Based on a literature overview, this paper summarizes the impact and legacy of the contributions of Wilfried Brutsaert and Jean-Yves Parlange (Cornell University) with respect to the current state-of-the-art understanding in hydraulic groundwater theory. Forming the basis of many applications in catchment hydrology, ranging from drought flow analysis to surface water-groundwater interactions, hydraulic groundwater theory simplifies the description of water flow in unconfined riparian and perched aquifers through assumptions attributed to Dupuit and Forchheimer. Boussinesq (1877) derived a general equation to study flow dynamics of unconfined aquifers in uniformly sloping hillslopes, resulting in a remarkably accurate and applicable family of results, though often challenging to solve due to its nonlinear form. Under certain conditions, the Boussinesq equation can be solved analytically allowing compact representation of soil and geomorphological controls on unconfined aquifer storage and release dynamics. The Boussinesq equation has been extended to account for flow divergence/convergence as well as for nonuniform bedrock slope (concave/convex). The extended Boussinesq equation has been favorably compared to numerical solutions of the three-dimensional Richards equation, confirming its validity under certain geometric conditions. Analytical solutions of the linearized original and extended Boussinesq equations led to the formulation of similarity indices for baseflow recession analysis, including scaling rules, to predict the moments of baseflow response. Validation of theoretical recession parameters on real-world streamflow data is complicated due to limited measurement accuracy, changing boundary conditions, and the strong coupling between the saturated aquifer with the overlying unsaturated zone. However, recent advances are shown to have mitigated several of these issues. The extended Boussinesq equation has been successfully applied to represent baseflow dynamics in catchment-scale hydrological models, and it is currently considered to represent lateral redistribution of groundwater in land surface schemes applied in global circulation models. From the review, it is clear that Wilfried Brutsaert and Jean-Yves Parlange stimulated a body of research that has led to several fundamental discoveries and practical applications with important contributions in hydrological modeling.@en

To analyze the influence of the spatial variability of the raindrop size distribution (DSD) on rainfall estimation using weather radar, a stochastic model is proposed in order to simulate range profiles of DSDs and consequently profiles of rainfall intensity, radar reflectivity and specific attenuation. A first application concerning the accuracy of attenuation correction algorithms is presented. Using a Monte Carlo technique, it is shown that an attenuation correction algorithm based on a forward implementation is more sensitive to the inadequacy of power-law relations than a backward algorithm, the latter being sensitive to the uncertainty of the path integrated attenuation (PIA). The sensitivity to spatial heterogeneity is limited for both algorithms. The corrected profiles remain uncertain because they are based on deterministic power-law relations which are not fully consistent with the stochastic nature of the DSD.@en

Microwave links can be used to estimate the path-averaged rain rate along the link when precipitation occurs. They take advantage of the near proportionality between the specific attenuation affecting the link signal and the rain rate. This paper deals with the influence of the spatial variability of rainfall along the link on the accuracy of the rainfall estimates. We focus on single-polarization single-frequency links operating at frequencies ranging from 5 to 50 GHz and with path lengths ranging from 500 m to 30 km. A stochastic simulation framework is used to investigate the frequency at which the linearity occurs (found to be about 30 GHz) for intense Mediterranean precipitation. In addition, the error associated with path-averaged rain rate estimates is being quantified. For instance, the bias is found to be in the order of -15% with an uncertainty of about 10% for microwave links described in the recent literature.@en

The raindrop size distribution (RSD) is a critical factor in estimating rain intensity using advanced dualpolarized weather radars. A new neural-network algorithm to estimate the RSD from S-band dual-polarized radar measurements is presented. The corresponding rain rates are then computed assuming a commonly used raindrop diameter speed relationship. Numerical simulations are used to investigate the efficiency and accuracy of this method. A stochastic model based on disdrometer measurements is used to generate realistic range profiles of the RSD parameters, while a T-matrix solution technique is adopted to compute the corresponding polarimetric variables. The error analysis, which is performed in order to evaluate the expected errors of this method, shows an improvement with respect to other methodologies described in the literature. A further sensitivity evaluation shows that the proposed technique performs fairly well even for low specific differential phase-shift values.@en

