Introduction

A time series model using the Autoregressive Integrated Moving Average with exogenous variables (ARIMAX) function was developed to predict impacts from groundwater pumping on Silver Springs discharge in Ocala Florida. This effort was conducted to determine the effects of groundwater withdrawal using the statistical relationship between rainfall and spring discharge at Silver Springs. Other statistical models were developed in previous work by both Southwest Florida Water Management Districts and St Johns River Water Management District. However, there were several opportunities for improvement including using consistent data, model calibration period, and residual periods. Additionally previous statistical methods included Multiple Linear Regression and Line of Organic Correlation methods. These methods did not account for autocorrelation that present in many time series analysis. Through inter-district collaboration, data was made consistent and new methods were explored. The ARIMAX model was explored in this paper and is useful for prediction when autoregressive patterns are present in model residuals that bias modeled coefficients.

Method

Autoregressive integrated moving average (ARIMA) models have long been used for univariate time series forecasting. The ARIMA method decomposes time series into several components and models the patterns assuming continuation into the future. The ARIMAX model extends this function by developing a transfer function from covariates in addition to its own signal.

The input covariates into the model were mean monthly rainfall data lagged up to 36 months. The nearest most complete rainfall data was collected from the NOAA Ocala station. Prior to 1946 rainfall at the Ocala station was not complete and contained many large gaps in the record. Gaps in the record were imputed by both SWFWMD and SJRWMD however there remained substantial differences. Since large differences were present, only data post 1946 was used to limit the confounding influence of different filled rainfall data.

The calibration period for the statistical model was assumed to start when the rainfall data was best in 1946-01-01 and continues through the relatively unimpacted period to 1969-12-31. The year 1969-12-31 was identified because pumping in the Silver Springs area has been documented to start increasing in the 1970’s due to growth in the city of Ocala and adjacent agricultural growth.

The model was used to predict unimpacted water levels at Sharpes Ferry from 1969-12-31 to 1999-12-31. The prediction period was ended at the end of 1999-12-31 since the relationship between the Sharpes Ferry Well and Silver Springs discharge substantially changed as indicated in (Baird et al. 2014). Additionally the USGS changed its rating at the end of 1999 because a new relationship was developed and the previous rating was no longer within the acceptable range of error. Additionally the Sharpes Ferry well was removed from data collection in 2002 due to well casing and recorder problems.

The best ARIMAX model was selected by developing many models and minimizing the AIC. Assuming no other confounding effects, the results of groundwater withdrawal on Sharpes Ferry well were considered the average residuals between the observed and modeled levels. Since the statistical models contain error, the average residual was taken over the period 1990-01-01 to 1999-12-31 to effectively smooth the model error.

Results

Model selection was determined by minimizing the AIC and the resulting unbiased coefficients are reported in Table . Iterations of various number of lagged rainfall variables were performed and the minimized number of lagged variables with the highest AIC was 28.

\begin{table}[ht] \centering \caption{ARIMAX model order and AIC, AICc, BIC values} \label{aicTable} \begin{tabular}{rrrrrrrrrr} \hline AR(p) & MA(q) & sAR(sp) & sMA(sq) & Seasonal & Dif(d) & sDif(sd) & AIC & AICc & BIC \\ \hline 2 & 0 & 0 & 0 & 12 & 1 & 0 & 264.32 & 272.64 & 381.43 \\ 2 & 0 & 1 & 0 & 12 & 1 & 0 & 262.17 & 271.04 & 382.94 \\ 2 & 1 & 1 & 0 & 12 & 0 & 0 & 261.89 & 271.89 & 390.10 \\ 2 & 1 & 1 & 0 & 12 & 1 & 0 & 263.97 & 273.41 & 388.39 \\ 1 & 0 & 0 & 0 & 12 & 1 & 0 & 267.28 & 275.06 & 380.73 \\ 1 & 1 & 0 & 0 & 12 & 1 & 0 & 264.62 & 272.93 & 381.72 \\ 1 & 0 & 1 & 0 & 12 & 1 & 0 & 265.99 & 274.30 & 383.09 \\ 1 & 2 & 1 & 0 & 12 & 1 & 0 & 258.89 & 268.33 & 383.31 \\ 1 & 2 & 1 & 0 & 12 & 0 & 0 & 261.88 & 271.88 & 390.08 \\ 1 & 1 & 1 & 0 & 12 & 0 & 0 & 260.08 & 269.49 & 384.62 \\ 1 & 1 & 0 & 0 & 12 & 0 & 0 & 263.21 & 272.04 & 384.09 \\ 1 & 1 & 0 & 1 & 12 & 0 & 0 & 260.06 & 269.47 & 384.60 \\ 2 & 0 & 0 & 0 & 12 & 0 & 0 & 266.24 & 275.07 & 387.11 \\ \hline \end{tabular} \end{table}

The best model had 28 lagged rainfall variables and the smallest AIC of 266.24 was calculated for the ARIMAX model with order (2,0,0) and seasonal order (0,0,0). The lagged coefficients for the rainfall variables were calculated and illustrated in Figure . ARIMA coefficients were also presented in Figure . Overall the lagged rainfall coefficients have similar characteristics to a unit hydrograph for storm events. The first order AR(1) process was calculated as the most influential in the residuals.

