A data-driven approach for modelling Karst spring discharge using transfer function noise models

In this study, the TFN model, proposed by von Asmuth et al. (2002) to simulate hydraulic heads, is adapted for the purpose of modelling spring discharge. The TFN model relates a system output time series (here, spring discharge) to an arbitrary number of system inputs or stresses (here, evapotranspiration and precipitation). Spring discharge \(Q_S(t) \; [L^3 T^{-1}]\) is modelled as the sum of all M contributions \(Q_m(t) \; [L^3 T^{-1}]\) from the different stresses m (\(S_m(t) \; [LT^{-1}]\)), a base flow component \(Q_b \; [L^3 T^{-1}]\) and the residual series \(r(t) \; [L^3 T^{-1}]\) (Eq. (1)). It is noted that the dimensions of the terms or variables may change, depending on the context of the problem studied. Residuals r(t) are calculated by subtracting simulated values from observed values. Furthermore, stresses may be used multiple times to compute different contributions as shown in Fig. 2.The contribution \(Q_m(t)\) is calculated by the convolution of a stress time series with an impulse response function (von Asmuth et al. 2002), given in discrete time aswhere \(\theta _m(t)\) is the impulse response function describing how the system reacts to an infinitely short pulse of stress \(S_m\).

A step response \(\Theta (t)\) can be obtained by integrating the impulse response function over time. The step response represents the system response due to a constant stress (e.g., continuous precipitation at a unit rate) starting at \(t = 0\).The step response eventually converges to a steady state response as \(t \rightarrow \infty\), giving the maximum gain of the response. The gain may be interpreted here as the resulting spring discharge contribution if it precipitates at unit rate infinitely in time. The response function represents the unit response of certain reservoirs or combinations of those (Denić-Jukić and Jukić 2003).

The karst system recharge process has to be represented using multiple response functions due to the corresponding duality (direct and indirect/diffuse recharge) (Pinault et al. 2001c, a; Ladouche et al. 2014). To make hydraulic heads non-linearly related to precipitation and potential evaporation, Peterson and Western (2014) proposed and tested the use of a soil–water balance approach for the calculation of recharge. Collenteur et al. (2021) tested this approach for hydraulic heads in Austria and obtained good results. The bucket-type non-linear recharge model developed in Collenteur et al. (2021) is extended in this study to represent the two flow components of recharge, i.e., the duality of the recharge process in karst aquifers. One response function is subsequently applied for each flow component, as done in previous studies (Pinault et al. 2001c, a; Ladouche et al. 2014). A schematic representation of the extended recharge model is presented in Fig. 2a.

A detailed description of the recharge model can be found in Collenteur et al. (2021), an explanation is given for the recharge model adaption here. After the maximum capacity of the interception storage \(S_{i, \max } \; [L]\) is reached, additional precipitation overflows as effective precipitation \(P_e \; [LT^{-1}]\). Instead of routing \(P_e\) entirely to the root zone storage as in the original formulation, \(P_e\) is divided into a quick recharge component \(R_{\text {quick}} \; [LT^{-1}]\) (see Eq. (10)) and a diffuse effective precipitation component \(P_{e, \text {diff}} \; [LT^{-1}]\) (see Eq. (7)). The latter is routed to the root zone storage and the division or fractionation is controlled by the factor \(\omega _f \; [-]\). The interception storage reservoir of the recharge model is governed by the following equations:where \(S_i \; [L]\) is the current water level in the interception storage, \(P \; [LT^{-1}]\) is the precipitation flux, \(E_i \; [LT^{-1}]\) is the interception evaporation flux, \(E_{\max } = k_v E_p, \; [LT^{-1}]\) is the vegetation-corrected evaporation flux with \(k_v \; [-]\) being the vegetation factor and \(E_p \; [LT^{-1}]\) being the potential evaporation flux. The root zone storage reservoir of the recharge model is governed by the following equations:where \(S_r \; [L]\) is the water level in the root zone storage, \(S_{r, \max } \; [L]\) is the storage capacity of the root zone storage reservoir, \(l_p \; [-]\) gives the fraction of \(S_{r, max}\) at which the soil evaporation is limited, \(K_s \; [LT^{-1}]\) is the saturated hydraulic conductivity of the soil, \(\gamma\) is the exponent controlling the flow non-linearity, and \(R_{\text {diff}} \; [LT^{-1}]\) is the slow or diffuse recharge component.