Many applications in urban areas require high-resolution rainfall measurements. Typical operational weather radars can provide rainfall intensities at 1-km 2 grid cells every 5 min. Opportunistic sensing with commercial microwave links yields path-averaged rainfall intensities (typically 0.1–10 km) within urban areas. Additionally, large amounts of urban in situ rainfall measurements from amateur weather observers are obtainable in real-time. The accuracy of these three techniques is evaluated for an urban study area of 20 × 20 km, taking into account their respective network layouts and sampling characteristics. We use two simulated rainfall events described in terms of drop size distributions on a 100-m grid and with a temporal resolution of 30 s. Accurate radar rainfall estimation with the Z-R relationship relies heavily on an appropriate choice of parameters, and a dual-polarization strategy is more suitable for higher intensities. Under ideal measurement conditions, the weather station network is the most promising, with a Pearson correlation coefficient above 0.86 and a relative bias below 4% for 100-m rainfall estimates at 5-min resolution. Microwave link rainfall observations contain the largest error, shown by a consistently larger coefficient of variation. The accuracy of all techniques improves when considering rainfall at larger scales, especially by increasing time intervals, with the strongest improvements found for microwave links for which errors are largely caused by their temporal sampling. Sparser networks are examined, showing that the decline in measurement accuracy only becomes significant when the link and station network density are reduced to less than half their levels in Amsterdam. @en

Microwave links can be used for the estimation of path-averaged rainfall by using either the path-integrated attenuation or the difference in attenuation of two signals with different frequencies and/or polarizations. Link signals have been simulated using measured time series of raindrop size distributions (DSDs) over a period of nearly 2 yr, in combination with wind velocity data and Taylor's hypothesis. For this purpose, Taylor's hypothesis has been tested using more than 1.5 yr of high-resolution radar data. In terms of correlation between spatial and temporal profiles of rainfall intensities, the validity of Taylor's hypothesis quickly decreases with distance. However, in terms of error statistics, the hypothesis is seen to hold up to distances of at least 10 km. Errors and uncertainties (mean bias error and root-mean-square error, respectively) in microwave link rainfall estimates due to spatial DSD variation are at a minimum at frequencies (and frequency combinations) where the power-law relation for the conversion to rainfall intensity is close to linear. Errors generally increase with link length, whereas uncertainties decrease because of the decrease of scatter about the retrieval relations because of averaging of spatially variable DSDs for longer links. The exponent of power-law rainfall retrieval relations can explain a large part of the variation in both bias and uncertainty, which means that the order of magnitude of these error statistics can be predicted from the value of this exponent, regardless of the link length.@en

Commercial microwave links (CMLs) in telecommunication networks can provide relevant information for remote sensing of precipitation and other environmental variables, such as path-averaged drop size distribution, evaporation, or humidity. The CoMMon field experiment (COmmercial Microwave links for urban rainfall MONitoring) mainly focused on the rainfall observations by monitoring a 38ĝ€¯GHz dual-polarized CML of 1.85ĝ€¯km path length at a high temporal resolution (4ĝ€¯s), as well as a co-located array of five disdrometers and three rain gauges over 1 year. The dataset is complemented with observations from five nearby weather stations. Raw and pre-processed data, which can be explored with a custom static HTML viewer, are available at 10.5281/zenodo.4923125 . The data quality is generally satisfactory for further analysis, and potentially problematic measurements are flagged to help the analyst identify relevant periods for specific study purposes. Finally, we encourage potential applications and discuss open issues regarding future remote sensing with CMLs.@en

In situ observations of snowfall over the Antarctic Ice Sheet are scarce. Currently, continent-wide assessments of snowfall are limited to information from the Cloud Profiling Radar on board the CloudSat satellite, which has not been evaluated up to now. In this study, snowfall derived from CloudSat is evaluated using three ground-based vertically profiling 24 GHz precipitation radars (Micro Rain Radars: MRRs). Firstly, using the MRR long-term measurement records, an assessment of the uncertainty caused by the low temporal sampling rate of CloudSat (one revisit per 2.1 to 4.5 days) is performed. The 10-90th-percentile temporal sampling uncertainty in the snowfall climatology varies between 30% and 40% depending on the latitudinal location and revisit time of CloudSat. Secondly, an evaluation of the snowfall climatology indicates that the CloudSat product, derived at a resolution of 1° latitude by 2° longitude, is able to accurately represent the snowfall climatology at the three MRR sites (biases<15 %), outperforming ERA-Interim. For coarser and finer resolutions, the performance drops as a result of higher omission errors by CloudSat. Moreover, the CloudSat product does not perform well in simulating individual snowfall events. Since the difference between the MRRs and the CloudSat climatology are limited and the temporal uncertainty is lower than current Climate Model Intercomparison Project Phase 5 (CMIP5) snowfall variability, our results imply that the CloudSat product is valuable for climate model evaluation purposes.@en

