction. For the unstructured mesh, three parameters were defined: the growth rate, the maximum element size of the whole volume and of the edges of the loops. Due to the small side length of the loops (25m), the volume of the model is 3 ∗ 108m3. There are about 700000 elements with 50000 boundary elements and 2800 edge elements. In comparison to the synthetic data, there are more elements in the modeling of the field data set as a result of the stronger refinement around smaller loops and the complicated built geometry. The degree of freedom is 1.5 · 106. The problem was the calculation of the magnetic fields taking the conductivity of the layers into account. The solver in COMSOL did not converge, because the geometry was small in comparison to the whole calculated area. As a first order approximation, it is valid to use an insulating homogeneous half space to derive the sounding curves. With a radius of 14.1m of an equivalent circular loop and a resistivity of 20Ωm in the aquifer, the penetration depth and the resolution depth of a homogeneous model does not vary S significantly if the conductivity is set to 0 m (after Muller¨ et al. [2006]). The 2D model with the modeled water content distribution for an insulating half space and the amplitudes of the modeled sounding curves can be seen in Figure 4.9 and in Figure 4.10 (black), respectively. The red curves in Figure 4.10 are the post-processed sounding curves. The measured and modeled phase is illustrated in the Figure 4.11. The phase of the coincident sounding in an insulating homogeneous half space is zero, because the third part of the kernel function and the phase due to the electromagnetic attenuation vanish. The phase of the modeled curves with separated loops is only influenced by the geometrical phase. Apart from the coincident soundings, the phase is in a good fit, although the adaptation is only modeled for the amplitude.

Figure 4.9: Model of the dune at the peninsula Darss with the modeled water content distribution. It was modeled with square loops with a side length of 25m with 2 turns, an ° S Earth’s magnetic ﬁeld of 49600nT at 69 N inclination and a conductivity of 0 m .

61 4 Field data set

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5

0 0 0

1 1 1

Rx 1 2 2 2

3 3 3 pulse moment [As] 4 4 4 0 150 300 0 150 300 0 150 300

0 0 0 0

1 1 1 1

Rx 2 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300

0 0 0 0 0

1 1 1 1 1

Rx 3 2 2 2 2 2

3 3 3 3 3 pulse moment [As] 4 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300 0 150 300 amplitudes [nV]

0 0 0 0

1 1 1 1

Rx 4 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 0 150 300 0 150 300 0 150 300 0 150 300 amplitudes [nV]

0 0 0

1 1 1 Rx 5 2 2 2

3 3 3 pulse moment [As] 4 4 4 0 150 300 0 150 300 0 150 300 amplitudes [nV] amplitudes [nV] amplitudes [nV]

Figure 4.10: The amplitudes of the ﬁeld data set after the post-processing (red), modeled amplitudes (black) and sounding curves after inversion (green).

62 4.3 Modeling of the dune

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5

0 0 0

1 1 1

Rx 1 2 2 2

3 3 3 pulse moment [As] 4 4 4 −180 0 180 −180 0 180 −180 0 180

0 0 0 0

1 1 1 1

Rx 2 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 −180 0 180 −180 0 180 −180 0 180 −180 0 180

0 0 0 0 0

1 1 1 1 1

Rx 3 2 2 2 2 2

3 3 3 3 3 pulse moment [As] 4 4 4 4 4 −180 0 180 −180 0 180 −180 0 180 −180 0 180 −180 0 180 phase [ ° ]

0 0 0 0

1 1 1 1

Rx 4 2 2 2 2

3 3 3 3 pulse moment [As] 4 4 4 4 −180 0 180 −180 0 180 −180 0 180 −180 0 180 phase [ ° ]

0 0 0

1 1 1 Rx 5 2 2 2

3 3 3 pulse moment [As] 4 4 4 −180 0 180 −180 0 180 −180 0 180 phase [ ° ] phase [ ° ] phase [ ° ]

Figure 4.11: The phase of the ﬁeld data set after post-processing (red) and and modeled phase (black) for an insulating half space.