Coefficients for ARIMAX results \label{Coefficients}

Coefficients for ARIMAX results

The resulting calibrated model was presented in Figure where the residuals have been adjusted for autocorrelation. Additionally only the structure of the ARIMX model is plotted in Figure . Relatively high correlation between the model and the observations is described by both the explanatory variables and the ARIMA residuals. Model calibration residuals were illustrated to explore if they meets time series modeling regression assumptions. Residuals were concluded to be homoscedastic and independent and can be visually examined in Figure . Autocorrelation and partial autocorrelation were considered negligible when near or below the 95% confidence intervals represented by dashed blue lines in Figure . Overall the ARIMAX model meets the assumptions of time series regression.

ARIMAX calibration results for Sharpes Ferry well with autocorrelation adjustment \label{Calibration Results}

ARIMAX calibration results for Sharpes Ferry well with autocorrelation adjustment

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment \label{Calibration results only structure}

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment \label{Calibration results only structure}

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment \label{Calibration results only structure}

ARIMAX calibration results for Sharpes Ferry well without autocorrelation adjustment

ARIMAX model calibration residuals \label{Calibration Residuals}

ARIMAX model calibration residuals

Model residual autocorrelation functions (ACF, PACF) \label{Autocorrelation}

Model residual autocorrelation functions (ACF, PACF)

The ARIMAX model was used to predict unimpacted water levels post 1969-12-31. These predicted unimpacted modeled levels were subtracted from the observed values to calculate residuals. The 80% and 95% confidence interval of the modeled levels are shown in gray and light gray respectively (Figure ). The confidence intervals provide the range of values based on the calibrated model error.

ARIMAX model prediction for Sharpes Ferry Well \label{Prediction Results}

ARIMAX model prediction for Sharpes Ferry Well

\begin{table}[ht] \centering \caption{Average residual over prediction period.} \label{avgResid} \begin{tabular}{lrr} \hline Period & Residual & Stnd\_Dev \\ \hline 1970-1999 & -0.51 & 1.21 \\ 1980-1999 & -0.59 & 1.15 \\ 1990-1999 & -0.80 & 0.99 \\ \hline \end{tabular} \end{table}
ARIMAX residuals Sharpes Ferry well \label{Prediction Residuals}

ARIMAX residuals Sharpes Ferry well

Rating curve for Sharpes Ferry well and Silver Springs discharge (1990-1999) \label{ratingCurve}

Rating curve for Sharpes Ferry well and Silver Springs discharge (1990-1999)

The mean residual over the period 1969-1999 was -0.51 ft and shown as a blue dash in Figure . The mean of the residual over the period of interest 1990-1999 was -0.8 ft and shown as a small red dash in Figure . Table summaries the different residual period averages. The standard deviation of the residual over the period of interest 1990-1999 was 0.986 ft. The average residual was then used to calculate flow using the rating curve between Sharpes Ferry well and discreate manual measurements at Silver Springs over the period 1990-1999. The USGS has a consistent rating curve suggesting that the relationship did not change during that period. The equation was developed from the regression in Figure follows: \[ y = 61.18 x -2153.9 \] The calculated flow reduction at Silver Springs was -48.8 cfs.

Conclusion

An statistical ARIMAX model was developed to determine the effects of groundater pumping on Silver Springs discharge. The model was calibrated over the period 1946-01-01 to 1969-12-31 and used to predict the period 1969-12-31 to 1999-12-31. The period after 2000 was culled from the analysis since the relationship between groundwater levels and Silver Springs discharge has changed and is undergoing investigation to identify the cause. The effects of groundwater pumping on water level at Sharpes Ferry over the period 1990-1999 is considered the average drawdown due to most recent pumping.

Using the ARIMAX method with a calibration period of 1946-1969 and the groundwater impacts taken as the average residual between 1990 and 1999, the drawdown due to groundwater withdrawal at Sharpes Ferry is -0.8 ft. Using the rating curve developed over the period 1990-1999 the average spring flow reduction at Silver Springs is -48.8 cfs.