The Water Residence Time distribution is the equivalent of the impulse response of a linear system allowing the propagation of water through a medium, e.g. the propagation of rain water from the top of the mountain towards the aquifers. We consider the output aquifer levels as the convolution between the input rain levels and the Water Residence Time, starting with an initial aquifer base level. The estimation of Water Residence Time is important for a better understanding of hydro-bio-geochemical processes and mixing properties of wetlands used as filters in ecological applications, as well as protecting fresh water sources for wells from pollutants. Common methods of estimating the Water Residence Time focus on cross-correlation, parameter fitting and non-parametric deconvolution methods. Here we propose a 1D full-deconvolution, regularized, non-parametric inverse problem algorithm that enforces smoothness and uses constraints of causality and positivity to estimate the Water Residence Time curve. Compared to Bayesian non-parametric deconvolution approaches, it has a fast runtime per test case; compared to the popular and fast cross-correlation method, it produces a more precise Water Residence Time curve even in the case of noisy measurements. The algorithm needs only one regularization parameter to balance between smoothness of the Water Residence Time and accuracy of the reconstruction. We propose an approach on how to automatically find a suitable value of the regularization parameter from the input data only. Tests on real data illustrate the potential of this method to analyze hydrological datasets.
The decision to benchmark against the cross-correlation is due to the fact that it is the preferred method for hydrologists in numerous recent articles: for determining transport of biological constituents in @cite_7 , or studying river-groundwater interaction with different types of measurements being cross-correlated like in @cite_27 . Cross-correlation is also used by @cite_16 for estimating mixing ratios and mean residence times, by @cite_23 for estimating the pure residence time curve. Therefore, the hydrology community is interested in a simple and fast method with minimal implementation time that gives a residence time curve estimation from different time series measurements. In the case of the cross-correlation method, one focuses on analyzing the position of the maximal amplitude and general shape of the curve. From this curve hydrologists extract the characteristics of interest for that particular channel (mean residence time, mixing ratios, etc.). In contrast to the cross-correlation method we offer positivity, smoothness and causality constraints that give a more precise curve and a similar computing time.
@cite_27 @cite_16 @cite_23 @cite_7
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A key parameter used in the assessment of bank filtration is the travel time of the infiltrated river water during the passage through groundwater. We analyze time series of electrical conductivity (EC) in the river and adjacent groundwater observation wells to investigate travel times of young hyporheic groundwater in adjoining channelized and restored sections of River Thur in North-East Switzerland. To quantify mixing ratios and mean residence times we perform cross-correlation analysis and non-parametric deconvolution of the EC time series. Measurements of radon-222 in the groundwater samples validate the calculated residence times. A simple relationship between travel time and distance to the river has not been observed. Therefore, we speculate that the lateral position and depth of the thalweg as well as the type of bank stabilization might control the infiltration processes in losing rivers. Diurnal oscillations of EC observed in the river and in nearby observation wells facilitate analyzing the temporal variation of infiltration. The diurnal oscillations are particularly pronounced in low flow situations, while the overall EC signal is dominated by individual high-flow events. Differences in travel times derived from diurnal and overall EC signals thus reflect different infiltration regimes. Summary We study the temporal variability of water transfer through the infiltration zone of a karst aquifer by estimating the impulse response of the system using cross-correlogram analyses between rainfall and piezometric level time series. We apply a sliding-window cross-correlation method, which calculates cross-correlograms on partially superposed short time series windows. We apply this method for rainfall and piezometric level time series at six boreholes in a fractured karstic aquifer located in Burgundy, France. Based on cross-correlogram functions, we obtain a time series of response time. At most of the boreholes, the cross-correlation functions change over time, and the response times vary seasonally, being shorter during the summer. This unusual structure can be partly explained by the seasonal variability in rainfall intensity, which is higher during the summer (May–September), inducing the seasonal behaviour of the epikarst. During the summer, when rainfall intensity is higher, the epikarst is more easily and quickly saturated. This induces an increase in lateral water transfer within the epikarst and an increase in concentrated fast flows. We also show that the response time seems to tend towards a limit which represents the maximum saturation of the epikarst. We propose a novel approach to understanding activities from their partial observations monitored through multiple non-overlapping cameras separated by unknown time gaps. In our approach, each camera view is first decomposed automatically into regions based on the correlation of object dynamics across different spatial locations in all camera views. A new Cross Canonical Correlation Analysis (xCCA) is then formulated to discover and quantify the time delayed correlations of regional activities observed within and across multiple camera views in a single common reference space. We show that learning the time delayed activity correlations offers important contextual information for (i) spatial and temporal topology inference of a camera network; (ii) robust person re-identification and (iii) global activity interpretation and video temporal segmentation. Crucially, in contrast to conventional methods, our approach does not rely on either intra-camera or inter-camera object tracking; it thus can be applied to low-quality surveillance videos featured with severe inter-object occlusions. The effectiveness and robustness of our approach are demonstrated through experiments on 330 hours of videos captured from 17 cameras installed at two busy underground stations with complex and diverse scenes. Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. The majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops. In this paper, a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure. A linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data. A link is assumed to be present between two time series if the weight of a time series, which is a linear time-invariant filter, is nonzero in the formation of the other. Reconstruction of the link connectivity structure under various scenarios is considered. It is shown that when the linear dynamic graph is allowed to admit non-causal weights, then the links structure can be recovered with the possibility of identifying spurious connections. However, it is shown that the spurious links remain local, where, a spurious link is restricted to be within one hop of a true link. Furthermore, strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed. The main tools for determining the network topology are based on variations of Wiener filtering. A significant insight provided by the article is that, in the class of network models considered in the paper, the Wiener filter estimating a stochastic process based on other processes remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated.