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Weather radar provides rainfall estimates at high resolution in both space and time, which is useful for many hydrological applications. Despite this, the radar rainfall estimation process introduces many sources of error, impacting the reliability of results obtained from the radar rainfall estimates. Key error sources include signal attenuation, radar calibration issues, ground clutter contamination, variability in the drop-size distribution and variation in the vertical profile of reflectivity. To gain an improved understanding of potential limitations, and the corresponding uncertainty of rainfall rates, the impact of these errors has been systematically investigated, developing a radar error model by inverting the rainfall estimation process.To this end, an ensemble of realistic rainfall events is simulated, and working backwards in a stochastic manner gives an ensemble of weather radar images, corresponding to each rainfall event, at each time step. The radar error model includes random noise effects, drop-size distribution errors, sampling estimation variance and importantly, attenuation effects. To allow for direct comparisons, standard radar processing methods are applied to each radar image, to obtain corrected ‘best guess’ rainfall estimates which would be obtained from each weather radar ensemble member in real world applications. The difference between the simulated and corrected rainfall for each ensemble member is then treated as the uncertainty corresponding to the radar rainfall estimation process.A simple measure is introduced, to help understand how often errors result in a rainfall signal completely irretrievable, referred to as ‘rainfall shadow’. Areas of rainfall that are ‘shadowed’ are defined as pixels where the simulated ‘true’ rainfall rate is significant, but the ensemble member has less than 10% of the original signal. This is equivalent to considering where a significant rainfall rate has been completely lost, and would therefore be irretrievable using standard correction methods, to quantify the frequency of occurrence in real-world radar rainfall applications. The impact of location of rainfall within images is considered, by introducing the second moment of area for radar images, in order to quantify the proximity of intense rainfall to the radar transmitter.Results show relationships between rainfall shadows and high bias and uncertainty in rainfall estimates, related to the amount of rainfall (i.e. proportion and rates) in images. More central rainfall also results in higher errors and higher variability. The minimum likelihood of occurrence of rainfall shadows showed that 50% of event images have at least 3% of significant rainfall shadowed. In addition, 25% of images had a shadowed area of over 45km2, with the minimum largest shadow in one area for 5% of images exceeding an area of 50km2. This gap would result in an underestimation of the impact of potential floods, showing that weather radar has potential for important information to be lost. A model framework for representing this uncertainty in the radar rainfall estimation process provides methodology for assessing the impacts of radar rainfall errors on hydrological applications.

In an attempt to get the best parameter estimations of the theoretically consistent IDF (Intensity Duration Frequency) models of rainfall intensity for the entire state of Baden Wuerttemberg, three well-defined optimization algorithms such as Differential Evolution (DE), Nelder Mead (NM), and TNC Truncated Newton (TNC) are taken into account for comparison.Seven-parametric IDF model contains mean intensity µ, intensity scale parameters λ1, λ2 , time scale parameter α, fractal/smoothness parameter Μ, Hurst parameter Η, exponent of the expression of probability dry θ, and tail index ξ, which are obtained by minimizing the error between empirical k-moments and model quantiles. Error metric focusing on distribution quantiles x(k,T) is thus minimized for all available scales k and a series of return periods T . Non-linear solver is chosen to perform this step as these errors are non-linear functions of the parameters.All results are demonstrated visually, and a final decision is made on the basis of precisely fitted parameter values to the model. This crucial step will also assist us in finding the optimum design values for stormwater and floods.

