Selection of Well Pairs

It is important that the index well selected for extending groundwater-level records is hydrologically similar to the study well and characterized by similar types of groundwater level fluctuations. As an indicator of hydrologic similarity, this study required that the index well and study well share the same physiographic province. The straight-line, Euclidean distance between individual study wells and index wells was calculated using geographic information system tools (tables 3 and 4). For this study, the maximum distance between paired wells for maintaining similar hydrologic characteristics was 30 miles (mi), which is about the average width of the drought regions in the Massachusetts drought management plan (Massachusetts Executive Office of Energy and Environmental Affairs and Emergency Management Agency, 2019). The paired wells more than 30 mi apart were flagged as suboptimal because the transfer of information from the index well to the study well may not be accurate over that distance (tables 3 and 4). In Massachusetts and its surrounding States, there are a very limited number of glacial till or bedrock wells. For this study, none of the well pairs in glacial till or bedrock aquifers were within a 30 mi span of one another and were therefore flagged as suboptimal.

The criteria for selecting an index well to pair with a study well were (1) a Pearson correlation coefficient of 0.8 or greater; (2) similar geologic characteristics; (3) similar topographic characteristics, if applicable; and (4) a concurrent record of 1 or more years. A correlation coefficient exceeding a threshold value of 0.8 or greater is indicative of a strong linear relation between the index well and the study well; this provides reliable estimates of groundwater levels. Correlation testing was performed using all 37 index wells for each study well. Only well pairs with similar geologic characteristics and a correlation coefficient of 0.8 or greater were considered for further analysis. When multiple index wells show the same or similar correlation coefficient to a single study well, distance between the wells determined the most suitable index well. Well pairs with a shorter distance between the wells were given preference over well pairs that were further apart. In addition, well pairs with a longer concurrent record were preferred over well pairs with a shorter concurrent record. A longer concurrent record improves the MOVE.1 method to estimate groundwater levels because the model parameters can be estimated more precisely (Vogel and Kroll, 1991).

Results from correlation testing on all possible pairs of wells indicated that 27 of 29 study wells have a Pearson correlation coefficient of 0.8 or greater to one or more index wells (table 4). The initial correlation testing resulted in a total of 75 possible pairs. Many of the possible pairs were dropped from further analysis because they did not meet the criterion for similarity in geologic and topographic characteristics. Two study wells—sites 420610071421402 (MA–NXW 54 Northbridge, Mass.) and 421410072081101 (MA–WUW 2R West Brookfield, Mass.)—had poor correlation coefficients (less than 0.8) and were dropped from further analysis. In several cases where the correlation coefficient is only slightly weaker at one or more index wells and the distance between the wells is similar, record extensions were performed on both wells and the performance metrics (described in the “Accuracy and Limitations of Groundwater Level Records From MOVE.1” section of this report) from the MOVE.1 method were evaluated to determine the best index well to use for the final record extension.

Use of MOVE.1 for Record Extension

The MOVE.1 method was used to estimate groundwater levels at 27 of 29 wells initially selected for extension. The MOVE.1 method is based on developing a linear relationship between the study well and the index well by assuming that the sample mean and sample variance of groundwater levels are maintained over time for the study well (Hirsch, 1982). Only the means and standard deviations for the concurrent record are used to define the MOVE.1 relation. The concurrent record is defined using continuous and discrete records. The MOVE.1 equation is then written as follows:

The programming language R, along with the R package smwrStats, version 0.7.5 (Lorenz, 2017a, b, 20221213), was used to perform the record extensions. R is an open-source programming language and software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing. The R script reads the station identifiers; extracts groundwater level data from the National Water Information System database for both the index well and the study well; loops through each index well associated with the study wells, performing the MOVE.1 method using the index and study wells; outputs estimated groundwater levels for the study wells; computes residual statistics; and generates model residuals.

Evaluation of MOVE.1 Models

Records were extended at 27 of the 29 wells initially selected for extensions. Plots of correlation, model fit, and statistical distribution of the residuals for the MOVE.1 models were evaluated for goodness of fit. The correlation plots of the concurrent records show the relation of index well to study well. The model-fit plots show the estimated groundwater levels and observed groundwater levels for the concurrent record. When extending records, the set of groundwater-level estimates and the statistical distribution of that set are important, rather than the individual estimates. Histogram plots were visually inspected and used to evaluate the data distribution. The distribution of residuals, representing the difference between the observed values and the estimated values, should be normally distributed with a mean of 0 for MOVE.1 modeling.

