WaterSubject

The methods behind the predefined impulse response function in continuous time (PIRFICT) time series model are extended to cover more complex situations where multiple stresses influence ground water head fluctuations simultaneously. In comparison to autoregressive moving average (ARMA) time series models, the PIRFICT model is optimized for use on hydrologic problems. The objective of the paper is twofold. First, an approach is presented for handling multiple stresses in the model. Each stress has a specific parametric impulse response function. Appropriate impulse response functions for other stresses than precipitation are derived from analytical solutions of elementary hydrogeological problems. Furthermore, different stresses do not need to be connected in parallel in the model, as is the standard procedure in ARMA models. Second, general procedures are presented for modeling and interpretation of the results. The multiple-input PIRFICT model is applied to two real cases. In the first one, it is shown that this model can effectively decompose series of ground water head fluctuations into partial series, each representing the influence of an individual stress. The second application handles multiple observation wells. It is shown that elementary physical knowledge and the spatial coherence in the results of multiple wells in an area may be used to interpret and check the plausibility of the results. The methods presented can be used regardless of the hydrogeological setting. They are implemented in a computer package named Menyanthes.

Transfer function noise (TFN) models are a convenient tool for modeling the evolution of a wide range of variables. The general theory of time series analysis originally stems from the statistical sciences. Because of their statistical background, the so-called autoregressive moving average (ARMA) time series models can be applied to all sorts of data, as long as the behavior of the system to be modeled is sufficiently linear or can be linearized by transforming the data. Time series models are especially useful for modeling systems whose behavior cannot, or not easily, be described in terms of physical laws and properties (e.g., economical data). In addition, TFN models are often used in hydrology and other sciences because they are relatively easy to construct and at the same time they can yield very accurate predictions.

When an ARMA type TFN model is applied to a data set, the so-called model order has to be specified. The model order includes the number of autoregressive and moving average parameters in both the deterministic and stochastic parts of the model and the delay time of the transfer function. Box and Jenkins devised an iterative model identification procedure to guide the modeler in finding the optimal model order. First, an initial model order is chosen based on statistical criteria like the cross-correlation function between the explained and explanatory variables. Second, the parameters of the model are estimated by minimizing the variance of the “innovations” or one-step-ahead prediction error using an optimization algorithm. Third, the adequacy of the model is checked diagnostically using statistical criteria like the auto- and cross-correlation functions of the innovations. If the model does not yet meet the criteria, the model order is updated and the procedure is repeated until the modeler finds the results satisfactory. A disadvantage of this approach is that the results of the model identification procedure can be ambiguous and the process itself is rather heuristic and can be knowledge and labor intensive.

presented the principles of a new type of TFN model that is optimized for use on hydrologic problems and operates in a continuous time domain. In this approach, the discrete transfer function used in ARMA models is replaced by a simple analytical expression that defines the impulse response function. The resulting class of models is referred to as predefined impulse response function in continuous time (PIRFICT). Showed that PIRFICT models overcome a series of limitations of ARMA TFN models, including the use of irregular or high-frequency data and the modeling of systems with a long memory. In addition, application of the PIRFICT model does not require a Box-Jenkins style model identification procedure. Since the transfer functions are confined a priori to physically plausible behavior, there is no need to identify the “order” of the transfer functions on statistical grounds. Therefore, application of the model is standardized, which facilitates implementation in a computer package such as Menyanthes. When the PIRFICT method was introduced , we restricted ourselves to the case of a single input/output series. Here, we will extend the method to cover more complex, real world situations where multiple stresses influence head fluctuations at one or multiple observation wells simultaneously.

