A method of transfer function-noise (TFN) modeling is presented that operates in continuous time and uses a predefined family of impulse response (IR) functions. The resulting class of models is referred to as predefined IR function in continuous time (PIRFICT). It provides a useful tool for standardized analysis of time series, as it can be calibrated using irregularly spaced data and does not require a model identification phase prior to calibration. In section 2, the discrete Box-Jenkins (BJ) model is presented and transformed into continuous time to obtain the PIRFICT model. The discrete transfer function of a BJ model, which is made up of a variable number of parameters, is replaced by a simple analytical expression that defines the IR function. From the IR function, block response functions are derived that enable the model to handle irregularly spaced data. In the example application, the parameter estimates and performance of the BJ and PIRFICT model are compared using a data set of 15 piezometers and a simulated series. It was found that the estimated transfer and BR functions of both models follow the same general pattern, although the BJ transfer functions are partly irregular. The performance of both models proves to be highly comparable for all piezometers.
 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 . 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.
 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.  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.
 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 , 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 . 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  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.
 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.
2. Method and Theoretical Background
2.1. In Discrete Time: The TFN Model of Box and Jenkins
 For linear, undisturbed phreatic systems that are influenced by precipitation surplus only, the following single input discrete TFN model can be used to model groundwater level fluctuations:
the observed groundwater level at time step t (with t ∈ ,(dimensionless)), relative to some reference level [L];
the predicted groundwater level at time step t attributable to the precipitation surplus, relative to d [L];
the residual series [L];
the level of ht* without precipitation, or in other words the local drainage; level, relative to some reference level [L];
transfer function (dimensionless);
backward shift operator (dimensionless), defined as Bipt = pt-i;
the precipitation surplus at time step t [L];
noise transfer function (dimensionless);
zero mean discrete white noise process [L].
In Box-Jenkins (BJ) models the transfer function Θ(B) is defined as a fraction, where the numerator is a so-called moving average (MA) function ω(B), and the denominator an autoregressive (AR) function δ(B), so that Θ(B) = ω(B)/δ(B). The weights Θ0, Θ1, Θ2…Θ∞ of the transfer function are normally referred to as the impulse response (IR) function, but from a continuous point of view this response is actually the response to an input series with the shape of a block (see section 2.3.). To avoid confusion, we will use the term transfer function in the discrete case and reserve the term IR function for the response to an actual instantaneous impulse. In the model identification phase of discrete TFN models, the model structure is defined by the choice of a delay time and the number of MA and/or AR parameters, which together form the nodes of the transfer functions in equations (2) and (3) and define their general shapes:
number of moving average parameters of the transfer model;
number of moving average parameters of the noise model;
number of autoregressive parameters of the transfer model;
number of autoregressive parameters of the noise model;
Thus the order of a BJ model is mostly specified as [nb nc nd nf nk]. Although it is often not explicitly mentioned, the modeler also has to specify the time discretization of the groundwater level series (th) and the precipitation series (tp), yet the options are normally restricted in practice, because of data availability and because th = tp for BJ models. We therefore specify the order of a BJ model as [nb nc nd nf nk][th = tp]. The overall effect of moving average parameters on the model structure is that they provide the transfer function freedom of shape with respect to real time from nk * tp to (nk + nb) * tp, while an additional AR parameter causes the tail of the IR function to be exponential (see Figure 1). In the BJ model identification procedure, the analyst identifies, by iteratively adding or removing MA parameters, the point from which the remainder of the transfer function can be adequately approximated by an exponential function, which often lies just beyond the peak of the transfer function. The number of parameters necessary to reach this point, however, is dependent on the observation frequency, as an MA parameter is necessary for every time step in between. Therefore the modeler also has to balance the observation frequency against the number of parameters, in order to get the best model result.