Initially, at \(t = 0\), the saturation in the root zone storage is set to \(S_r (t=0) = 0.5 \; S_{r, \max }\) and the interception storage is empty (\(S_i (t=0) = 0\)). After the calculation of the recharge fluxes, each recharge flux gets convoluted according to Eq. (2) with the corresponding response function for the quick or diffuse flow component, respectively.

Predefined impulse response functions are used to transform the computed recharge components (fast and slow flow) from the recharge model to the final discharge contributions (see also Eq. (2)). As such, the response functions account for the effect of the porous medium. Predefined response functions have the advantage of a smaller number of parameters compared to models using discrete non-parametric response functions (von Asmuth et al. 2002; Neuman and De Marsily 1976; Dreiss 1989). Using response functions in continuous time allows to calculate the system response at arbitrary time steps (von Asmuth et al. 2002). These aspects are also part of the well-established general theory on system identification and the reader is referred to Ljung (1999) for more information.

While the medium and late recession behaviour of karst spring discharge can be represented by an exponential function with a corresponding recession coefficient (Maillet 1905; Kovács and Sauter 2008), the early recession behaviour may be better represented following a power-law function (Hergarten and Birk 2007; Birk and Hergarten 2010). Following this, we use a scaled exponential function (Eqs. (11), (13)) and a scaled Dagum probability density function (Kleiber 2008) (Eqs. (12), (14)), which follows a power-law, as impulse response functions. The response functions are shown with typical parameter values in Fig. 3.where \(A_1 \; [L^3T^{-1}]\) and \(A_2 \; [L^3]\) are the scaling parameters and \(a_1 \; [T]\), \(a_2 \; [T]\), \(b_2 \; [-]\), and \(c_2 \; [-]\) are shape parameters.

In the modelling approach used here, the step responses are used for the computations, which are given in Eqs. (13) and (14).

The physical interpretability of Eq. (13) is given by the relation of \(a_1\) to the recession coefficient of the system (Maillet 1905; Kovács and Sauter 2008). Eq. (14) can be physically interpreted as well, where \(A_2\) controls the overall scale of the response, \(a_2\) controls the overall length of the response, \(b_2\) controls the lag time of the response and \(c_2\) is a damping parameter. However, no analogy to a certain reservoir type can be given as in Denić-Jukić and Jukić (2003); Ladouche et al. (2014).

All previously mentioned models and methods were implemented in an adapted version of the time series analysis software Pastas (Collenteur et al. 2019), based on version v0.16.0 (Collenteur et al. 2020). The code and scripts are available as supplementary materials (see Supplementary information).

To represent the karst system conceptually, we employ an approach similar to Denić-Jukić and Jukić (2003), representing the response to diffuse and direct recharge individually. To account for conceptual or structural variability of the model, we employ two parallel recharge models (see Fig. 2) with two response functions for each recharge model, where one recharge model gets attributed with two exponential response functions (one for direct, one for diffuse discharge behaviour) and the other recharge model gets attributed with two Dagum response functions (one for direct, one for diffuse discharge behaviour).

Because all response functions feature a scale parameter (\(A_1, A_2\) in Eqs. (11), (12)), the overall contribution of processes related to exponential or power-law behaviour is variable. Furthermore, the contribution of diffuse or direct system behaviour is correspondingly variable as well, enabling the representation of complex recession behaviour that dynamically changes from a power-law to an exponential. Therefore, although the structure of the recharge model is fixed, the parallel setup together with variable representation of system behaviour allows for a high degree of flexibility. The model structure is shown in Fig. 2b.