As rainfall constitutes the main source of water for the terrestrial hydrological processes, accurate and reliable measurement and prediction of its spatial and temporal distribution over a wide range of scales is an important goal for hydrology. We investigate the potential of ground-based weather radar to provide such measurements through a theoretical analysis of some of the associated observation uncertainties. A stochastic model of range profiles of raindrop size distributions is employed in a Monte Carlo simulation experiment to investigate the rainfall retrieval uncertainties associated with weather radars operating at X-, C-, and S-band. We focus in particular on the errors and uncertainties associated with rain-induced signal attenuation and its correction for incoherent, non-polarimetric, single-frequency, operational weather radars. The performance of two attenuation correction schemes, the (forward) Hitschfeld-Bordan algorithm and the (backward) Marzoug-Amayenc algorithm, is analyzed for both moderate (assuming a 50 km path length) and intense Mediterranean rainfall (for a 30 km path). A comparison shows that the backward correction algorithm is more stable and accurate than the forward algorithm (with a bias in the order of a few percent for the former, compared to tens of percent for the latter), provided reliable estimates of the total path-integrated attenuation are available. Moreover, the bias and root mean square error associated with each algorithm are quantified as a function of path-averaged rain rate and distance from the radar in order to provide a plausible order of magnitude for the uncertainty in radar-retrieved rain rates for hydrological applications.@en

At short wavelengths, especially C-, X-, and K-band, weather radar signals are attenuated by the precipitation along their paths. This constitutes a major source of error for radar rainfall estimation, in particular for intense precipitation. A recently developed stochastic simulator of range profiles of raindrop size distributions (DSD) provides a controlled experiment framework to investigate the accuracy and robustness of attenuation correction algorithms. The work presented here focuses on the quantification of the influence of uncertainties concerning radar calibration, the parameterization of power law relations between the integral variables (radar reflectivity Z and specific attenuation k), and total path integrated attenuation (PIA) estimates at X-band. The analysis concerns single frequency, incoherent and nonpolarimetric radar systems. Two attenuation correction algorithms, based on a forward and a backward implementation respectively, are studied. From DSD range profiles, the corresponding profiles of integral radar variables are derived. Using a Monte Carlo approach, the accuracy and robustness of the two algorithms are quantified for the different sources of error previously mentioned. This framework of realistic DSD variability provides a robust way to confirm that, under realistic assumptions concerning the PIA estimation uncertainty, the forward algorithm outperforms the backward algorithm for PIA values below 10 dB.@en

Weather radar systems are very suitable tools for the monitoring of extreme rainfall events providing measurements with high spatial and temporal resolution over a wide geographical area. Nevertheless, radar rainfall retrieval at C-band is prone to several error sources, such as rain path attenuation which affects the accuracy of inversion algorithms. In this paper, the so-called rain profiling techniques (namely the surface reference method FV and the polarimetric method ZPHI) are applied to correct rain path attenuation and a new neural network algorithm is proposed to estimate the rain rate from the corrected measurements of reflectivity and differential reflectivity. A stochastic model, based on disdrometer measurements, is used to generate realistic range profiles of raindrop size distribution parameters while a T-matrix solution technique is adopted to compute the corresponding polarimetric variables. A sensitivity analysis is performed in order to evaluate the expected errors of these methods. It has been found that the ZPHI method is more reliable than FV, being less sensitive to calibration errors. Moreover, the proposed neural network algorithm has shown more accurate rain rate estimates than the corresponding parametric algorithm, especially in presence of calibration errors.@en

This paper presents a first assessment of the hydrometeorological potential of a C-band doppler weather radar recently installed by the Royal Meteorological Institute of Belgium near the village of Wideumont in the southern Ardennes region. An analysis of the vertical profile of reflectivity for two contrasting rainfall events confirms the expected differences between stratiform and convective precipitation. The mean areal rainfall over the Ourthe catchment upstream of Tabreux estimated from the Wideumont weather radar using the standard Marshall-Palmer reflectivity-rain rate relation shows biases between +128% and -42% for six selected precipitation events. For two rainfall events the radar-estimated mean areal rainfall is applied to the gauge-calibrated (lumped) HBV-model for the Ourthe upstream of Tabreux, resulting in a significant underestimation with respect to the observed discharge for one event and a closer match for another. A bootstrap analysis using the radar data reveals that the uncertainty in the hourly discharge from the ∼1600 km2 catchment associated with the sampling uncertainty of the mean areal rainfall estimated from 10 rain gauges evenly spread over the catchment amounts to ±25% for the two events analyzed. This uncertainty is shown to be of the same order of magnitude as that associated with the model variables describing the initial state of the model.@en

[1] The matter of the efficient and parsimonious parameterization of hillslope subsurface flow remains an important issue in catchment hydrological studies (Brutsaert, 1995). Insights into the influence of the shape and hydraulic characteristics of hillslopes is required to further our understanding and our ability to model catchment hydrological processes. Recently, Troch et al. (2003) introduced the hillslope-storage Boussinesq (HSB) equation to describe subsurface flow and saturation along geometrically complex hillslopes. The HSB equation can be linearized and further reduced to an advection-diffusion equation for subsurface flow in hillslopes with constant bedrock slopes and exponential width functions. This paper presents a dimensional analysis of the latter equation in order to study the moments of the characteristic response function (CRF), corresponding to the free drainage of this type of hillslope. These moments, in a dimensionless form, can be expressed as functions of a similarity parameter, hereafter called the hillslope Péclet number, and a group of dimensionless numbers accounting for the effects of the boundary and initial conditions. The analytical expressions for the first four central CRF moments are derived for two types of initial conditions. The analysis of their respective influences shows that the hillslope Péclet number is an efficient similarity parameter to describe the hillslope subsurface flow response. Moreover, comparison between the CRF moments predicted by means of our similarity analysis and empirical moments derived from outflow measurements for different types of laboratory hillslopes shows good agreement.@en

