# mGEODAR--A Mobile Radar System for Detection and Monitoring of Gravitational Mass-Movements
## 2 Materials and Methods
### FMCW Design Variables  
A linear pulse in an FMCW radar is given by the bandwidth \(B\) and the pulse duration \(\tau\), or the so-called chirp rate \(\alpha=\frac{B}{\tau}\). The target distance \(r\) is then a linear relation between this chirp rate, the signal propagation velocity \(c\) and the frequency \(f\) of the deramp signal  
\[r=\frac{c\,\tau}{2\,B}\cdot f=\frac{c}{2\,\alpha}\cdot f\,. \tag{1}\]  
The range resolution \(\Delta r\) is the distance between two discernible targets and is related to the bandwidth \(B\) by \(\frac{c}{2\,B}\). Approximating \(c\) with the speed of light gives \(\Delta r=0.375\,\mathrm{m}\) for the above mentioned mGEODAR settings. As avalanches are usually distributed targets with reflection returning continuously from the complete flow lengths, the range resolution equals the spatial sampling or so-called range gates along the mountain slope [5].  
The maximum unambiguous range for an FMCW radar depends mainly on the largest resolvable frequency of the analog-to-digital converter (ADC), e.g., half the sample rate \(f_{adc}\) gives a maximum range of \(r=937\,\mathrm{m}\) for the given system that can be adopted to longer ranges easily. Furthermore, the transmitted signal strength must be sufficient as it decays with \(r^{-4}\) for point targets and \(r^{-3}\) for distributed targets like avalanches [9]. Additionally for long range radars, the pulse duration \(\tau\) needs to be longer than the two-way propagation time \(\frac{2\,r}{c}\). The temporal resolution \(\Delta t\) depends on the pulse repetition rate that is limited by the pulse duration \(\tau\). When pulses are transmitted continuously \(\Delta t\) equals \(\tau\). Averaging over consecutive pulses for signal-to-noise improvements or sending pulse trains with different chirp rates reduces \(\Delta t\) accordingly.  
A further constraint on the pulse parameters \(B\) and \(\tau\) is imposed by the expected velocities \(v\) of the target. The target should not move between two range gates during a single pulse, that is, the maximal velocity should by smaller then \(\frac{\Delta r}{\tau}\), otherwise the frequency of the deramp signal is smeared between several range gates similar to effects caused by pulse non-linearities [12]. Additionally, a moving object with radial velocity \(v\) causes a Doppler shift proportional to the speed and wavelength \(\lambda\). As Doppler processing is difficult for distributed targets, this frequency shift \(\frac{2\,v}{\lambda}\) manifests as a ranging error in Equation (1). A target approaching with \(20\,\mathrm{m/s}\) imposes a ranging error of \(0.52\,\mathrm{m}\).  
The wavelength \(\lambda\) of the transmitted signal is \(0.03\,\mathrm{m}\) (\(10\,\mathrm{GHz}\)) and resembles the minimal size of objects that cause reflections rather than scattering. A powder cloud consists of single millimeter sized snow crystals and is therefore transparent to X-Band signals. Significant reflections are received from the flowing dense avalanche underneath the cloud [1]. Furthermore, X-Band radar permits avalanche observations independent of weather and visibility conditions.