63 4 Field data set

The sounding sensitivity of the survey is presented in Figure 4.12, summarizing the 2D kernel function over all pulse sequences. The kernels of the coincident measurements are located on the main diagonal of the plot. In the upper and lower half of the figure, the transmitter Tx (blue) is situated in the south-southeast and north-northwest of the receiver Rx (red), respectively. For separated loop measurements, the main part of the signal is displaced towards the receiver and is decreased with increasing separation. The 2D profile sensitivity of the model is shown in Figure 4.13. It summarizes the modeled 2D kernel functions for an insulating homogeneous half space over all pulse moments and all configurations of the loops. The sensitive part of the signal is located in the first 30 meters. The highest profile sensitivity is found at the edges of the loops. The Baltic sea is situated in an area where the 2D profile sensitivity is small. Neglecting the Baltic sea in the modeling and setting the water content to zero, the maximal deviation is 1.3% at the first loop position for the coincident sounding (Tx1Rx1) and 1.0% for the half-overlapping measurement Tx2Rx1. For all other loop configurations the deviation decreases below 1.0%. That indicates, that the influence of the Baltic sea with a water content of 100% is neglectable in the modeling.

64 4.3 Modeling of the dune

Figure 4.12: 2D sounding sensitivity of the survey. The arrows show the midpoints of the transmitter (blue) and the receiver (red) of the involved loops.

65 4 Field data set

Figure 4.13: 2D profile sensitivity of the whole survey. Modeling was realized by square loops with a side length of 25m, an electrically insulating homogeneous half space, an Earth’s magnetic field of 49600nT with an inclination of 69°N. The profile runs from north-northwest to south-southeast.

4.4 2D inversion

The 2D water content distribution is determined using the 2D inversion scheme, taking the topography into account and including the adaptation error for the amplitude, obtained by the nonlinear least square fitting algorithm, that is given for every pulse moment of the soundings. The data were inverted with the kernel of an insulating half space. Generally, the amplitude of the sounding decreases with increasing conductivity and/or decreasing water content. The influence of the conductivity on the determination of the water content was assessed by Braun and Yaramanci [2003]. Conductive structures in and above an aquifer result in an underestimation of the water content, whereas the inversion for a model with a conductive layer below the aquifer leads to a higher water content in the aquifer. The importance of the electrical conductivity for a three layer model for the determination of the water content distribution can be seen in Figure 4.14. The data were modeled for a circular loop with a radius of 14.1m and 2 turns, with an Earth’s magnetic field of 49600nT at 69°N and with an increasing conductivity with increasing depth. The first 3000Ωm layer with a thickness of 6m and a water content of 5% is followed by a 9m thick layer of 20Ωm

66 4.4 2D inversion with 25% water content. For the underlying half space below 15m, a resistivity of 5Ωm and a water content of 12% is assumed. The 1D water model (blue) is shown in Figure 4.14. These conditions are nearly valid for the coincident sounding of loop 3 at the top of the dune. The modeled data were inverted with an insulating kernel to estimate the error of the determination of the water content for the field data set. Here, the error of the adaptation is very low (0.38%). The figure shows that the water content for the homogeneous half space is underestimated by 2% and the thickness of the aquifer increased (red). The agreement between the model (blue) and the inversion results for the water content distribution with depth is very good for the first 15m. Since the resistivity as well as the water content influence the sounding and both are unknown values, the error can only be estimated. Finally, it can be assumed that it is possible to obtain a good and reliable estimation for the 2D water content distribution for the whole 2D survey even using the kernel calculated for an electrically insulating half space.

subsurface model SNMR signal rel. phase 0 0 0

5 0.5 0.5

10 1 1

15 1.5 1.5

20 2 2 depth [m]

25 pulse moment [As] 2.5 pulse moment [As] 2.5

30 3 3

35 3.5 3.5 measured measured inversion inversion rms 0.38081 % 40 4 4 0 10 20 30 100 150 200 250 −5 0 5 10 water content [vol. %] amplitude [nV] phase [deg]

Figure 4.14: On the left side, the water model (blue) and the results of the 1D inversion (red) for a coincident sounding and in the center and on the right side, the adaptation to the amplitude and the phase of the complex MRS signal can be seen.