The large variability of the raindrop size distribution (DSD) in space and time must be taken into account to improve remote Ser..sing of precipitation. The ability to simulate a large number of 2-D fields of DSDs sharing the same statistical properties provides a very useful simulation framework that nicely complements experimental approaches based on DSD ground measurements. These simulations can be used to investigate radar beam propagation through rain and to evaluate different radar retrieval techniques. The proposed approach uses geostatistical methods to provide structural analysis and stochastic simulation of DSD fields. First, the DSD is assumed to follow a Gamma distribution with three parameters. As a consequence, 2-D fields of DSDs can be described as a multivariate random function. The parameters are normalized using a Gaussian anamorphosis and simulated by taking advantage of fast Gaussian simulation algorithms. Variograms are used to characterize the spatial structure of the DSD fields. The generated fields have identical spatial structure and are consistent with the obSer.vations. Because intermittency cannot be simulated using this technique, the size of the simulation domain is limited to the meso-7 scale (2-20 km). To assess the proposed approach, the method is applied to data collected during intense Mediterranean rainfall. Taylor's hypothesis is invoked to convert time Ser.ies into 1-D range profiles. The anisotropy of the fields is derived from radar measurements. Simulated and measured reflectivity fields are in good agreement with respect to the mean, the standard deviation, and the spatial structure, demonstrating the promising potential of the proposed stochastic model of DSD fields.@en

The raindrop size distribution (DSD) is a statistical description of the number and size distribution of raindrops within a specified volume of air. DSDs play a central role in radar remote sensing and are essential for understanding the scattering and absorption of electromagnetic radiation as it travels through the air and interacts with falling rain. They form the mathematical backbone for linking radar observations to physical quantities such as rainfall intensities, liquid water content, and kinetic energy. Yet, in most remote sensing applications, the DSD in the target volume is likely to be unknown. Therefore, parameterized DSD models have been created to act as substitutes for direct measurementswhen observations are unavailable, incomplete, or impractical. Over time, a large number of models have been proposed, each with their own intricacies and constraints. Understanding these mathematical constructs and the assumptions behind them is essential for interpreting radar measurements and improving quantitative rainfall estimation.@en

Recently, Troch et al. [10] introduced the hillslope-storage Boussinesq (hsB) equation to describe subsurface flow and saturation along complex hillslopes. They demonstrated that numerical solutions of the hsB equation account explicitly for plan shape of the hillslope, by introducing the hillslope width function, and for profile curvature through bedrock angle and the hillslope soil depth function (see also Paniconi et al. [7] and Hilberts et al. [4]). This paper presents an analytical solution of the linearized hsB equation applicable to hillslopes with constant bedrock slope and exponential width functions. Then, the first four moments of the impulse response function (travel time distribution) of such hillslopes are derived using the Laplace transform of the partial differential equation describing the groundwater flow. These moments are explicitly related to the hydraulic (porosity, conductivity) and geometric (bedrock slope, width function, and soil depth) parameters of a hillslope. The zeroth moment can obviously be interpreted as the total outflow volume, and the (normalized) first moment as the mean response time of the hillslope. The second, third, and fourth moments have interpretations in terms of the relative width, asymmetry, and peakedness, respectively, of the hydrograph. The paper examines the effect of the geometric properties of hillslopes on the travel time distribution in a dimensionless context.@en

Knowledge of the raindrop size distribution (DSD) is essential for understanding the physics of precipitation and for interpreting remotely sensed observations of rain. Disdrometer measurements of DSDs are affected by uncertainties due to the limited sampling volumes or areas of the sensors. Determining this sampling error directly from disdrometer observations is of primary importance for the practical application of DSD analyses. Gage et al. (2004) proposed an estimator of the sampling error affecting the radar reflectivity estimates based on pairs of collocated disdrometers. We provide an interpretation of this estimator and assess its accuracy through controlled experiments using a Monte Carlo framework. Our simulation model of the disdrometer sampling process closely mimics the observations reported by Gage et al. (2004). Using this model, we demonstrate that the estimator proposed by Gage et al. (2004) provides a reliable quantification of the reflectivity sampling error. However, we also show that its accuracy depends on the ratio between the length of the disdrometer time series involved and the characteristic time scale of the rainfall.@en