GPR theory

In principle, radar systems emit electromagnetic radiation following the propagation as described by Maxwell’s equations, where the electric component is orthogonal to the magnetic component (Reynolds, 1997; Daniels, 2004). Ground-penetrating radar is a ultrawide band technique, which emits very short pulses with a typical pulse repetition frequency of 50’000 times per second (50 kHz) (Reynolds, 1997). The transmitted electromagnetic waves are scattered and reflected by layer boundaries and objects and recorded by a receiver (Fig. 1.1). The receiver is set to scan at a fixed time rate, depending upon the used system. The time range of record for the receiver can be adjusted and therefore the possible depth range of the scan determined. Antennas can be used either in monostatic or bistatic mode. Monostatic mode is defined as when one antenna device is used as transmitter and receiver. Bistatic mode describes the record technique with two separated antennas: one serves as a transmitter and the other as a receiver. In the context of this study, I utilized mainly a bistatic mode of antennas, with the transmitter and receiver arranged in a short, fixed distance with a nominal frequency of 400 MHz – 2000 MHz, according to the requirements of the application. These antennas are shielded in one radiation direction, which resulted in a limited emitted radiation pattern and record constraints in the direction of interest (Fig. 1.1). This improves the interpretability of the recorded reflections significantly.

The propagation of radar waves in media is dependent on the properties of the penetrated ma-terials. Some materials such as polar ice are virtually transparent to electromagnetic waves, whereas other materials such as seawater or water-saturated clay degrade the propagation of

1.2 Background in radar waves the signal to such an extent or reflect the signal at the surface in a way that they are virtually non-transparent. The transparency of radar waves defines the measurement scale for GPR ap-plications in a material. While in polar ice the measurement scale ranges from centimeters to several kilometers, applications in e.g. soil science can be found on scales between a few meters to several tens of meter using standard devices. The possible penetration depth depends on the used frequencies and on the clay, water, iron and salt content of the penetrated medium.

High surface salt concentrations can totally prohibit any GPR application. The propagation of electromagnetic waves in one dimension in z-direction can be represented by:

∂²E

∂z² =µε∂²E

∂t² , (1.1)

withEthe electrical field. The magnetic permeabilityµ;µ=µ₀µ_rand the dielectric permittivity ε;ε=ε0εr consist of the relative part according to the penetrated material (with the subscript r) and magnetic susceptibility or electric permittivity constants of free space (subscript0). The velocity of propagationv is

v= 1

√µε (1.2)

and the velocity of light in free space c

c= 1

√µ₀ε₀. (1.3)

The permeability is a quantity of the magnetization of the penetrated material, which isµ_r= 1for nonmagnetic materials (Daniels, 2004). The velocity of propagation in a constant nonmagnetic material simplifies therefore to

v_r= c

√ε_r. (1.4)

As mentioned above, within snow and ice formations, the parameters influencing the emitted waves are limited, mostly to the three phases of water and the fraction volume of air, which are all nonmagnetic materials. The relative dielectric permittivity values of the respective contributions to snow are the permittivity of air εa = 1, the permittivity of ice εI = 3.1−3.2 (depending on the pureness and density) and the permittivity of waterε_w= 81(forT = 20°C). The contrast in relative dielectric number or dielectric permittivity between adjacent layers in radiation direction causes reflection. At a boundary between two media, parts of the emitted energy of the radar waves will be reflected and the remaining parts transmitted. The reflected field strength is dependent of the impedances Zi of the two media (medium 1 and 2) at the boundary and described by the reflection coefficientR.

R= Z₂−Z₁ Z2+Z1

, (1.5)

whileZ = ^µ_ε. As the permeability is set toµ_r= 1for the here investigated media Rsimplifies

is the complex permittivity, with ε^p the ordinary permittivity,ε^q the dielectric loss factor, which is equal to the electrical conductivity σ divided by the circular frequency ω and with i=√

−1.