Ultimately, the goal of inversion in this study is to obtain the most probable simulations during calibration and testing periods together with associated uncertainty bounds. This can be achieved via Bayesian inversion, which is, however, computationally intensive. Multiple calibration data sets are available for the studied case, and thus different data-model-combinations exist. It is of interest (especially for practical applications) to avoid the computationally costly Bayesian inversion for all data-model-combinations. This is achieved by performing less computationally costly least-squares calibration first for all data-model-combinations and selecting the most promising data-model-combination afterwards. Then, Bayesian inversion is only performed for this most promising data-model-combination (see “Bayesian inversion and uncertainty quantification”). This general workflow is shown in Fig. 4.

Three different data-model-combinations are tested during least-squares calibration, using identical model structures (see also Fig. 9): for data-model-combination 1, system input and output data is used from 1993-01-01 00:00:00 to 1994-12-31 23:00:00; for data-model-combination 2, system input and output data from 2014-01-01 00:00:00 to 2015–12-31 23:00:00 is used; data-model-combination 3 considers both previously mentioned data sets. In the remainder, these three different data sets are termed calibration data.

Because the water levels in the reservoirs of the recharge models are initialized at the beginning of the simulation to an arbitrary level, a warmup period is used. Three full cycles of calibration data are prepended to the calibration period as a warmup period. Subsequently, a warmup period of 3650 days is used for all models during calibration. For data-model-combination 1 and data-model-combination 2, mean values are used to fill the remaining beginning of the warmup period.

For convenience, all spring discharge data units are transformed as \((m^3 s^{-1}) \rightarrow (10^3 \; m^3 s^{-1})\) before calibration. The total number of calibration parameters is \(\textbf{m}=28\) for all data-model-combinations (\(2 \times 7\) recharge model parameters, \(2 \times 4\) parameters for the Dagum response function, \(2 \times 2\) parameters for the exponential response function, 1 base flow parameter, 1 noise model parameter; see Table 3).

Previous studies with the Pastas package and the examples from the package documentation (Collenteur et al. 2020) suggest that it may have aggravating effects to simultaneously optimize all model parameters at once, starting from initial values, or to optimize model parameters together with the noise model parameter. Therefore, the least-squares calibration is divided into three consecutive steps. First, a run without a noise model is performed, having the parameters \(\omega _f = 0.1, S_{r, \max } = 250.0 \; \text {mm}, l_p = 0.25, S_{i, \max } = 2.0 \; \text {mm}, k_v = 1.0\) fixed in both parallel recharge models. This is motivated by substantial correlation between the parameters and the corresponding aggravating effects of non-uniqueness and equifinality (Beven 2006; Gupta et al. 2009). In the second step, the calibration is continued reusing the optimized parameter set from the first step but varying all model parameters, still without a noise model. In the third and final step, the results from step 2 are used again together with a noise model, where the noise model parameter is optimized.

Afterwards, the calibration results from the third step of least-squares calibration are evaluated according to fit metrics for the calibration period for all data-model-combinations. Bayesian inversion (see “Bayesian inversion and uncertainty quantification”) is then performed with the most promising data-model-combination (see “Model fit metrics”) to obtain final simulations for the calibration and testing periods. Testing period performance for 2016 is finally assessed by the main author of KMC (P.-Y. Jeannin) and given in this study (Jeannin et al. 2021).

The sum of the squared residual series from Eq. (1), r(t) , is minimized during least-squares optimization, obtaining a set of calibrated parameters. A noise model can be employed on r(t) to better meet statistical assumptions for parameter inference and uncertainty quantification (von Asmuth et al. 2002; von Asmuth and Bierkens 2005; Nocedal and Wright 2006; Vrugt 2016; Vrugt et al. 2009; Evin et al. 2013, 2014; Sullivan 2015; Ghanem et al. 2017). The equation for the autoregressive lag-1 (AR1) noise modelled residual series can be written as (e.g., von Asmuth et al. 2002)where \(\nu (t)\) is the noise or future prediction error, i.e., the effect of random noise on the residuals between the time steps \(t - \Delta t\) and t. There, \(\Delta t\) is the time step length and \(\alpha\) is a decay constant of the effect, which has to be calibrated.