 It can be readily seen that the limitations of discrete TFN models with regard to irregularly observed time series follow directly from equations (1)–(3). In these equations, time is considered to be a dimensionless index t ∈ , such that each time step is equal to one, regardless of the discretization in real time. Furthermore, the same index is used for ht and pt, so the sample intervals of the input and output series have to be identical. Consequently, the analyst is forced to lower the frequency of all time series to the lowest one available and thus disregard relevant information about the distribution of the input series in between the time steps.
2.2. In Continuous Time: The PIRFICT Model
 Most processes modeled with TFN models, such as precipitation and groundwater level fluctuations, are in reality not discrete but continuous. In continuous time, the transformation of a time series with a linear, time-invariant transfer function is given by a convolution integral [Quimpo, 1971]. Thus equations (1)–(2) can be written in continuous time as
Unlike the transfer function of discrete TFN models, the IR function θ(t) of a convolution integral does not depend on the observation frequency of the input series. It describes the dynamical response of a system to an instantaneous impulse and is time-invariant and an integral property of a specific system [Jury and Roth, 1990; Maas, 1994].The IR function, in hydrological terms, describes the way in which the water table responds to an instantaneous impulse of precipitation surplus. In that respect it is similar to the instantaneous unit hydrograph used in surface water hydrology [e.g., Dooge, 1973]. Equation (3) can be transformed to continuous time by replacing the discrete white noise process at by a continuous white noise process. The residual series n(t) can then be modeled as a colored stochastic process, which is given by the following stochastic integral:
with W(t) a continous white noise (Wiener) process [L], with properties:
To allow for discretely available data in continuous time with nonequidistant intervals we use t in real time, and index t with i instead. A series of N discretely available observations of a continuous process, such as groundwater level fluctuations, can then be written as:
The drainage level d in equation (5) is usually unknown, but can be eliminated from the equations by summing both sides of equation (5) for all time steps (from t = t0 to tN), dividing its result by N, and assuming that the drainage level is constant, which gives:
Combining equations (5) and (9) gives an overall equation for a continuous TFN model of which the components are all centered about their respective temporal averages (the overbar symbols), and in which d is no longer present:
 In the continuous case, the model order is defined by choosing continuous mathematical functions to represent the IR functions. The mathematical functions are selected on physical grounds, by an iterative procedure of model identification, estimation and diagnostic checking, or with the use of automatic model selection criteria. There are, however, several important differences with the discrete model identification procedure. First of all, when chosen carefully, a continuous IR function can have a flexible shape and be equivalent to a series of AR/MA transfer functions. Secondly, the model identification procedure is simplified, because the model frequency does not interfere with the model order. Thirdly, a continuous IR function can be objectively chosen as the function that represents the physics of the analyzed system best. A physically based IR function on the one hand reduces the sensitivity of the model to coincidental correlations in the data, but on the other hand it can reduce the fit if for some reason the physical assumptions prove incorrect. In BJ TFN models the model order can be chosen on physical grounds [Young and Beven, 1994], but with models that contain MA parameters the exact shape of the transfer function cannot.
 Here we choose the Pearson type III distribution function (PIII df), with an extra parameter that adjusts the area, to describe the response of the water table to precipitation surplus:
with a, n, b, and A being parameters. The physical basis of the PIII df lies in the fact that it describes the transfer function of a series of coupled linear reservoirs [Nash, 1958], the parameter n denoting their number, a equaling the inverse of the reservoir coefficient normally used, and b being the delay time. The extra parameter A is necessary because in the case of equation (6), where a precipitation surplus series is transformed into a groundwater level series, the law of conservation of mass does not apply. Mathematically, n is not restricted to integer values, which further increases the flexibility of the PIII df IR function. The PIII df can take shapes gradually ranging from steeper than exponential, via exponential to Gaussian (see Figure 2). In discrete terms, it can be equivalent to an AR(1) model, but also to all ARMA models of which the weights of the transfer function together form one of the curves in the PIII df family. Knotters and Bierkens  showed that an ARX(1,0) model is a discretized version of a single linear reservoir, and subsequently identified it as the most appropriate linear time series model for time series of groundwater level. Their choice of the ARX model, however, was based on a simple physical model of a one dimensional soil column, discarding lateral flow and the functioning of the unsaturated zone. In this sense the PIII df forms an extension to their method, as it includes the ARX model but can for example also describe the combined response of a saturated and layered unsaturated zone.