67 4 Field data set

The inversion result in Figure 4.15 shows a fairly good conformity with the 2D water model derived from the geoelectrical and IP data and modeled with an electrically insulating half space (see Figure 4.9). For the modeling of the aquifer, horizontal structures were assumed. The 2D inversion shows more topology in the aquifer, as it can also be seen in the section of the resistivity and the IP data in Figure 4.4. There are areas with higher conductivity as well as with higher chargeability in the aquifer. These areas are also indicated in the water content distribution. The increased water content in the south-southeast below loop 4 and 5 is linked with increased conductivity. The body of the dune is very dry with a water content of 2-5% followed by an aquifer with a water content of about 23%. The area below 12m has a water content of approximately 12%. A wedge from the lower half space associated with a higher chargeability (at ∼40m) leads to a small restriction of the aquifer. The 2D profile sensitivity in Figure 4.13 is the cumulative kernel function of all measurements and gives a good approximation of the sensitive area of the inversion. For the field data, this sensitive region is smaller due to the higher conductivity resulting in higher attenuation with increasing depth. The penetration depth is assumed under 30m. The Baltic sea in the north-northwest is outside of the sensitive area of the profile and is only indicated. The field data are also inverted with a horizontal insulating kernel. The inversion result is illustrated in Figure 4.16 (top). The 2D water content distribution is interpolated on a regular grid and is shifted vertically to the altitude of the dune dependent on the size of the inversion grid (see section 3.6). Here, the inversion grid is finer than the grid for the synthetic data set, resulting from the smaller loops, the noise-influenced data and the topography of the dune. The shifted data can be seen in Figure 4.16 (bottom). It shows more or less two separated lenses being larger and depper as compared to the result of the inversion with the real topography (Figure 4.15). The water content in the aquifer is raised, but for the body of the dune, it is nearly equal to the results of the inversion with the consideration of the topography. Finally, the inversion with neglecting the topography and the shifting afterwards leads to an obviously wrong water content, which is raised and in the wrong depth. Table 4.1 gives the adapation error of the inversion with and without the consideration of the topography. It is the average root mean square error of all soundings. The adaptation to the field data is bad, due to the fact that the whole 2D survey had to be considered in the inversion. But it reveals also that the inversion with a horizontal surface is not consistent.

model rms [%] inversion with topography 151.9 inversion without topography 277.1

Table 4.1: Adaptation error of the inversion with and without the consideration of the topography.

68 4.4 2D inversion

Figure 4.15: The result of the inversion including the error of the adaptation of the amplitudes. For comparison, the black lines show the 2D model of the modeling with COMSOL.

69 4 Field data set

Figure 4.16: Inversion results of the ﬁeld data set with a homogeneous half space without S any topography and a conductivity of 0 m (top). The data are shifted not continuously depending on the inversion grid (bottom).

