In this section, an example of a practical field application of the proposed entropy approach is presented. Although this example is focused on designing sampling strategies for nitrate contamination in groundwater, the following approach is generic enough in nature for designing the optimal sampling strategy to track groundwater quality. The first step in any monitoring well network design requires collecting preliminary data on potential sites or hot spots of nitrate in aquifers. Various approaches have been suggested in the literature for identifying strategic locations for monitoring groundwater. For example, Al-Zabet [

51] suggested a monitoring strategy based on aquifer vulnerability to contamination potential [

51], whereas Scot [

52] developed a random site-selection approach [

52], and others have used an overlay model using multiple geographic information system (GIS) data layers [

53] for monitoring. These approaches are data intensive (e.g., land use land cover data, water well density, aquifer vulnerability) and usually general for all contaminants. However, different contaminants show varying levels of mobility and reactivity in groundwater [

47]; a general approach cannot describe hot spots of each contaminant. In contrast, the entropy approach presented in this paper is specific for each contaminant and does not require multiple GIS data layers. In this study, the entropy approach identified hot spots of nitrate contamination in groundwater across various scales (

Figure 3,

Table 1). These hot spots can be used as strategic sampling locations.

The second step in designing the optimal sampling strategy is to determine sampling frequency for various contaminants. Because data collection, processing and analysis can be expensive, it is desirable that the sampling frequency be cost effective. There are various criteria, such as land use change and groundwater fluctuations, that are used to determine sampling frequency for monitoring. However, it is essential to find redundancies in observations and to outline the sampling frequency without losing any significant information. To test the redundancy in the sampling frequency, we computed

$NME$ in Ogallala and Trinity Aquifers for the 2000-onwards decade at the fine scale by removing nitrate data points. We chose the 2000-onwards decade at the fine scale because both time series (nitrate time series in Trinity and Ogallala Aquifers for 2000-onwards) had more than 500 nitrate values. We removed part of the data (10%, 20%, 50%, 75%) from the nitrate time series and computed

$NME$. To make sure that there is no systematic bias in the analysis, part of the data was removed randomly from the time series using “randperm” function in MATLAB. Subsequently, for comparison, we also removed one alternate and two alternate data points from both time series and computed

$NME$. As shown in

Table 2, for the Ogallala Aquifer, when sample points were removed by 50% (removing one alternate sample),

$NME$ did not change. However, when sample points were removed by 66% (removing two alternate samples),

$NME$ changed significantly. In addition, when sample points were removed by 10 to 50%,

$NME$ values did not change much from the base case. However,

$NME$ values changed significantly, when samples were removed randomly by 75%. Therefore,

$NME$ values suggest that sampling frequency can be reduced by 50% in future sampling without losing much information in the Ogallala Aquifer. On the contrary,

$NME$ values showed little change when samples were removed randomly by 10 to 75% or one or two alternate samples in the Trinity Aquifer. These results are also substantiated by

Figure 3. It is evident from

Figure 3 that Trinity Aquifer shows higher persistence as compared to the Ogallala Aquifer at the fine scale. Therefore, removing sample points (>50%) from the time series of the Ogallala Aquifer changed

$NME$ significantly, but removing sample points (up to

$75\%$ ) from the time series of the Trinity Aquifer did not change

$NME$ values significantly. These results suggest that persistence and

$NME$ together can be an effective approach to design the optimal sampling strategy. We also want to emphasize that each monitoring program has different objectives; therefore, the reader can creatively apply this entropy-based method in designing their own sampling strategies.