Climate models and their future projections, are normally provided in coarse spatial resolutions which makes them an imprecise source of information for certain hydrological purposes. Finding the proficient means of downscaling such data is one of the open questions of climate research. Previous research has shown that, the rainfall extremes show self-similarity in time and that a relatively similar behavior exists in regard to the spatial scale as well (Veneziano et al 2002). This study aims at determining the spatial scaling relationship of the rainfall extremes by using fine grids of radar datasets and upscaling them. In an empirical manner by aggregating the radar rainfall cells in space and for different cell sizes with a = 1, 2, 3, …12 km and for different durations of d = 5 min, 15 min 30 min, 1 hr, 2 hr, 4 hr, …, 24 hr the Annual Maximum Series are extracted. Using the AMS of different spatial and temporal scales and applying the Koutsoyiannis et at. 1998 method for rainfall extreme value analysis, the probability distribution function is fitted. Assessing the changes of the PDF parameters with the scale, with a logarithmic transformation on both variables; ln(parameter) vs. ln(scale), can show the sought relationship. The preliminary results of the study show definable non-linear relationships for location and scale parameters of the GEV distribution and the eta parameter of the Koutsoyiannis et al. 1998 parametrization. Koutsoyiannis, D. Kozonis, and A. Manetas, A mathematical framework for studying rainfall intensity-duration-frequency relationships, Journal of Hydrology, 206 (1-2), 118–135, doi:10.1016/S0022-1694(98)00097-3, 1998.Veneziano, Daniele; Furcolo, Pierluigi (2002): Multifractality of rainfall and scaling of intensity-duration-frequency curves. In Water Resour. Res. 38 (12), 42-1-42-12. DOI: 10.1029/2001WR000372.

Dependencies between variables are often very complex, and may for high values be different from that of the low values. As the normal distribution and the corresponding copula behaves symmetrically for low and high values the frequent application of the normal copula for the description of the dependence may be inappropriate. In this contribution a new way of defining high dimensional multivariate distributions with changing correlations is presented. The method can also be used for a flexible definition of tail dependence. Examples of copulas with linear changing correlations illustrate the methodology. Parameter estimation methods and simulation procedures are discussed. A five dimensional example using groundwater quality data and another four dimensional using air pollution data are used to illustrate the methodology.

In this paper the question of how interpolation of precipitation in space by using various spatial gauge densities affects the rainfall-runoff model discharge if all other input variables are kept constant is investigated. This was done using a physically-based model as the reference with a reconstructed spatially variable precipitation and a conceptual model calibrated to match the reference model output. Both models were run with distributed and lumped inputs. Results showed that all considered interpolation methods resulted in underestimation of the total precipitation volume and that the underestimation was directly proportional to the amount. The underestimation was very severe for low observation densities and disappeared only for very high density precipitation observation networks. This result was confirmed by using observed precipitation with different observation densities. Model runoffs showed worse performance for their highest discharges. Using lumped inputs for the models showed deteriorating performance for peak flows as well even when using simulated precipitation.

This research addresses a strong need to precisely improve the statewide seasonal precipitation intensity duration frequency (IDF) estimates at ungauged locations. In order to obtain IDFs at unvisited sites, IDFs observation locations are interpolated. Therefore, different deterministic and geostatistical approaches include Inverse Distance Weighted (IDW), Ordinary kriging (OK), Regression Kriging (RK), Co-Kriging (CoK), Kriging with External Drift (KED), and Functional Kriging (FK) have been taken into account for comparison. Apart from visual assessment, a cross-validation approach is used to compare these methods to judge their prediction accuracy.Annual or intra-annual IDF calculations across the state is not well correlated with other variables except elevation, thus directionally smoothed altitude is only considered as a covariate that offered a significant reduction in bias. All results indicate that IDW interpolation is incapable of improving the regional point IDF approximations provided by kriging algorithms except in the case of annual IDF predictions at shorter scales where its performance is more or less similar to OK. Whereas summer IDF observations are well predicted by KED that also exhibits good behavior for longer duration extremes of all seasons. Moreover, the shorter duration winter IDF guesstimates are best achieved with CoK. From now, it can be noticed that the accuracy of the interpolator changes according to the hydrological seasons and storm durations.Overall, this study ensures to design of a well-planned map in advance for the entire state of Baden Wurttemberg on the basis of accurate forecasting of seasonal IDF estimates of precipitation extremes at unsampled sites. Hence, this crucial step will surely help us to tackle the natural disasters due to climate change before time.