To optimize the model parameters, the residual or noise series (see “General methodology”) is then minimized (Eqs. (16), (17))where \(\textbf{x} \in \mathcal {X} \subseteq \textrm{R}^{\textbf{m}}\) is the vector of model parameters where the subscripts opt, res, and noise denote the optimal solution, residuals, and noise, respectively. \(\textbf{m}\) is the number of model parameters and \((b_l, b_u)\) are the vectors of lower and upper bounds of the parameters (see Table 3), respectively. A non-linear least-squares algorithm, as implemented in the Python library Scipy (version 1.9.0) (Virtanen et al. 2020), is used to minimize the respective series.

Four fit metrics are used to evaluate the model performance during the calibration period for each data-model-combination according to criteria suggested in the KMC (Jeannin et al. 2021). The volume conservation criterion (\(\overline{VCC}\), see Eq. (20) in the Appendix) is the ratio of the simulated flow volume and the observed one, where a value of 1 is the optimal outcome. The Nash-Sutcliffe efficiency (NSE, see Eq. (21) in the Appendix) incorporates the mean squared error as well as the variance and ranges between \(- \infty < NSE \le 1\), where a value higher than \(\approx 0.75\) indicates a reliable model (Nash and Sutcliffe 1970; Jeannin et al. 2021). The linearized version of the NSE (\(NSE_{lin}\), see Eq. (22) in the Appendix) is used as well, which is not as sensitive to outliers as the original NSE (Legates and McCabe Jr. 1999). The Kling-Gupta efficiency (KGE, see Eq. (23) in the Appendix) is related to NSE but normalized to the range \(0 \le KGE \le 1\) (Gupta et al. 2009). After the three-step least-squares calibration described in “General methodology”, the best performing data-model-combination is selected according to the combined \(\overline{VCC}, NSE, NSE_{\text {lin}} \& \; KGE\) criteria and considered for Bayesian inversion.

Contrary to the deterministic approach of finding a single optimal parameter set with least-squares calibration, Bayesian inversion is a statistical approach based on Bayes’ theorem (Eq. (18)). There, the prior parameter distribution \(p(\textbf{x})\) describes the belief about model parameters before taking any observed data, here \(\textbf{D} = \{ Q_{S, t} \}_{t=0}^N\), into account. \(p(\textbf{x})\) is then conditioned on observed data via the likelihood function \(p(\textbf{D} \mid \textbf{x})\), which represents the probability of the model parameters when comparing the corresponding model simulation with observed data. This conditioning results in the posterior parameter distribution \(p(\textbf{x} \mid \textbf{D})\). Therefore, parameter uncertainty can be directly quantified and model structural uncertainty is indirectly accounted for by using two parallel model parts (see “Model structure”). The present noise model furthermore indirectly introduces effects of data uncertainty (Vrugt 2016; Vrugt et al. 2009).Markov-chain Monte Carlo (MCMC) sampling is a highly popular method to sample from the (unnormalized) posterior density (e.g., Vrugt 2016; Vrugt et al. 2009; Teixeira Parente et al. 2019; Sullivan 2015), which is also used in this study. The general mechanism of MCMC is that a Markov-chain is generated in the parameter space (characterized by the prior distribution) by successively generating or proposing a new state (or sample) based on the previous state of the chain. A proposed state is then either accepted (resulting in a posterior sample) or rejected (resulting in a new proposal being made). Numerous MCMC algorithms are available; Pastas uses the emcee package (Foreman-Mackey et al. 2013) as given by the lmfit wrapper package (Newville et al. 2020). A parallelized affine-invariant ensemble sampler (AIES) algorithm is used (Goodman and Weare 2010), which runs multiple chains simultaneously. A uniform prior \(p(\textbf{x})\) is assumed and the parameter prior ranges are given in Table 3. It is also assumed that the residuals are independent and follow a normal distribution, leading to the log-likelihood function \(\textrm{ln} \mathcal {L}(\textbf{x}, \textbf{D}) = \textrm{ln}p(\textbf{D} \mid \textbf{x})\):where \(\sigma _t^2 \; [L^6T^{-2}]\) is the data variance. Using the noise series in place of the residuals ensures that the effect of noise is incorporated in Bayesian inversion. Therefore, the assumption of independent and identically (normally) distributed residuals can be met more accurately when using the noise series \(\nu (\textbf{x}, t)\) compared to using the residual series \(r(\textbf{x}, t)\) (Vrugt 2016; Vrugt et al. 2009; Evin et al. 2013, 2014).

