## 1. Introduction

[2] Time series models can be used to model the dynamical behavior of a wide range of variables. Once a time series model is calibrated to a limited set of observations, it is thought to represent the dynamical properties of the system that generated the observations. The model can then be used to make real time forecasts of future values of the output variable and its uncertainty, to predict values at nonobserved periods and to quantify and separate the influence of different variables on the output signal. An extensive overview of time series modeling theory was first published by *Box and Jenkins* [1970]. In univariate time series models, the modeled time series is assumed to be generated by a linear transformation of a random input signal. The input signal is considered to be a discrete white stochastic process, while the characteristic response of the analyzed system is estimated by minimizing the likelihood function, or a least squares criterion, of the noise series. Transfer Function-Noise (TFN) time series models are used whenever a time series can be modeled by linearly transforming one or more deterministic input series, while the residuals of the transfer model are autocorrelated. As these conditions are often met in hydrology, TFN models have been used on many hydrologic variables, including time series of groundwater head [*Tankersley et al.*, 1993; *Gehrels et al.*, 1994; *Van Geer and Zuur*, 1997]. Most time series can also be modeled by mechanistic models that operate according to the physical laws of the analyzed system, for example by a transient distributed groundwater model such as MODFLOW in the case of time series of groundwater head. The use of TFN models is, however, often preferred over the use of mechanistic models, because TFN models often yield more accurate predictions and are less complicated than mechanistic models [*Hipel and McLeod*, 1994]. In addition, because of their stochastic nature, TFN models are well equipped to model the behavior and uncertainty of phenomena that are not well explained by physical laws only. Apart from these favorable properties, there are also drawbacks to the use of TFN models as described by Box and Jenkins. First, TFN models require very specific knowledge from the analyst, as he or she has to be able to perform an iterative model identification procedure. Furthermore, traditional time series models can only operate on time series that are equally spaced in time and are noninterrupted, while the frequency of all input and output variables is coupled and has to be equal.

[3] In time series literature [e.g., *Box and Jenkins*, 1970; *Hipel and McLeod*, 1994; *Ljung*, 1999] little attention is often paid to the fact that time is a continuous phenomenon. Most time series models simply divide time into fixed portions, and only perform calculations with time series that are observed in accordance with their time discretization. For many applications, this does not pose a serious problem, as the frequency of the measuring equipment can easily be adapted to match the demands of the time series model. In hydrological practice, however, the analyst will often have to make do with the data available. Time series of the groundwater level, for example, are often collected manually and tend to be nonequidistant and contain missing data. *Bierkens et al.* [1999] published a possible solution to this problem. In their article they show that an ARX(1,0) model (i.e., an autoregressive-exogenous variable model [*Hipel and McLeod*, 1994]) that operates on a daily frequency can be fitted on irregularly observed time series with the aid of a Kalman filter, because the Kalman filter estimates the variance of the one-step ahead prediction error or innovation series at nonobserved time steps. Although this “KALMAX” approach to a large extent alleviates this problem for simple exponential systems, it does not offer a satisfactory solution for time series of slow systems with a nonexponential response, e.g., large groundwater systems with thick unsaturated zones [*Gehrels et al.*, 1994]. An adequate daily frequency TFN model for such complex, slow systems would require an ARMAX (i.e., an autoregressive-moving average-exogenous variable [*Hipel and McLeod*, 1994]) or full scale TFN model with too many parameters to be calibrated smoothly, as the number of parameters is coupled to the model frequency. Moreover, using the KALMAX approach, all input series still have to be available with equidistant, but small, time steps. Another consequence of time discretization is that the model order and parameter values depend on the frequency of observation. Because of this dependency, a transformation is necessary to couple time series models with different frequencies [*Koutsoyiannis*, 2001]. Problems can also occur with the scaling of possible autoregressive (AR) parameters, as they asymptotically approach a value of 1 when the frequency increases.

[4] The topic of model identification has received much attention in time series literature, as it is an important and difficult part of time series analysis. In general, the approaches and methods proposed by several authors are very diverse. Following *Knotters* [2001], they can be subdivided into three main categories. The first category is formed by iterative procedures of identification, calibration and diagnostic checking. Such procedures are normally based on stochastic realization theory and/or statistical hypothesis testing, and include the traditional and still much practiced procedure that was proposed by *Box and Jenkins* [1970]. A general disadvantage of this approach is that the results of the model identification procedure can be ambiguous [*Hipel and McLeod*, 1994] and the process itself is rather heuristic and can be very knowledge and labor intensive [*De Gooijer et al.*, 1985]. Secondly, the model order can be identified using automatic model selection procedures, using information measures, Bayesian methods and/or the one-step-ahead prediction error. Among the most used criteria are Akaike's final prediction error (FPE) criterion, information criterion (AIC) and Bayes information criterion (BIC) [*Akaike*, 1970, 1974, 1979]. Both AIC and FPE, however, are not consistent but asymptotically overestimate the true order of time series models [*Shibata*, 1976]. Although many of the available measures have desirable characteristics, there is not one definitely superior method, and a coherent and systematic framework which indicates when to use a particular method is lacking [*De Gooijer et al.*, 1985]. A third category of model identification procedures is formed by procedures that use physical insight into the system, in contrast to the above mentioned methods that are based only on the data itself. In the data-based mechanistic modeling methodology [e.g., *Young and Beven*, 1994; *Price et al.*, 2000] physical analysis is combined with statistical model identification. TFN models are identified directly from the data but are only accepted as a reasonable representation of the system if they have a valid physical interpretation. *Knotters and Bierkens* [2000] went one step further and used physical analysis to limit the set of candidate TFN models to one. In their study, the ARX(1,0) model is chosen as the most appropriate linear time series model for time series of the groundwater level on the basis of a water balance of the phreatic groundwater zone.

[5] In this paper we present a method of transfer function-noise modeling in continuous time, which uses predefined impulse response (IR) functions. The resulting class of TFN models is referred to as PIRFICT models (PIRFICT stands for predefined IR function in continuous time), and circumvents a number of limitations of discrete TFN models linked to time discretization and model identification. The paper is organized as follows. First, the theory and basic equations of discrete TFN models are given and subsequently transformed into continuous time to obtain the PIRFICT model. Next, the implementation of the PIRFICT model is elucidated step-by-step, and is summarized at the end of section 2. In the example application, the performance of the PIRFICT model is compared to that of traditional Box-Jenkins TFN models and finally, discussion and conclusions are given. For reasons of simplicity, the theory is developed and illustrated using a single input TFN model with the groundwater level as output series and precipitation surplus as input series, but the equations can be readily extended to include multiple input series.