 As the residuals, which are thought to be the output of the noise model, are the result of a variety of causes (e.g., errors in the observations of the input and output series, errors in the model parameters, simplifications or errors in the model concept), it is difficult to make a clear choice of the noise IR function on physical grounds. We therefore choose a simple AR(1) noise model, and rely on diagnostic checks to test its adequacy. The choice of an AR(1) model equals the choice of an exponential IR function in continuous time [e.g., Box and Jenkins, 1970; Chatfield, 1989]. In order to get an exponential noise model with an appropriate innovation variance function, here we use an IR function of the following form (J. R.Von Asmuth and M. F. P. Bierkens, Modeling irregularly spaced residual series as a continuous colored stochastic process, submitted to Water Resources Research, 2002) (hereinafter referred to as Von Asmuth and Bierkens, submitted manuscript, 2002):
with the parameter α determining the decay rate of ϕ(t) and σn2 denoting the variance of the residuals.
2.3. Model Evaluation, Parameter Estimation and Diagnostic Checking
 When the IR functions have been chosen, the PIRFICT model is identified and can be evaluated. In the following, we will describe the different steps in the implementation of the PIRFICT model. First, the available time series have to be transformed to continuous series, as most time series are not available as continuous series due to the discrete observation process. Many collected time series, however, can be regarded as the change in the primitive function of some underlying continuous process, as is the case with precipitation surplus:
When precipitation surplus data is only available at discrete intervals, the continuous series p(τ) cannot be reconstructed exactly, but it can be approximated by assuming that the distribution of p(τ) is uniform during the period ti−1 to ti [Ziemer et al., 1998]. Equation (13) can then be written as:
In this way, the higher the frequency of observation, the more ti − ti−1 approaches zero and the better the approximation of the continuous precipitation surplus series p(τ) (with units [LT−1]) will be. Note that the error made by this assumption will vary in time when nonequidistant precipitation surplus observations are used, which will introduce nonstationarity in the model residuals. However, we assume that, when the time steps are not too large and irregular, this effect is small when compared to the other sources of model error and can therefore be neglected.
 Secondly, as equations (6) and (7) contain time from −∞, we have to define an initial condition to be able to evaluate the equations with observations starting from t = 0. Here we define the initial condition of p(τ) from τ = −∞ to τ = 0 to be , the temporal average of the precipitation surplus series used for the simulations (but when available a historical average could also be used). With the aid of equation (14), the transfer model (equation (6)) can now be evaluated using the block response (BR) function. The BR function Θ(t) can be obtained by convoluting the IR function with a “block” of precipitation surplus with unit intensity over a period Δt, which equals:
To make the BR function equivalent to the discrete transfer function, it has to be divided by Δt in order to scale it to a block input with unit area. In Figure 3 some example BR functions of a single linear reservoir are plotted for different time steps. The groundwater level series h*(t) can be obtained by adding the responses of all “blocks” of precipitation. Because Θ(t) is a continuous function, h*(t) itself is also continuous, and for every observation of h(t) a sample of the residual series n(t) can be obtained.