Figures 3 through 5 not only show how accurately the models represent the groundwater-level range and data distribution at the study well but also provide insight into how well the models estimate groundwater levels and statistics for drought management. The three figures are examples of models ranging in quality from excellent (fig. 3) to fair (fig. 5), with the distance between the pared wells spanning 2, 17, and 66 mi. Visual inspection of the model fit and the interquartile range of the residuals were considered when evaluating the quality of the model. The aquifer systems of the three well pairs are stratified drift, which is the primary aquifer type of the study wells. The mean and distribution of the residuals for all the models indicated no clear bias in the estimated records.

Figure 3 shows the correlation, model fit of the concurrent continuous data, and distribution of the residuals from the MOVE.1 method of the study well at site 414205070334701 (MA–BHW 506–0206 [MW–21M3] Bourne, Mass.) and the index well at site 414219070313601 (MA–SDW 525–0109 [MW–126S]). The paired wells are 2 mi apart with 2.5 years of concurrent data. The degree of correlation between study well and index well is considered very strong at 0.998 (fig. 3A). The model estimated daily groundwater levels from January 2002 to June 2018 (about 16 years of record). The model fit of the concurrent continuous data is considered excellent (fig. 3B). The residuals are very small, ranging from −0.37 to 0.22 foot (ft) with a mean of 0 ft and an interquartile range from −0.07 to 0.08 ft. The magnitude of the range of the groundwater data based on the record extension is 10.88 ft (a 6.08-ft change in range).

Figure 4 shows the correlation, model fit of the concurrent continuous data, and distribution of the residuals from the MOVE.1 method of the study well at site 413930070190901 (MA–A1W 306 Barnstable, Mass.) and the index well at site 414632070014901 (MA–BMW 22R Brewster, Mass.). The paired wells are 17 mi apart with 6.6 years of concurrent data. The degree of correlation between study well and index well is considered strong at 0.910 (fig. 4A). The model estimated daily groundwater levels from December 2001 to November 2015, along with several days in July 2016 and January 2019 (about 14 years of record). The estimated groundwater levels are considered a good model fit to the observed water levels (fig. 4B), with 50 percent of the estimated data within 0.5 ft of the observed data. The residuals range from −2.95 to 1.75 ft with an interquartile range of −0.46 to 0.53 ft. The magnitude of the range of the groundwater data based on the record extension is 8.35 ft (a 0.70-ft change in range).

Figure 5 shows the correlation, model fit of the concurrent continuous data, and distribution of the residuals from the MOVE.1 method of the study well at site 423810072435401 (MA–CSW 8R Colrain, Mass.) and the index well at site 431540071452801 (NH–WCW 1 Warner, N.H.). The paired wells are 66 mi apart with 4.1 years of concurrent data. Because the wells span more than 30 mi, the paired wells are considered nonideal for record extension. The model estimated daily groundwater levels from March 1999 to December 2017 (about 18 years of record). The degree of correlation between the paired wells is considered good at 0.866 (fig. 5A). Figure 5B shows the estimated water levels appear as a fair model fit to the observed water levels with 50 percent of the estimated data within 0.8 ft of the observed. However, the plot does show the estimated water levels typically 1 to 1.5 ft deeper than the observed water levels during dry periods. The residuals range from −2.51 to 2.24 ft with an interquartile range from −0.84 to 0.80 ft. The magnitude of the range of the groundwater data based on the record extension is 16.41 ft (a 6.53-ft change in range).

Accuracy and Limitations of Groundwater Level Records From MOVE.1

The MOVE.1 models were developed from statistical variables (sample means and sample standard deviations of groundwater levels) that best explain the variability of groundwater levels and are subject to the limitations of the concurrent data used to develop the models. If the concurrent data used to calculate the sample mean and standard deviations are not representative of the population (true) mean and sample deviations of the two sites used in the record extension, the estimated values may not be very accurate. How well the estimated values represent the observed true values, or the accuracy of the MOVE.1 model, is an important consideration in the application of the model and the interpretation of the results.

The accuracy of the estimates of groundwater level records from the MOVE.1 model is indicated by two statistics: the root mean square error (RMSE) and the Nash-Sutcliffe efficiency coefficient (NSE). The RMSE represents the mean of the absolute distance between the observed and estimated values. A lower RMSE indicates a better fit between the observed and estimated data. The units of RMSE are the same units as the groundwater levels. NSE is a normalized statistic that provides a measure of how well model output matches measured data. NSE ranges from negative infinity to 1.0, with a value of 1.0 indicating a perfect match between model and observed data and a value of 0.0 or less indicating that the mean of the observed values is a better predictor than the model-estimated values. Both RMSE and NSE were reviewed collectively and used to determine whether the estimates of groundwater-level records are acceptable for the purpose of computing percentiles for drought determination. The Pearson correlation coefficient and the residual-regression statistics also provided a quantitative measure of the predictive value of data for each well. The Pearson correlation coefficient is a statistic that was used to determine how closely the concurrent data from index well and study well are related (table 4).