Based on the statement of Eisen (2002) that for the upper parts of an ice body (upper 100 m), the generation of electromagnetic reflections is dominated by discontinuities in the ordinary part of the dielectric permittivity. I disregarded the imaginary part and the dielectric anisotropy of the crystal fabric as sources of reflections. For conditions with a low electrical conductivity σ (σ < 0.1 S/m; Daniels, 2004), the effect of ε^p is commonly disregarded and it is sufficient to replaceε^p by the relative dielectric permittivityε_r. Equation (1.6) simplifies for media with low σ values to

The sign of the reflection coefficient changes depending on whether the waves passes into a medium with higher or lower permittivity. This is apparent in the phase structure of the re-flected signal in relation to the transmitted wavelet (Arcone et al., 2005). The analysis of the phase structure of specific reflections is used for interpretation of related layer transitions in the snowpack in chapter 4 and 5. In dry snow conditions, the contributing factors to the reflec-tion coefficient are limited to the fracreflec-tion volumes of the two contributing materials air and ice.

Therefore, a contrast from air to ice will result in the highest amount of reflected radiowave energy. However, liquid water appearances, with a relative dielectric permittivity of more than 20 times the permittivity of ice, dominate the signal reflections in a recorded snowpack and will further increase the reflection coefficient. Additionally, the penetration of the signal will be strongly attenuated due to added absorption as the electrical conductivity cannot be neglected anymore.

In the following, to confirm the disregard of the electrical conductivity in dry snow and to demon-strate the influence of water, I will calculate the theoretic extent of the gradient in conductivity to cause reflections in the snowpack, based on the experiences of the results in chapter 4 and 5. Previously, I describe the maximal extent of conductivity for a dry snow cover. Basically, dry snow consists of a mixture of ice particles and air and therefore, the electrical conductivity is set by the mixture between the fraction of ice and air. The resulting conductivity of snow is approximately located in the range between the conductivity of ice σ_I ≈ 10⁻⁸S/m (Hobbs, 1974), and the conductivity of airσ_a= 2.5·10⁻¹³Sm⁻¹ (Lide, 1996), (see Heilig et al., 2008 for

1.2 Background in radar waves details). In addition, to calculate the theoretical extent for the change in electrical conductivity in snow, I utilize the approximated power reflection coefficientRp derived from Paren (1981)

Rp = (1

4∆(tanδ))² (1.9)

with ∆ tanδ = ∆^ε_ε^q_p, the dielectric loss tangent. The following assumptions shall be given: the relative dielectric permittivity at a layer boundary is constant and a reflection only caused by an effective reflectivity value of about Ri ' −70 dB (Heilig et al., in press –Paper 3). This results in a change in electrical conductivity of about ∆σ_I = 10⁻² S/m for a reflection in ice and ∆σ_s = 1.5·10⁻² S/m for a typical permittivity of snow (appendix B). These values are in the range of a change in electrical conductivity from air to tap water (Hobbs, 1974; Daniels, 2004) and far above the given values for ice and air. This theoretical determination confirms the assumption that the electrical conductivity is negligible in dry snow conditions to cause reflections in the snowpack and demonstrates that even if the dielectric permittivity of water would not be different to snow, the change in electrical conductivity can cause remarkable reflections in moist snow conditions.

Regarding the objective of this study, which is the seasonal snow cover, the relative permeability in equation (1.1) is set to µr = 1 for snow as a nonmagnetic material and the electrical con-ductivity is insignificant for dry-snow conditions. Therefore, I conclude that snow and air are low-loss materials for electromagnetic waves, with the consequence, that the one-dimensional propagation of radar signals with time is only dependent on the dielectric relative permittivity of the penetrated material. Hence, and in agreement with previous studies (Kovacs et al., 1995;

Mätzler, 1996), I assume for dry snow conditions, that the change in snow density within the snowpack is the sole parameter causing reflections of electromagnetic waves, conditionally on the absence of conducting impurities such as dust layers or buried objects.