 Next, the noise model (equation (7)) is evaluated in order to obtain a series of innovations ν(t). To evaluate the noise model without having to use a Kalman Filter (which is computationally expensive) we will derive a direct relation between the residuals n(t) and the innovations ν(t). Consider the series ν(t) as the nonequidistantly sampled change in the solution to the Ito stochastic integral describing the residual series:
 Subsequently, the parameter set β = (A,a,n,b,α) of the IR functions has to be estimated from the data. By adjusting the value of the parameters the time series model can be calibrated on a set of observations of the groundwater level series h(ti) by minimizing a certain objective function. Bierkens et al.  use a log likelihood function [Schweppe, 1973] as objective function for the innovations of a Kalman filter with an observation error variance of 0, to get a maximum likelihood estimate of the model parameters (under the assumption that the innovations are Gaussian). However, for reasons of efficiency, we seek an objective function that can be expressed in terms of individual innovations. From (12) and (16) we have
With (19) the likelihood function can be approximated by the following weighted least squares criterion (Von Asmuth and Bierkens, submitted manuscript, 2002):
with ν2 (tj, β) calculated from the residual series using equation (18). From equation (20), a Jacobian matrix can be easily obtained, so it can be minimized with respect to β using a Levenberg-Marquardt method. This makes the parameter estimation problem much more efficient than using a Kalman filter in conjunction with a log likelihood function and some global optimization algorithm.
 Finally, the accuracy and validity of the model results is checked by examining the autocorrelation and cross-correlation functions of the innovations, the covariance matrix of the model parameters and the variance of the IR functions. The covariance matrix of the parameters C(p) is estimated using the Jacobian matrix obtained from the calibration routine and σν2. From the covariance matrix, also the correlation between the parameters is calculated. The variance of the IR function θ(t) can be obtained by
Assuming a normal distribution, a confidence interval for θ(t) can be plotted as ± 2σ. As in discrete TFN models, serious model inadequacy can be detected by examining the autocorrelation function of the innovation series ν(t) (which indicates whether the white noise assumption holds) and the cross-correlation function of ν(t) and the input series p(t) (which indicates whether there are still patterns left in the innovation series that could be explained by the input series). The autocorrelation and cross-correlation functions at lag k are defined in the same way as in discrete TFN models, but because of the nonequidistant sampling a tolerance around lag k of ±0.5k is implied. A similar approach is used in the field of Geostatistics to obtain spatial variograms [Journel and Huijbregts, 1978]. An example of the autocorrelation functions of a BJ and a nonequidistant PIRFICT model is given in Figure 4.
2.4. Summary of Method
 In summary, the method described above consists of the following steps. First, for every input series an IR function is chosen, which in principle can be any continuous function or combination of functions, but in practice will often be based on the physical laws of the analyzed system. The input series are assumed to be uniformly distributed in between the time steps and transformed into continuous series using equation (14). The transfer convolution integral (equation (6)) can now be evaluated using block response functions for every block pulse, to obtain a continuous prediction of the output series. Using equation (18), a sample of the innovation series is obtained for every observation of the output series, whether or not equidistant. An estimate of the model parameters is made with the aid of a Levenberg-Marquardt algorithm, which numerically minimizes a weighted least squares criterion (equation (20)) that is based on the likelihood function of the innovations. Finally the accuracy and validity of the model is checked using the auto and cross-correlation functions of the innovations, the covariance matrix of the model parameters and the variance of the IR functions.
3. Example Application
3.1. Description of Setup and Data Set
 The example application is devised to illustrate the performance of the PIRFICT model in practice by comparing its calibration and validation results and parameter estimates with those of Box-Jenkins (BJ) models fitted on the same data. For this purpose, groundwater level series from 15 piezometers are selected, all lying in a dune reserve in the province North-Holland, Netherlands, near the town of Egmond. Piezometers are selected at locations that are little disturbed by the groundwater abstraction in the area [Rolf and Lebbink, 1998], thus allowing the groundwater level series to be modeled with precipitation surplus as a single input series. The groundwater level in all piezometers is well observed in the same period, with observations taken manually about the 14 and 28 of every month, so that the model results are little influenced by a difference in length of the calibration period or the number of observations in that period. For the calibration process, observations of the groundwater level from 1-1-1990 until 1-1-2001 are used and observations of the precipitation surplus (precipitation minus potential evaporation) starting from 1-1-1987. The precipitation series is available on a daily basis and is observed by the Provincial Water Company of North-Holland in the dunes near the town of Castricum, whereas the daily potential evaporation series originates from a station of the Royal Dutch Meteorological Institute near the town of De Kooy.