Two important aspects of dealing with complex data sets are addressed in this paper. The first one is the treatment of different types of stresses within the model. Different stresses require different parametric impulse response functions. Used the Pearson type III function for modeling the effect of precipitation surplus. Here, we will introduce analytical solutions of elementary hydrogeological schematizations as guides to develop appropriate impulse response functions for other stresses. We will also show that from a physical point of view, different stresses do not always have to be connected in parallel and get a separate transfer function, as is the standard procedure in ARMA TFN modeling. The second aspect deals with the interpretation and checking of the plausibility of the results. While the time series literature commonly involves the analysis of individual time series, using the PIRFICT approach, one can analyze and process all available series of heads in an area in batch. Given that they are part of the same hydrological system, the results of neighboring observation wells may show a spatial coherence, which yields valuable extra information as regards the properties of the system and the plausibility of the results. In this paper, however, all series are still modeled separately and the resulting spatial patterns are analyzed a posteriori. Future research will include methods to impose spatial coherences a priori in the model.

This paper is organized as follows. First, we discuss how different types of stresses are dealt within the model. We illustrate the approach by analyzing a single series being influenced by precipitation, evaporation, ground water withdrawal, and river-level fluctuations. Second, results are presented for a case where data are available from multiple observation wells. A discussion and conclusions are given at the end of the paper.

In this paper, the PIRFICT model for time series analysis was extended to handle multiple inputs. For stresses other than areal recharge, analytical solutions of simple hydrogeological schematizations were used as a guide to develop appropriate impulse response functions. Different stresses are not necessarily connected in parallel in the model, as is the standard procedure in ARMA models. A case with a single observation well was used to illustrate how the model can effectively decompose head series into partial series that each show the effect of an individual stress. In the example with multiple observation wells, it was shown that, next to its a priori use in defining the impulse response functions, physical knowledge is also valuable in checking the consistency and plausibility of the model results a posteriori. The parameter values should fall within a range that is physically plausible. The spatiotemporal patterns observed in the variables supply important and independent feedback on the results, as there is no spatial dependency imposed on the models. By focusing on the model residuals, missing stresses, processes, or other sources of error may be readily identified. High error levels are a possible source of bias in the estimates, as other stresses may partly compensate missing stresses when their influence is correlated. In this case, the main source of error was the fact that the behavior of the individual pumping wells was not accounted for. In spite of this, the overall drawdown pattern of the wellfield in total, which is often the factor of interest, was represented well. The spatial distribution showed that the estimates for the evaporation series were clearly biased. Such a problem may be corrected by incorporating data on the missing factors in the model (in this case, the pumping at individual wells). If such data are not available (as in this case), the bias can be reduced by constraining the optimization problem to a realistic range based on the values and patterns in the surrounding observation wells. Future research will include methods to impose a spatial coherence a priori in the model, so that the individual dynamics of a high number of pumping wells can be incorporated in the model and the results of neighboring observation wells are linked and remain plausible.

The presented time series model may be applied to decompose series of head fluctuations into partial series, representing the influence of individual stresses. This enables the evaluation of individual effects of a stress, such as hydrological interventions, pumping wells, climatic changes, and surface water levels. Furthermore, the presented model may be used for forecasting, gap filling, scenario studies, trend analyses, and optimization and control of hydrogeological systems (using more or less standard time series analysis methods; e.g.,. The PIRFICT approach is particularly suited for the batch processing of many series, it uses a small number of parameters, and it is not limited by irregular or high-frequency data. Although here we specifically focus on ground water heads, the approach presented can also be applied to (contaminant) transport problems or for that matter to hydrological problems in general. In the field of transport, the Pearson III function has proven to be well usable also (it matches the convection-dispersion equation, whereas in other fields, care has to be taken to select appropriate impulse response functions and methods for the processes and stresses occurring there.

An attractive feature of time series models is that they are based on relatively few assumptions and the fits are generally high; the model lets the data more or less speak for itself. As such, time series models are a valuable tool for preprocessing ground water level series before calibrating a ground water model. Using time series models, missing stresses or series that are influenced by hydrological interventions may be readily identified. Also, series may be identified that are not suitable for model calibration, for example, because they represent a hydrological feature that is not incorporated in the model, such as perched water tables. One step further, transient ground water models may be calibrated directly on the moments of the impulse response functions estimated with time series analysis , which requires the calibration of steady models only. Moments of the impulse response functions can also be modeled spatially with the analytic element method, creating a possibility for transient modeling with analytic elements.