70 5 Conclusions

The aim of this thesis was the investigation and assessment of the influence of the topography on MRS data in comparison to the flat earth with a horizontal surface. The application of an extended modeling scheme based on the finite element algorithm allows to model MRS soundings for any topographical situation. The modeling of a dune structure with different slopes of the dune, a homogeneous half space of 100Ωm and an assumed water content of 100% shows only the effect of the topography. The modeling was performed for different separations of the loops, for coincident, half-overlapping and edge-to-edge measurements. Due to the fact that the sensitive area for separated loop measurements is confined to a zone under the receiver, the measurements contain different information for reverse transmitter and receiver configuration. With increasing loop separation, the area of the main signal contribution is located at successively shallower depth. The deviation to the flat earth increases with increasing separation of the loops. For coincident sounding it is less important to consider the topography. The deviation between the flat earth with a horizontal surface and the topographical case with a maximum slope of 30° considering coincident soundings is below 2% which is quite small. Coincident soundings are more influenced for smaller pulse moments and thus for shallower depth. Nevertheless, for slopes from 30° on, the consideration of the topography has to be taken into account. It has been demonstrated that the resulting angle between the Earth’s magnetic field and the loop especially affects the soundings. The amplitudes decrease with increasing resulting angle. But the effect of the reduced or increased volume of investigation also influences the measurements. A reduction of the volume generally leads to smaller amplitudes. It is the opposite for an increased volume. For the discussed edge-to-edge measurements three kinds of loop layouts can be distinguished. Generally, for concave structure, the amplitudes become smaller with increasing slope of the dune. The common area of the sensitive volume of the receiver and the volume of maximum amplitudes of the transmitter decreases, thus the amplitudes decrease with increasing slope. In the case of a convex structure, the amplitudes increase, because the sensitive area of the receiver coincides with the area of the maximum amplitude of the transmitter. The third kind of loop layout is a configuration with one convex and one concave bended loop. In this case the amplitudes are unchanged compared to the flat earth. The influence of the topography is always much stronger, if the transmitter lies in the north of the receiver. The mean deviation between the flat earth and an edge-to-edge measurement over a dune with a slope of 5° is already 7%, which is very high for even small slope angles. The deviation increases up to 30% for the edge-to-edge measurement and a slope of 30°.

71 5 Conclusions

As it has been shown in this thesis, the effects which influence the soundings are the same for a profile in the north-south and the east-west direction, but they are always stronger for a north-south profile direction, due to the inclination of the Earth’s magnetic field. The 2D inversion of a synthetic data set for separated loop surveys on a profile taking the topography into account gives reliable results. Two water models were defined. A model with a layered aquifer and a model with a water bearing lens below the dune with a slope of 10°. For the inversion neglecting the topography with a shifting afterwards, noticeable wrong inversion results are detected and the original structure is not reflected. It shows higher differentiations in the homogeneous half space and the deviations to the model already occur directly under the surface. The water content is raised in and around the water bearing structures. For the 2D water model the adaptation without topography is really bad. To really test the improvement of the incorporating the topography into the data interpretation, a MRS survey across a dune at the Baltic sea has been realized. Two surveys with IP and geoelectrical data were measured, that confirm the detailed and reasonable model of the water content distribution, received from the inversion scheme. The consideration of the topography improves the inversion results and a better adaptation to the data is achieved. All the effects which influences the soundings explained in this thesis, are not considered in the inversion with neglected topography. This leads to the conclusion that the topography has to always be considered in the inversion as well as in the modeling of a 2D survey. For the modeling with COMSOL to save computational time, the definition of the boundary condition has to be improved to minimize the calculated 3D volume of the problem. The idea is to define the analytic solution on the boundaries. For later work, to get away from the grid generator of the program COMSOL, another grid generator with a more consistent grid will be searched and possibly could be imported to COMSOL. For the 2D inversion scheme it is desirable to fix internal boundaries and integrate known water content distributions for structures into the inversion, as it is the case for the Baltic sea with its water content of 100% and known location.

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74 A Appendix

75 A Appendix

76 A.1 Appendix to Chapter 2

A.1 Appendix to Chapter 2

Figure A.1: 2D kernel function for four selected pulse moments with the same modeling conditions as in Figure 2.2. The transmitter is situated in the east of the receiver. The midpoints of the loops are marked with arrows.

77 A Appendix

Figure A.2: 2D kernel function for four selected pulse moments with the same modeling conditions as in Figure 2.2. The transmitter lies in the south of the receiver.