 As the performance of models may be dependent on the properties of a dataset, the example application is not solely focused on assessing the performance of both models for this specific dataset, but also on clarifying the mechanisms which influence model performance. First of all, we will illustrate the performance of a range of BJ models, their dependence on user defined choices such as model order and time discretization, and the dependence of the results of PIRFICT models fitted on the same data from an example series. For this purpose, piezometer 19AZW246_1 is selected. At this location the water table showed the slowest response to precipitation surplus, which makes it especially suited for illustrating the beneficial properties of the PIRFICT model. Next, we will identify a single BJ model order for all 15 piezometers, and compare the calibration results with those of the PIRFICT model. For six of the piezometers, observations that were taken before the calibration period are available, and for these series a validation is also performed. Unfortunately, the observations in this period are only available on a quarterly basis and not suited for a cross-validation. Finally, a small simulation experiment is performed to confirm some hypotheses based on results from the real world data.
3.2. Comparison of Calibration Results Using Data From Piezometer 19azw246_1
 In order to be able to calibrate a BJ model on a specific time series, the model order has to be identified first. As Knotters and Bierkens  have shown, the response of the saturated zone to precipitation, under the assumption of vertical flow, shows exponential decay and therefore equals an ARX model. However, the first ordinates of the response function can also be nonexponential and the response can be delayed when horizontal flow and the functioning of the unsaturated zone are taken into account. Therefore we only consider BJ models with order [nb 0 1 1 1] [th = tp], i.e. models in which the transfer and noise model contain one autoregressive parameter, a delay time and a variable number of moving average parameters. The number of moving average parameters nb is chosen to be 5, 10 and 20, also dependent on the time discretization, which covers the range of probable model orders. The time discretization is chosen in accordance with the observation frequency and such that the response of the system can be described adequately with a reasonable number of parameters. In the case of these time series this means that th = 15.2, 30.4 or 60.8 days, so that exactly 24, 12 or 6 groundwater level observations are available every year. As BJ models require equidistant time series for their calibration and the groundwater level measurements are taken approximately the 14th and 28th of every month, the groundwater level series has to be made equidistant. In this application, an equidistant series starting from 1-1-1990 is obtained by linearly interpolating the groundwater level between the two nearest observations. Observations of the precipitation surplus are available on a daily basis, and an equidistant series is obtained by taking the sum of the daily series for every time interval.
 As both the input and output time series are modified by the interpolation and resample operations required for BJ models, PIRFICT models are fitted for every data set obtained this way. To make a distinction between the results thus obtained, the PIRFICT models are tagged as [th tp] and as (non)equidistant, which in this case does not indicate a difference in model order but a difference in the data set used. The calibration results of the different BJ models and the PIRFICT model are given in Table 1, expressed in the form of a number of criteria. In Table 1, first the root mean squared error (RMSE) and root mean squared innovation (RMSI) are given, which form a measure for the error of the transfer model and the variance of the noise process, respectively. Because the variance of the groundwater level series and the residual series can be influenced by interpolation and resampling operations, also the explained variance percentage (EVP) is given. The EVP is defined as
A logical way of comparing model results seems to be the use of automatic model order selection criteria such as AIC and FPE. However, both criteria use the innovation variance or their likelihood for determining the best model order, which are influenced by the sample frequency, so these criteria cannot be used to compare models with different sample frequencies or datasets.