All 28 parameters are varied using 100 walkers (chains), 12,000 steps, a burn-in period of 4000 steps together with a thinning of 4. MCMC sampling thus produces an array of samples with dimensions \([\mathrm {n_{samples}} \times \mathrm {n_{walkers}} \times \mathrm {n_{parameters}}] = [2000 \times 100 \times 28]\), giving a total of \(2 \cdot 10^5\) samples from the parameter space. A final model simulation is then performed for the testing period using the parameter maximum likelihood estimates (MLE) from the MCMC sampling.

The final simulation for the testing period was performed using the parameter MLE; fit metrics are shown in Tables 4 and 5. It is noted that in Table 4, the overall score is computed as \(score = 0.4 \cdot KGE + 0.4 \cdot \overline{VCC} + 0.2 \cdot NSE\). The effort in Table 4 refers to the time needed to develop and calibrate the model (Jeannin et al. 2021); model development took approx. 12 h, least-squares calibration took approx. 30 min for all models combined, and MCMC sampling took approx. 9 h, resulting in an effort of approx. 1 day. Having a total score of 0.90, the TFN model of this study outperformed all other models from the Karst Modelling Challenge. This could be attributed to the good volume conservation performance of the TFN model compared to other models. Volume conservation was especially well met in rising limbs and flood recession (see Table 5), while base flow and undetermined flow were still overestimated. Similar characteristics were observed for the KGE, while NSE values did not indicate a superior performance. Table 5 suggests that the TFN model performs especially well for dynamic periods, while base flow cannot be represented as accurately.

To incorporate the uncertainty from Bayesian inversion into the analysis of model simulation during the testing period, simulations were carried out for the testing period using 1000 randomly chosen posterior samples. Mean and median spring discharge were computed for every time step as well as a 5–95%-quantile ranges (Fig. 8). Mean and median simulated flow were generally higher compared to the MLE simulation, which was especially evident during recession periods. The width of the uncertainty interval increased with the duration of flood recession. Peak flows were generally estimated similarly for MLE simulation, mean, and median simulated flow. The peak flow event in May 2016 was estimated higher with mean values of flow and lower with median values. Subsequent peak flow events in June 2016 were estimated to be larger by the MLE simulation compared to mean values. Those deviations of the MLE simulation results all lie within the 5–95%-quantile range of simulated flows, and can thus be said to be reliable estimates.

It has been shown with least-squares calibration that the choice of the calibration data set affects the (initial) model performance assessment (see “Least-squares calibration”). It has furthermore been shown that the chosen three step optimization scheme for the least-squares calibration was not sufficient to find the (single) parameter set, which best reflects the observations during calibration. This is supported by the improvement of calibration fit metrics show in Table 2. Bayesian inversion with MCMC resulted in a single best parameter set used for final simulation during the testing period, which performed better than least-squares estimates. Furthermore, MCMC sampling provided rigorous estimates of parameter and simulation uncertainty without the limiting assumptions of a linear model or a pre-defined parameter (posterior) distribution type. Particularly in the light of equifinality, a strong argument is made that UQ should always be included in a thorough modelling study involving TFN models (e.g., Gupta et al. 2009). Even though the computational cost for MCMC sampling is magnitudes larger compared to least-squares optimization, the small computational cost of a single model run motivates Bayesian inversion for a single data-model-combination. Computational cost may be reduced by shortening the warmup period but still ensuring a physically viable state of the recharge model prior to the calibration period.