Rainfall is highly variable in space and time. The knowledge of precipitation variability is very important for design or for uncertainty assessment of models. In this contribution two different aspects of variability are investigated – the treatment of zero observations for spatial interpolation and the problem of high order dependence. The finer the temporal resolution of precipitation observations the more zeros have to be considered. Should one include all zeros for the description of the spatial variability (for example variograms)? Examples corresponding to different time aggregations are show that zeros need a specific treatment. High order dependence is investigated using time series observed at multiple sites. Results are compared to a meta-Gaussian approach. A large high-resolution dataset from South-West Germany is used to demonstrate the problems and the different approaches.

Partitioning a dataset in multivariate analysis is one of the key points to better understanding the hydrological process. Different regions of a catchment may bring floods variously due to distinct types of floods or their simultaneous occurrence. Therefore, it is needed to determine the spatial extent floods brought together. In a multidimensional space, it is demanding to investigate floods. It is not clear which kind of clustering methods or dimension reduction techniques are appropriate for visualizing initial similarities among measured peaks. Two methods are applied to reduce dimensions and compare their differences in this research. Multidimensional scaling (MDS) and t-distributed Stochastic Neighbor Embedding (tSNE) are the employed models for 55 years of extreme floods in the Neckar catchment. MDS is based on Principal Component Analysis (PCA), which is a linear technique. While tSNE is a non-linear dimensionality reduction method. In theory, tSNE can handle outliers and perplexity and preserve the local structure of data. As a result, compared to the MDS, both methods react similarly in soliciting an additional algorithm to cluster data in 2D space. It is another challenge that has to be investigated in future research. Due to the fatter and heavier tails, the t-student distributions have a greater chance for extreme values than normal distributions. Therefore, tSNE can better visualize data in a high-dimensional space for assessing extreme events. However, these algorithms must be run in different climates and deal with distinct hydrometeorological variables.

It is well known that vegetation shows the apparent spatial distribution characteristics in mountainous terrain at fine scales (tens of meters to kilometers). The micrometeorological data, like radiation and temperature, are intensely governed by local topography. The relationship between terrain and the distribution of vegetation, water, energy, and carbon fluxes at fine scales in terrestrial ecosystems is still unclear. This study aims at analyzing the eco-hydrological process at tens of meters scales in a typical humid hilly area, with varying altitudes, slopes, aspects, and soil textures causing the corresponding uneven micrometeorological conditions. We use the radiation and temperature data corrected by the micro-topography data to drive the eco-hydrological model (Ecosystem demography model version 2). Results showed that different regions have different micrometeorological conditions, the distribution of vegetation, water, energy, and carbon fluxes. Furthermore, the topographic heterogeneity, giving rise to the uneven soil texture and micrometeorological conditions, directly or indirectly affects the distribution of vegetation, water, energy, and carbon fluxes. The findings will improve our understanding of the eco-hydrological processes.

We propose a new methodology for faster calibration of multiparameter models using the Fourier transform. The HBV (rainfall-runoff) model is used to demonstrate the concept. It requires precipitation, temperature and potential-evapotranspiration as input time series with a total of 10 calibration parameters. By changing each parameter, the resulting runoff series is affected in a unique way. We observed that these changes are independent of others in the frequency domain. Parameters that show combined effects with others may be calibrated together. Indirectly, we can also find out model parameters that produce unique effects and the ones that are dependent on each other. The advantage is that it is easier to do multiple one-dimensional calibrations as compared to a single high-dimensional one.

