## 6 Local Oscillator Stability  
The second issue we raise when using a commercial GPR for probing an acoustic transducer is the local timebase stability. Piezoelectric-based transducers are classically used as frequency selective components to design oscillators: in such a case, the resonator is designed to exhibit a frequency as stable as possible as the environmental conditions are varying. For example, design techniques include selecting locally temperature insensitive piezoelectric substrate cuts and wave modes, as well as a packaging preventing stress transfer or pressure variation that might significantly affect the acoustic wave velocity and, hence, the phase, yielding varying Barkhausen conditions in the oscillator and, thus, the output frequency. When designing a sensor, the opposite design considerations are applied: the acoustic velocity is expected to vary significantly with one given quantity, while the packaging aims at preventing other physical parameters from significantly affecting the acoustic velocity. Nevertheless, even when selecting strongly dependent material cuts and wave mode to quantities under investigation, the stability of single crystal substrates rarely yields sensitivities above the hundreds of ppm (part per million) range. Thus, the timebase reference should drift by a minute amount with respect to these expected variations for the measurement to be stable and hardly affected by a baseline drift, which cannot be attributed to the instrument or the sensor measurement. We have reported [49] the excessive timebase drift of a commercial GPR, Mala's ProEx. As a reminder of the core result of this past investigation, we have observed when probing a SAW delay line in a fixed environment that due to the instrument timebase drift, the time delay between echoes appeared to be shifting by 15,000 ppm (1.5%, as observed through a 4.5-ns drift for echoes separated by 300 ns). Lithium niobate YXI/128\({}^{\circ}\) typically exhibits a temperature sensitivity in the 70 ppm/K range: the GPR timebase drift makes such a sensor unusable in the narrow temperature range expected in sub-surface measurements, where a few ppm acoustic velocity variations are expected for sub-K temperature variations. The cause of the excessive drift is the means by which the stroboscopic timebase is generated. Indeed, with a GPR typically operating in the 100 to 2000-MHz range, sampling rates above several GHz are targeted to properly identify the pulse round trip. Few direct sampling analog to digital converters allow such large sampling rates, so that a stroboscopic approach is often adopted in which multiple measurements are repeated, under the assumption of a stationary environment during the measurement, with each new sampling delayed by \(N\times dt\), \(N\) the sample index and \(dt\) a fixed time interval set so that \(1/dt\) is an equivalent sampling frequency. The stability of generating the programmable \(N\times dt\) delay is of utmost importance to the stability of the sampling rate, on which all subsequent digital processing is based. In this past publication [49], the timebase drift issue was identified: we will here demonstrate a solution to this issue by replacing a core element of the GPR receiver. In [PERSON]'s ProEx, the timebase is defined by feeding a comparator with two saw-tooth-shaped signals, as classically designed in a voltage to time converter [50]. One slowly varying saw-tooth shaped signal sweeps the voltage span of the comparator input during the measurement duration needed to collect the \(N\) samples (Figure 8a, top), while the second saw-tooth signal spans the whole input voltage range during the time needed to collect a single sample. The PRR (Pulse Repetition Rate) determines the largest distance at which a target can be detected and is determined in the current scheme by the ramp reset signal period. As an example, for [PERSON]'s ProEx, the PRR is 100 kHz, so that the furthest target is located at 1500 m in air or 850 m in ice.