Table 1. Calibration Results of BJ Models With Different Orders and PIRFICT Models Calibrated on the Same Data From Piezometer 19AZW246_1
[nb nc nd nf nk] [th = tp]
[5 0 1 1 0] [30.4]
[10 0 1 1 0] [30.4]
[20 0 1 1 0] [30.4]
[20 0 1 1 0] [15.2]
[5 0 1 1 0] [60.8]
 From the results in Table 1 it can first of all be seen that the differences between the fit of both models are small. Although the PIRFICT model does show the lowest RMSE and highest EVP, the variation caused by differences in observation frequency and model order is larger than the differences between both model types. The optimal result for the BJ model appears to be given by the [10 0 1 1 0] [30.4] model, so we can identify this for the moment as the optimal model order. As expected, the RMSI of both models varies with the time lag between the observations of the output series. When th = 15.2, the RMSI lies in the order of 7 to 8 cm, whereas the RMSI is 9 to 10 cm when th = 60.8. Furthermore, the results show that the RMSI of the BJ model decreases when the number of parameters increases, while the RMSE shows an optimum for the [10 0 1 1 0] model for th = 30.4. This phenomenon could well be attributed to the fact that the models are calibrated by minimizing the RMSI (which is linked to the noise part of the model) rather than the RMSE (which is linked to the transfer model). Because of this, adding extra parameters and thereby overfitting the data will result in a gradually improving fit of the noise model, but can at the same have a negative effect on the fit of the transfer model. Overfitting behavior, or the generally higher number of parameters of a BJ model, could also explain the fact that the RMSI of the BJ model is lower than that of the PIRFICT model, while its RMSE is higher.
 The parameter estimates of the different models can best be compared by plotting the estimated BR functions. In Figure 5a, the transfer functions of three [nb 0 1 1 0] [30.4] Box-Jenkins models are plotted, with nb = 5, 10 and 20. The results appear to be significantly influenced by the model order, as the transfer functions of the nb = 10 and 20 model lie partly outside the confidence interval of that of the nb = 5 model. The number of MA parameters for the case nb = 5 is apparently too low to model the slow response of the system well. According to Figure 5 and the RMSE, in this case 10 MA parameters just about suffice to model the first part of the response of the system, while the remainder of the response function can be described adequately by a single AR parameter. From Figure 5b it can be seen that the response of the PIRFICT model follows the response of the [10 0 1 1 0] [30.4] BJ model rather closely. The results of both models should therefore be highly comparable, as the only difference in the transfer functions is the irregular pattern of the BJ model around the smooth curve of the PIRFICT model. In Figure 5c the order of the BJ model, in the traditional sense, is kept constant while the sample frequency is varied, resulting in three [10 0 1 1 0] [th = tp] BJ models with th = tp =15.2, 30.4 and 60.8. As expected, the parameter estimates of the BJ model prove to be also significantly influenced by the sample frequency, which interferes with the model order when time is used as a unitless index. Finally, in Figure 5d, the BR functions of three PIRFICT models are plotted, calibrated on the same data as the BJ model (i.e. th = tp = 15.2, 30.4 and 60.8). From the results, the first two BR functions prove to be almost identical, while the model apparently has difficulties estimating the first part of the BR function correctly for th = tp = 60.8, although it does not differ significantly. This effect is probably caused by the increasing time interval and/or decreasing number of observations, as the BJ [60.8] model estimates the transfer function in about the same way.
3.3. Comparison of Calibration and Validation Results From 15 Piezometer Series
 For a broader comparison between the Box-Jenkins and PIRFICT model, both models are calibrated on time series from 15 piezometers with observations ranging from 1-1-1990 until 1-1-2001. For reasons of comparability and objectivity, no Box and Jenkins style iterative model identification procedure is performed for each separate piezometer to obtain the most appropriate model order (in our case defined by nb, nk and th). Instead, a [10 0 1 1 nk] [30.4] BJ model is calibrated on all piezometers, which proves to give the transfer function just about enough MA parameters to model the slowest response of the 15 piezometers well (see section 3.2). For all models, a delay time is applied, which has shown to improve both the calibration and validation results. As expected, the improvements are greatest for the PIRFICT model because of its predefined shape. On the basis of a manual model identification procedure, the delay time for the 15 piezometers is chosen to be nk = [12 6 0 0 0 3 1 13 13 6 6 8 1 5 0] days. It is applied by shifting the precipitation series along the time axis. Shifting the input series rather than the response function makes it possible to apply delay times which are smaller than the discrete time interval unit, and by doing so the delay time could be kept equal for the BJ and PIRFICT model.
