# 6.1 Fitting Models

Fitting models have been used in the field of hydrology for a long time. For this discussion, fitting models involve the matching of field data to either a statistical distribution function or a single- or multiple-linear regression equation. Additionally, the broad category includes other forms of linear systems modeling in which the karst aquifer is treated as a filter, in that complex mathematical functions (linear kernels) are used with inputs like recharge and pumpage in order to generate an output of spring flow. The simplest application of linear systems in hydrology has been in rainfall-runoff models that employ what is called the unit hydrograph method (Sherman, 1932). These and similar techniques have been applied to karst aquifers (Neuman and de Marsily, 1976; Dreiss, 1982; Wicks and Hoke, 2000). To some extent, these are considered “black-box” methods because detailed knowledge about the physical system (for example, locations of conduits, transmissivity, and storage properties) is not required.

Fitting methods have advantages and disadvantages similar to those of lumped parameter models. In fact, statistical models are similar to the lumped parameter approach described in Section 6.2, except they lack a physically based differential equation to describe the aquifer. Instead, complex mathematical functions (sometimes based on statistical distributions) are used with time offsets and shape terms in order to take input(s) and create an output (response) that mimics the output (response) observed in the field. Another example of a fitting model is application of artificial neural networks, which is a form of pattern recognition, to karst aquifers (Hu et al., 2007; Trichakis et al., 2009). For neural network models, limited understanding of the karst aquifer system is needed for selection of appropriate inputs, such as recharge and pumpage, which are physically related to the outputs, such as spring flow(s) and/or water level(s).

Simple regression analysis has been used to predict spring discharge given water levels in a nearby well by developing a linear regression equation using historical data. This technique has also been used for tidally affected springs by developing a multiple-linear regression using both water levels at a nearby well and tidal stage data to estimate spring discharge (Wanakule, 1988; Knochenmus and Yobbi, 2001). Development of regression equations requires numerous spring discharge measurements over the full range of conditions that are anticipated to be simulated in predictive models, but does not involve time-series analysis. The discharge measurements must be paired with a water level in a nearby well (and if tides influence the system, then a tidal gage) from the same time as the discharge measurement. Figure 64, which is from Knochenmus and Yobbi (2001), shows the simplest form of a fitting model and their report presents the details of developing more complex multi-linear regressions for tidally affected springs. In this example, there is a known relationship between spring discharge and nearby groundwater level as well as tidal level at tidally effected springs, so the regression equations work well for accurately representing discharge at most of the springs.

Figure 64  Example of the simplest type of fitting model. The linear regression lines are developed for estimation of spring discharge from a groundwater level in a well in the Upper Floridan aquifer for each spring using the paired groundwater level for each discharge measurement at each spring location. These are non-tidally affected springs and the wells are close to the spring, such that the stage (groundwater level) to discharge relationship is linear through the range of discharge. This method is commonly applied for stable springs as it is less expensive than constructing a weir at the spring. Once the range of flow has been established and the stage-discharge measurements completed in order to create the regression equation, then fewer discharge measurements are required to establish that the regression equation remains accurate. Modified from Knochenmus and Yobbi (2001).

The advantage of fitting models is that they are easy to apply and calibrate. The disadvantage is that they are specific to the respective catchment and model results are highly uncertain if prediction simulations require input or output variables that exceed those of the historic calibration period.

# Example Application of Complex Fitting Model Using Stable Isotopes

Stable-isotope samples were collected at about 6-week intervals over a 6-year period at a streamflow-loss zone that recharges the karstic Madison aquifer in South Dakota (Figure 65) and at a nearby well located close to or within a main groundwater flow path (Long and Putnam, 2002, 2007). Time-series analysis of the isotope data indicates that the well responds rapidly to recharge from a sinking stream during wet periods. The hydraulic connection between the stream’s loss zone and the well is primarily through karst conduits. During dry periods when streamflow is low, isotopes in the well samples are primarily aquifer-matrix water that has been stored for months or years.

Figure 65  Location of the study area for collection of stable-isotope data in the karst Madison aquifer, South Dakota, USA. From Long and Putnam (2002).

The data were analyzed by correlation analysis and linear-systems analysis for a 34-month period of high recharge. The stable-isotope and water-level data sets exhibit the highest correlation when the data between the losing-reach of the stream and the well are lagged by 22 days, which may approximate the travel time from the loss zone to the well. Linear-systems analysis resulted in an estimated travel time to the well of about 15 days with a system memory of 2 to 3 years resulting from diffuse matrix flow. Based on these analyses, the conduit-flow velocity was estimated at 380 to 800 ft/day (120 to 240 m/day). The data used to develop the model are shown in Figure 66a. A log-normal distribution approximated the observed distribution of travel times for conduit flow. Figure 66b shows the final model and the log-normal distribution used as the transfer function for estimating the amount of stable oxygen-18 isotope in the spring discharge based on its presence in the stream water. The narrow peak of the transfer function reflects the roughly 2-week lag time.

Figure 66  Statistical regression model of stable oxygen-18 isotope. a) δ18O data and Spring Creek streamflow recharge to the Madison aquifer from 1996 to 2001. The A sections indicate periods of near-maximum recharge, whereas B sections indicate periods of lower recharge. The streamflow recharge has a maximum estimated rate of 21 ft3/s (~0.6 m3/s) (Hortness and Driscoll, 1998). b) Results of linear-systems analysis including the computed δ18O data for the sampled well and the transfer function used in the analysis. Unit conversions: 1 cubic foot per second ~ 0.03 m3/s. From Long and Putnam (2002, 2007).