Different calibration strategies were followed by the two modelling approaches of the KMC which are compared in detail to the TFN modelling approach in this study (see Tables 4 and 5). While for the Gardenia model (lumped reservoir model), an iterative optimization—similar to least-squares optimization—was used, for the Varkarst model (semi-distributed model), an MCMC algorithm was adopted (see the supplementary materials in Jeannin et al. (2021)). Conceptually, those approaches have different backgrounds and in the present study, a combination of these approaches was used. This suggests that the choice of the specific calibration methodology (i.e., least-squares-type or Bayesian) not necessarily pre-determines performance of a modelling approach during a testing period. It is possible, however, that the performance of the VarKarst and Gardenia models could be improved by implementing the same parameter estimation strategy used in the present study.

The TFN model presented in this study has substantially more parameters compared to the Gardenia (\(\textbf{m} = 9\)) and Varkarst (\(\textbf{m} = 8\)) models. Other lumped modelling approaches presented in the karst modeling challenge use a similar or even larger number of parameters compared to the TFN model. Generally, there is no clear correlation between a decreasing number of parameters and increasing model performance. With the present TFN model, a trade-off is being made by introducing more parameters but enabling the model to represent more complex system and recession behaviour, which has been shown to produce good results.

The resulting MCMC posterior 5–95%-quantile range of flow simulation (Fig. 12) during the calibration period failed to cover several peak flow events and recession periods, which was especially evident from mid 2014 to mid 2015. It was observed in that period that the lower and upper quantile range limits followed a power-law during late recessions, while late recession behaviour in other periods with generally less total spring discharge were exponential. The quantile range limits therefore do not represent the general behavioural limits (e.g., exponential or power-law related recession) of the model in periods with generally more total spring discharge, but are biased towards power-law related behaviour. This was alleviated for periods with generally lower discharge, e.g., from mid 2015 to late 2015, where the upper limit followed an exponential law during late recession and the lower limit followed a power-law recession for longer. Therefore, the dynamic deviation from power-law to exponential recession is critical in representing spring discharge dynamics, which could in theory not be represented with one single instance of the non-linear recharge model. These observations motivate the (implicit) incorporation of structural variability in the TFN model structure with two recharge models, as this source of uncertainty could not have been incorporated in a structurally rigid model.

While the 5–95%-quantile range around the mean and median simulated flow are narrow in the calibration and testing periods, the uncertainty in flow simulation generally increased with the duration of recession periods, indicating uncertainty in the tail of the response functions. This was previously observed in similar studies (e.g., Denić-Jukić and Jukić 2003; Jukić and Denić-Jukić 2006) and can hardly be alleviated in practical applications as all response functions represent aggregated system behaviour and as linearity as well as time invariance is assumed.

The TFN model furthermore suggests a diffuse recharge regime for both recharge models after Bayesian inversion (see “Bayesian model calibration and uncertainty quantification”), which can be supported by field measurements (Perrin et al. 2003; Jeannin et al. 2021). During early recession, power-law related behaviour is dominant and subsequently followed by exponential behaviour during later recession. This representation of hydrological processes is in agreement with the findings of Hergarten and Birk (2007); Birk and Hergarten (2010). In the TFN model output, discharge contributions related to diffuse processes show fluctuations with lower frequency and smaller magnitude compared to their quick-flow counterparts, which agrees with general karst system functioning (see section (Bakalowicz 2005; Hartmann et al. 2014)). Consequently, the TFN model can be said to give a conceptually feasible and realistic system representation. This is furthermore supported by the fact that the base flow parameter \(Q_b\) was inferred to be virtually zero (see Table 3), indicating an autogenic recharge regime, which corresponds to field observations (Jeannin et al. 2021).

It has been shown in Fig. 7 that noise modelling greatly reduced residual autocorrelation, better meeting the assumption of independent residuals during least-squares calibration and Bayesian inversion. However, the noise model was not capable of making the residual series normally distributed, although the distribution showed a smaller variance compared to the raw residuals (see Fig. 7). Still, the reduced autocorrelation was assumed to improve model calibration and inversion robustness by better meeting the assumptions of residual independence. Independently of the modelling approach used, noise modelling can always be performed to improve model calibration or inversion robustness and rigor (von Asmuth et al. 2002; von Asmuth and Bierkens 2005) and our results encourage the utilization of noise modelling in future studies.