 The PIRFICT model is first calibrated using the same data as the BJ model, which is therefore denoted as equidistant [30.4 30.4], and secondly using nonequidistant [30.4 1] data. The results should therefore show the combined effect of the interpolation operations needed to make the data equidistant and the effect of summing the daily precipitation surplus series into 30.4 day totals. In addition, for six of the piezometers, a validation is performed on the observations taken before 1-1-1990. A time plot of the available observations in both the calibration and validation period for piezometer 19azw246_1 is shown in Figure 6, along with predictions of a [10 0 1 1 0][30.4] BJ model and a PIRFICT model using the same data. Figure 6 clearly shows that the observation frequency in this period is not equal to that of the calibration period, but has been changed at the end of 1989 from 4 times a year into 24 times a year, on average. Because the frequency of the observations is much lower than the frequency of the predictions, the predicted values are linearly interpolated to match the dates of the observations, and not vice versa as in the calibration routine. For the PIRFICT model, the simulated values do not have to be interpolated, as the simulated water table depth is continuously defined.
 The average results of all 15 piezometers are summarized in Table 2. As in Table 1, the RMSE, RMSI, and EVP are given, along with the validation RMSE. Again, the differences are small as the average RMSE of the [10 0 1 1 nk] [30.4 30.4] BJ model, with a difference of only 0.8 millimeter, almost equals the RMSE of the nonequidistant [30.4 1] PIRFICT model. In 7 out of the 15 piezometers the fit is slightly better when the PIRFICT model is used, whereas 8 piezometers show a better result with the BJ model. At the same time, however, the RMSI of the BJ model is lower for all piezometers, whereas the opposite is true for the validation RMSE. The lower RMSI is probably to a certain extent caused by the interpolation operations that were carried out on the data to make it equidistant, which tend to smoothen high-frequent variations of a signal. This is corroborated by the fact that the PIRFICT model yields a RMSI that is 2.4 millimeter smaller for the equidistant data than for the nonequidistant data. However, the RMSI of the BJ model is on average 4.5 millimeter smaller than that of the [30.4 30.4] PIRFICT model while the VRMSE of the BJ model is higher. This, again, points to overfitting behavior of BJ models. The BJ model is able to produce a lower value of the objective function using the same data, but apparently the correlations fitted are to some extent coincidental as they are accompanied by a higher VRMSE. When the response functions of both models are compared, the smooth continuous BR functions prove to fit the general shape of the more irregular discrete transfer functions well for all piezometers, indicating that the PIRFICT model can adequately model phreatic systems with a relatively fast, intermediate and slow response (Figures 7a, 7b, and 5b).
Table 2. Average Calibration and Validation Results of the BJ and PIRFICT Model for 15 and 6 Piezometers Respectively
BJ [10 0 1 1 0] [30.4]
PIRFICT [30.4 30.4] Equidistant
PIRFICT [30.4 1] Nonequidistant
3.4. A Small Simulation Study
 Because we were not able to perform a cross-validation and cannot determine the “true” response of the real world phreatic systems, there is only indirect evidence that the irregular shape of the transfer function of BJ models is to some extent coincidental or due to overfitting behavior. In order to have more direct evidence, we performed a short study in which we simulated a groundwater level series by transforming a daily frequency precipitation surplus series in a synthetic system with a known response (a PIII IR function with A = 1500, a = 0.002, n = 1.5 ). To the series thus obtained, we added a residual series that was generated by convoluting a “discrete white noise” series of normally distributed random numbers with a known, exponential noise IR function (α = 25, σa = 2.5 cm on a daily frequency). From the series thus obtained the parameters that we should have retrieved using the TFN models are known exactly. The parameter estimates of a [10 0 1 1 0] [30.4] BJ model and a PIRFICT model are depicted in Figure 8 as BR functions, along with the true BR function of the synthetic system. The estimated parameters of the noise model are [α = 27.9, σν = 2.43 cm] and [α = 29.6, σν = 2.39 cm] for the PIRFICT and BJ model respectively. As was the case with the real world data, the BJ transfer function shows a partly irregular pattern and in this case even falls partly outside the confidence interval of the PIRFICT response. When comparing the estimates of both models with the true response function the deviation of the continuous BR function is less than that of the discrete transfer function, as is the case for the parameters of the noise model.
