Modeling in-ice radio propagation with parabolic equation methods

We investigate the use of parabolic equation (PE) methods for solving radio-wave propagation in polar ice. PE methods provide an approximate solution to Maxwell’s equations, in contrast to full-field solutions such as finite-difference-time-domain (FDTD) methods, yet provide a more complete model of propagation than simple geometric ray-tracing (RT) methods that are the current state of the art for simulating in-ice radio detection of neutrino-induced cascades. PEs are more computationally efficient than FDTD methods, and more flexible than RT methods, allowing for the inclusion of diffractive effects and modeling of propagation in regions that cannot be modeled with geometric methods. We present a new PE approximation suited to the in-ice case. We conclude that current ray-tracing methods may be too simplistic in their treatment of ice properties, and their continued use could overestimate experimental sensitivity for in-ice neutrino detection experiments. We discuss the implications for current in-ice Askaryan-type detectors and for the upcoming Radar Echo Telescope, two families of experiments for which these results are most relevant. We suggest that PE methods be investigated further for in-ice radio applications.

Figure 1

An example of radio propagation modeled by the parabolic solver, using the index of refraction profile from the south pole, derived from SPICE [29] core data in the top 100 meters and a functional fit below. For this figure we model a 350 MHz continuous-wave radio from a dipole source 100 m below the surface.

Figure 2

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(100,-25)\mathrm{m}$, with a functional $n(z)$ profile corresponding to a fit to South Pole data. The two frequency domain plots correspond to the direct (left) and reflected (right) pulses in the time-domain signal.

Figure 3

A comparison for discrete range steps for the maximum value of the received field, for PE (solid line) and FDTD (dashed line). This is for a continuous-wave 135 MHz signal, with the direct and reflected interference pattern evident. From left to right, range steps of 200, 275, and 350 m are shown. The curves have been normalized to the peak $x$-axis value to assist comparison.

Figure 4

The $n(z)$ profile (top panel) as measured by the two SPICE core [29] firn holes (core 1, thick solid line, core 2 thin solid line), along with the functional fit (dashed line) to these data used for ray-tracing codes at the South Pole. Residuals are shown below.

Figure 5

The residuals of the $n(z)$ profile and the functional fit $f(z)$, as measured by the two SPICE core firn holes (core 1, thick solid line, core 2 thin solid line), along with profiles linearly interpolated at $1/4$ (dashed) and $3/4$ (dotted) interval between these two, used in the interpolated $n(x,z)$ case.

Figure 6

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(100,-25)\mathrm{m}$, with an $n(z)$ profile corresponding to the first of 2 SPICE core firn samples. The two frequency domain plots correspond to the direct and reflected pulses in the time-domain signal.

Figure 7

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(100,-25)\mathrm{m}$, with an $n(z)$ profile corresponding to the second of 2 SPICE core firn samples. The two frequency domain plots correspond to the direct and reflected pulses in the time-domain signal.

Figure 8

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(100,-25)\mathrm{m}$, with an $n(x,z)$ profile corresponding to a linear interpolation between the first SPICE core at $x=0$ and the second SPICE core at $x=300\mathrm{m}$.

Figure 9

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-100)\mathrm{m}$ and a receiver at $x,z=(250,-2)\mathrm{m}$, with an $n(z)$ profile corresponding to the first of 2 SPICE core firn samples.

Figure 10

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-100)\mathrm{m}$ and a receiver at $x,z=(250,-2)\mathrm{m}$, with an $n(z)$ profile corresponding to the second of 2 SPICE core firn samples.

Figure 11

Time domain (top) and frequency domain (bottom) comparisons between FDTD (thin solid line), PE (thick solid line), and RT (dashed line) for a source at $x,z=(0,-100)\mathrm{m}$ and a receiver at $x,z=(250,-2)\mathrm{m}$, with an $n(x,z)$ profile corresponding to a linear interpolation between the first SPICE core at $x=0$ and the second SPICE core at $x=300\mathrm{m}$.

Figure 12

Time domain (top) and frequency domain (bottom) comparisons between PE (thick solid line) and FDTD (thin solid line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(250,-2)\mathrm{m}$, with a functional $n(z)$ profile for the South Pole. The columns in the time and frequency domain plots correspond to the first and second signals, respectively.

Figure 13

Time domain (top) and frequency domain (bottom) comparisons between PE (thick solid line) and FDTD (thin solid line) for a source at $x,z=(0,-30)\mathrm{m}$ and a receiver at $x,z=(250,-2)\mathrm{m}$, with $n(x,z)$ corresponding to a linear interpolation between the SPICE core 1 at $x=0$ and SPICE core 2 at $x=300$. The columns in the time and frequency domain plots correspond to the first and second signals, respectively.

Figure 14

Time domain (top) and frequency domain (bottom) comparisons between PE (thick solid line) and RT (thin solid line) for a source at $x,z=(0,-1050)\mathrm{m}$ and a receiver at $x,z=(1350,-120)\mathrm{m}$, with a $n(z)$ profile corresponding to the second of 2 SPICE core firn samples. The columns in the time and frequency domain plots correspond to the direct and reflected signals, respectively.

Figure 15

Time domain (top) and frequency domain (bottom) comparisons between PE (thick solid line) and RT (thin solid line) for a source at $x,z=(0,-1050)\mathrm{m}$ and a receiver at $x,z=(1350,-2)\mathrm{m}$, with a $n(z)$ profile corresponding to the second of 2 SPICE core firn samples.

Figure 16

Spectrum of the first 100 ns of a received signal for varying receiver (RX) depth using parabolic equations. This is for a transmitter buried 1050 m in the ice, displaced 1 km in range from the receivers. The scale is dB (power) relative to the transmitter output.

Figure 17

Spectrum of the first 100 ns of a received signal for varying receiver (RX) depth using ray tracing. This is for a transmitter buried 1050 m in the ice, displaced 1 km in range from the receivers. The scale is dB (power) relative to the transmitter output.

Figure 18

Spectrum of the first 100 ns of a received signal (using PE) for a receiver at $(x,z)=1350$, $-2\mathrm{m}$ as a function of apparent arrival angle, measured with respect to the local horizontal of the receiver.

Figure 19

Spectrum of the first 100 ns of a received signal (using PE) for a receiver at $(x,z)=1350$, $-200\mathrm{m}$ as a function of apparent arrival angle, measured with respect to the local horizontal of the receiver.

Figure 20

Comparison of peak power received for the functional (top) and data-driven (bottom) $n(x,z)$ profiles, for 350 MHz continuous-wave radio transmitted from a source 10 m beneath the surface of the ice.

Figure 21

Comparison of peak power received for the functional (top) and data-driven (bottom) $n(x,z)$ profiles, for 350 MHz continuous-wave radio transmitted from a source 1 m beneath the surface of the ice.