78 A.1 Appendix to Chapter 2

79 A Appendix

A.2 Appendix to Chapter 3

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 5 5 5 −40N S Rx 1 10 10 10 −20 3 4 5 6 7 15 15 15 0 1 2 8 9 0 3K 0 3K 0 3K 20

0 0 0 0 elevation [m] 40 5 5 5 5 100 50 0 −50 −100 Rx 2 distance [m] 10 10 10 10 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 3 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 4 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 5 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 6 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 0 5 5 5 5 5 Rx 7 10 10 10 10 10 15 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 3K 0 0 0 0 5 5 5 5 Rx 8 10 10 10 10 15 15 15 15 0 3K 0 3K 0 3K 0 3K 0 0 0 5 5 5 Rx 9 10 10 10 15 15 15 0 3K 0 3K 0 3K

Figure A.3: The calibration curves (red) of a survey over the dune with a slope of 5°. The black line shows the corresponding sounding for the ﬂat earth. 80 A.2 Appendix to Chapter 3

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 5 5 5 −40N S Rx 1 4 5 6 10 10 10 −20 3 7 1 2 8 9 15 15 15 0 0 3K 0 3K 0 3K 20

Figure A.4: The calibration curves (red) of a survey over the dune with a slope of 30°. The black line shows the corresponding sounding for the ﬂat earth.

81 A Appendix

Tx 1 Tx 2 Tx 3 Tx 4 Tx 5 Tx 6 Tx 7 Tx 8 Tx 9

0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 1 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 2K 0 2K 0 3K 0 2K 0 2K 0 1K 0 0.5K 0 0.5K 0 0.5K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 2 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 2K 0 2K 0 2K 0 3K 0 2K 0 2K 0 1K 0 0.5K 0 0.5K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 3 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 3K 0 2K 0 2K 0 2K 0 3K 0 2K 0 2K 0 1K 0 0.5K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 4 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 2K 0 3K 0 2K 0 2K 0 2K 0 3K 0 2K 0 2K 0 1K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 5 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 2K 0 2K 0 3K 0 2K 0 2K 0 2K 0 3K 0 2K 0 2K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 6 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 1K 0 2K 0 2K 0 3K 0 2K 0 2K 0 2K 0 3K 0 2K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 7 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 0.5K 0 1K 0 2K 0 2K 0 3K 0 2K 0 2K 0 2K 0 3K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 8 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 0.5K 0 0.5K 0 1K 0 2K 0 2K 0 3K 0 2K 0 2K 0 2K 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 Rx 9 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 0 0.5K 0 0.5K 0 0.5K 0 1K 0 2K 0 2K 0 3K 0 2K 0 2K

Figure A.5: Whole synthetic data set of the dune structure with a slope of 10° and all 81 possible conﬁgurations.

82 A.2 Appendix to Chapter 3

Figure A.6: Presentation of the 2D kernel functions in the north-south direction. The transmitter (Tx 3) is situated in the north of the receiver (Rx 5). The rows show the 2D kernels for diﬀerent slopes of the dune (0°-30°) for the 13th pulse moment at q=8.79As. The columns show the three parts of the kernels and the full kernel function. 83 A Appendix 2K 2K 1K 1K Tx8Rx9 Tx9Rx8 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 1K 1K Tx7Rx8 Tx8Rx7 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 30° 1K 1K Tx6Rx7 Tx7Rx6 0 0 10° 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 1K 1K 5° Tx5Rx6 Tx6Rx5 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 2° amplitude [nV] amplitude [nV] 1K 1K Tx4Rx5 Tx5Rx4 0 0 1° 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 1K 1K 0° Tx3Rx4 Tx4Rx3 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 1K 1K Tx2Rx3 Tx3Rx2 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 16 10 12 14 16 2K 2K 1K 1K Tx1Rx2 Tx2Rx1 0 0

0 2 4 6 8 0 2 4 6 8

10 12 14 16 10 12 14 16

pulse moment [As] moment pulse [As] moment pulse Tx north of Rx Tx south of Rx

Figure A.7: The ﬁgure shows the amplitudes of all half-overlapping measurements over the dune (1-9) at diﬀerent slopes (0°-30°) of the dune.