The RadAlp experiment aims at developing advanced methods for rainfall and snowfall estimation using weather radar remote sensing techniques in high mountain regions for improved water resource assessment and hydrological risk mitigation. A unique observation system has been deployed since 2016 in the Grenoble region of France. It is composed of an X-band radar operated by Meteo-France on top of the Moucherotte mountain (1901m above sea level; hereinafter MOUC radar). In the Grenoble valley (220m above sea level; hereinafter a.s.l.), we operate a research X-band radar called XPORT and in situ sensors (weather station, rain gauge and disdrometer). In this paper we present a methodology for studying the relationship between the differential phase shift due to propagation in precipitation (Phi(dp)) and path-integrated attenuation (PIA) at X band. This relationship is critical for quantitative precipitation estimation (QPE) based on polarimetry due to severe attenuation effects in rain at the considered frequency. Furthermore, this relationship is still poorly documented in the melting layer (ML) due to the complexity of the hydrometeors' distributions in terms of size, shape and density. The available observation system offers promising features to improve this understanding and to subsequently better process the radar observations in the ML. We use the mountain reference technique (MRT) for direct PIA estimations associated with the decrease in returns from mountain targets during precipitation events. The polarimetric PIA estimations are based on the regularization of the profiles of the total differential phase shift (Psi(dp)) from which the profiles of the specific differential phase shift on propagation (K-dp) are derived. This is followed by the application of relationships between the specific attenuation (k) and the specific differential phase shift. Such k-K-dp relationships are estimated for rain by using drop size distribution (DSD) measurements available at ground level. Two sets of precipitation events are considered in this preliminary study, namely (i) nine convective cases with high rain rates which allow us to study the phi(dp)-PIA relationship in rain, and (ii) a stratiform case with moderate rain rates, for which the melting layer (ML) rose up from about 1000 up to 2500ma.s.l., where we were able to perform a horizontal scanning of the ML with the MOUC radar and a detailed analysis of the phi(dp)-PIA relationship in the various layers of the ML. A common methodology was developed for the two configurations with some specific parameterizations. The various sources of error affecting the two PIA estimators are discussed, namely the stability of the dry weather mountain reference targets, radome attenuation, noise of the total differential phase shift profiles, contamination due to the differential phase shift on backscatter and relevance of the k-K-dp relationship derived from DSD measurements, etc. In the end, the rain case study indicates that the relationship between MRT-derived PIAs and polarimetry-derived PIAs presents an overall coherence but quite a considerable dispersion (explained variance of 0.77). Interestingly, the nonlinear k-K-dp relationship derived from independent DSD measurements yields almost unbiased PIA estimates. For the stratiform case, clear signatures of the MRT-derived PIAs, the corresponding phi(dp) value and their ratio are evidenced within the ML. In particular, the averaged PIA/phi(dp) ratio, a proxy for the slope of a linear k-K-dp relationship in the ML, peaks at the level of the copolar correlation coefficient (rho(hv)) peak, just below the reflectivity peak, with a value of about 0.42 dB per degree. Its value in rain below the ML is 0.33 dB per degree, which is in rather good agreement with the slope of the linear k-K-dp relationship derived from DSD measurements at ground level. The PIA/phi(dp) ratio remains quite high in the upper part of the ML, between 0.32 and 0.38 dB per degree, before tending towards 0 above the ML.