4. Discussion and Conclusions
 The method presented has shown to circumvent a number of the limitations of discrete Box-Jenkins Transfer Function-Noise models. First, the PIRFICT model can be calibrated on data at any frequency available because it operates in a continuous time domain and the time steps of the output variable are not coupled to the time steps of the input variables. Thus also the frequency of the input series can be irregular. Second, compared to the combined ARX model and Kalman filter, the PIRFICT model offers a further extension of the possibilities of calibrating TFN models on irregularly spaced time series, because the shape of the transfer function is not restricted to an exponential. Third, using the PIRFICT model, the model identification process is simplified because the model frequency does not interfere with the model order and parameter values, and the flexibility of a single continuous IR function can be such that it comprises a range of ARMA transfer functions. Furthermore, the model can be readily identified using physical insight. Although the method is presented in the form of a single input TFN model for time series of groundwater head, it is in fact quite general, and could be used for a variety of single or multi input, (non) hydrological problems (i.e. for the same purposes that Box-Jenkins TFN models are used, see introduction). The continuous time approach probably offers most advantages in cases where the above mentioned limitations occur, where the analyst is interested in the functioning of a system on different or small timescales, in the automated analysis of large quantities of time series, or in assessing time invariant response characteristics of systems. While the PIRFICT model probably performs best when the dynamical behavior of a system can be expressed in the form of a simple analytical formula, this should not pose a restriction to the use of the model, as the shape of the IR function can also be generalized by using sums of PIII df (or other) functions.
 It was shown in the example application that the PIRFICT model yielded comparable parameter estimates when the sample interval was varied, as long as the series of observations was sufficiently long and dense. As expected, the results of the BJ model proved to be influenced by user defined choices such as model order and time discretization. As for model performance, the PIRFICT model gave results comparable to those of the discrete BJ TFN models for all series analyzed. A first indication of this can be found in the fact that the estimated BR and transfer functions of both models show the same general behavior (but the BJ transfer function is partly irregular while the PIRFICT BR function is smooth). Secondly, the average calibration RMSE of the nonequidistant [30.4 1] PIRFICT model of the 15 piezometers was on average only slightly higher than that of the BJ model, whereas the validation RMSE was actually slightly lower. Although the differences are small, they could well be explained by the different structure of the transfer function of both models, which influences their overfitting behavior. As the shape of the transfer function of BJ models is partly free, the model is also free to fit coincidental cross-correlations between the input and output series, which will result in a lower calibration RMSE, a higher validation RMSE and a partly random pattern of the transfer function. Because of the use of a predefined IR function, the estimates of the PIRFICT model are forced to follow certain physical laws. This on the one hand makes the model less sensitive to overfitting behavior (next to its generally lower number of parameters) and can therefore yield better estimates, but can on the other hand also negatively influence the results when the physical assumptions prove incorrect or are to rough. These hypotheses were corroborated by the results of the simulation experiment, in which the response of the synthetic system was a PIII distribution and therefore equal to that of the predefined IR function in the PIRFICT model. For all series analyzed, the PIII df, or a system of serially coupled linear reservoirs, has shown to model the response of the water table to precipitation surplus adequately.
 The authors wish to thank H.L.M. Rolf of the Provincial water supply company of North-Holland for supplying the piezometer data and doing some exploring calculations. The anonymous reviewers are thanked for their valuable comments.