Overall, the performance metrics indicate the majority of MOVE.1 models estimated groundwater levels with reasonable accuracy (table 4). The interquartile range of the residuals shows that the groundwater level estimates are within 0.5 ft of the observed levels at about 20 wells. RMSE values ranged from 0.097 to 2.292 ft with a median of 0.536 ft. NSE values ranged from 0.623 to 0.996, with a median value of 0.759.

In general, performance of the MOVE.1 model improves the closer the index well and study well are and the more geomorphological characteristics the wells share. RMSE values were generally smaller for study wells located closer to their index wells and larger for study wells located further from their index wells. For example, four wells—sites 414205070334701 (MA–BHW 506–0206 [MW–21M3] Bourne, Mass.), 414455070325701 (MA–BHW 507–0253 [MW–67M1] Bourne, Mass.), 414405070312104 (MA–SDW 483–0170 [MW–17M3] Sandwich, Mass.), and 415808070024301 (MA–TSW 257–0034 Truro, Mass.)—have RMSE values less than 0.3 ft and are less than 6 mi from the index wells. Conversely, two wells—sites 423810072435401 (MA–CSW 8R Colrain, Mass.) and 415229070554301 (MA–LKW 14R Lakeville, Mass.)—have RMSE values greater than 1 ft and are 66 and 38 mi from the index wells, respectively. The RMSE and NSE values for site 421923072451001 (MA–WXW 20 Westhampton, Mass.) indicate low model accuracy. The poor performance of the model may be attributed to hydrologic differences and the distance from its index well (72 mi apart).

Because of the number of suitable index wells is limited, some index wells were used at more than one study well. Two index wells were used three or four times to extend the groundwater record at the study wells. One implication of using the same index well for multiple record extensions is the increase the intergroundwater level correlation in subsequent statistical analyses. When the estimates from the record extensions are compared to one another based on the same index well, the comparisons and inferences are unreliable. In any case, using the same index well for multiple record extensions will result in estimates that are all similar and introduce regional bias in the estimates.

Comparison of Groundwater Level Records From MOVE.1 and Observed Records

By using this method of record extension, the records for study wells—particularly those with very short records—were able to be substantially increased (table 3). Groundwater-level records averaged 4.0 years prior to the record extension and 18.1 years after the record extension. These longer records resulted in a wider range of groundwater levels for 23 of 27 wells (fig. 6; table 5). Higher (shallower) groundwater levels occurred at 18 wells and lower (deeper) groundwater levels occurred at 13 wells. Four wells—sites 421438071165601 (MA–DVW 10R Dover, Mass.), 422058072085101 (MA–HHW 1R Hardwick, Mass.), 415453070434901 (MA–PWW 22 Plymouth, Mass.), and 423715072042801 (MA–TMW 3R Templeton, Mass.)—showed no change in the range of groundwater levels after the record extensions. Although the records of these four wells are relatively short, the extreme highest and lowest groundwater levels were measured in the observed record. Even when the record extension results in no change in the range of groundwater levels, the extended records help minimize bias when certain data are overweighted, such as extremes in short-data records. With the additional data of an extended record, extremes are contextualized as outliers. The range of groundwater levels increased moderately in 16 wells, by between 0.1 and 3.0 ft, and increased substantially in 7 wells, by between 3.1 and 8.0 ft. Overall, the record extensions resulted in a much broader range of groundwater levels often associated with long-term conditions (fig. 6; table 5).

Future Studies for Estimating Monthly Percentiles

Estimated groundwater levels of several wells can be substantially overestimated or underestimated for some months, as shown by the monthly boxplots of MOVE.1 residuals (fig. 8). For example, monthly boxplots of the MOVE.1 residuals for study well at site 415453070434901 (MA–PWW 22 Plymouth, Mass.) and the paired index well at site 415228070554601 (MA–LKW 14 Lakeville, Mass.) show that 8 of 12 months contain a bias; months January to April and July to October show the interquartile range of the residuals had either overestimated or underestimated values (fig. 8). For this study, a single MOVE.1 model for each study well was developed to estimate the historical daily groundwater levels for the period of record. For future consideration, seasonal or month-specific MOVE.1 models could be developed to compute monthly percentiles more accurately. A recent study by Dudley and others (2018) estimating groundwater level records based on their relation to hydrologic and meteorological variables found including such variables in statistical models yielded successful models.