The number of active gauges with open‐data policy for discharge monitoring along rivers has decreased over the last decades. Therefore, spaceborne measurements are investigated as alternatives. Among different techniques for estimating river discharge from space, developing a rating curve between the ground‐based discharge and spaceborne river water level or width is the most straightforward one. However, this does not always lead to successful results, since the river section morphology often cannot simply be modeled by a limited number of parameters. Moreover, such methods do not deliver a proper estimation of the discharge's uncertainty as a result of the mismodeling and also the coarse assumptions made for the uncertainty of inputs. Here, we propose a nonparametric model for estimating river discharge and its uncertainty from spaceborne river width measurements. The model employs a stochastic quantile mapping scheme by, iteratively: (a) generating realizations of river discharge and width time series using Monte Carlo simulation, (b) obtaining a collection of quantile mapping functions by matching all possible permutations of simulated river discharge and width quantile functions, and (c) adjusting the measurement uncertainties according to the point cloud scatter. We validate our method over 14 different river reaches along the Niger, Congo, Po Rivers, and several river reaches in the Mississippi river basin. Our results show that the proposed algorithm can mitigate the effect of measurement noise and also possible mismodeling. Moreover, the proposed algorithm delivers a meaningful uncertainty for the estimated discharge and allows us to calibrate the error bars of in situ discharge measurements., Plain Language Summary: Water scarcity affects the life of more than 2 billion people around the world. The situation will likely worsen as populations and the demand for freshwater grows. Since the demand for freshwater is increasing year by year, the importance of detecting and monitoring the Earth's freshwater is addressed in actions about ensuring the availability and sustainable management of water. However, due to a decline in the number of active gauges, the dynamic of surface water storage has been hardly quantified. The recent breakthroughs in spaceborne geodetic techniques enable us to overcome the lack of comprehensive measurements of freshwater resources, which is a major impediment for a realistic understanding of the hydrological water cycle. Several approaches have been developed for estimating river discharge from space. Among them, developing an empirical rating curve function between ground‐based discharge observations and spaceborne measurements of a physical quantity of the corresponding river reach like water level or river width is the most straightforward one. To release from the bottleneck of characterizing the river section geometry through a limited number of model parameters, we have introduced a nonparametric approach to estimate river discharge from spaceborne river width measurements., Key Points: The relation between river width and discharge is modeled by a stochastic quantile mapping function. Being nonparametric, our discharge estimation mapping function does not require the selection of a model for discharge estimation. The method allows to calibrate uncertainties of in situ discharge., Deutsche Forschungsgemeinschaft (DFG) http://dx.doi.org/10.13039/501100001659, http://hydrosat.gis.uni-stuttgart.de/

With modern data acquisition technology, Electrical Resistivity Tomography (ERT) surveys can be deployed easily and run cost-efficiently over large areas and over long periods. Inversion of the electric potential data to image the subsurface resistivity distribution is a key step in the workflow. In contrast to deterministic inversion commonly used in ERT, geostatistical inversion methods construct a variety of possible realizations of resistivity distributions and allow for a better interpretation of results and a quantification of the uncertainty. In this paper we present a geostatistical approach for the inversion of ERT survey data based on Random Mixing. The method of Random Mixing had been developed and tested for inverse groundwater modelling and is applied here to ERT as an example of geophysical inversion for the first time. Random Mixing is based on the idea of constructing realizations of the unknown electric conductivity field by combining random fields representing the spatial correlation of conductivity such that direct measurements of conductivity if available and indirect dipole measurement of electric potentials at electrodes are matched. Implementation is based on the three-dimensional finite element method to solve the forward model where properties in a core region are updated through the Random Mixing iteration. To reduce computational costs for the evaluation of the misfit cost function one-dimensional interpolation is applied. The results of Random Mixing for a synthetic example and a field survey inversion are discussed and compared with deterministic inversion results.

A new method for simulating conditional spatial fields is presented which improves on linear geotatistics currently used in commercial petroleum software. It preserves the continuity of extreme high and low value regions and provides an assessment of uncertainty based on the dependence of the magnitude of adjacent values. Moreover, this method includes flexibility to handle a variety of conditioning constraints, including non-linear constraints, integral equalities and inequalities.