Fiber-optic distributed temperature sensing (FO-DTS) increasingly is used to map zones of focused groundwater/surface-water exchange (GWSWE). Previous studies of GWSWE using FO-DTS involved identification of zones of focused GWSWE based on arbitrary cutoffs of FO-DTS time-series statistics (e.g., variance, cross-correlation between temperature and stage, or spectral power). New approaches are needed to extract more quantitative information from large, complex FO-DTS data sets while concurrently providing an assessment of uncertainty associated with mapping zones of focused GSWSE. Toward this end, we present a strategy combining discriminant analysis (DA) and spectral analysis (SA). We demonstrate the approach using field experimental data from a reach of the Columbia River adjacent to the Hanford 300 Area site. Results of the combined SA/DA approach are shown to be superior to previous results from qualitative interpretation of FO-DTS spectra alone.
 Groundwater/surface-water exchange (GWSWE) plays an important role in hydrologic and ecological processes such as water-level fluctuations, stream/river discharge, transfer of contaminants, temperature regulation, and nutrient cycling [Brunke and Gonser, 1997]. Heat exchange across the groundwater/surface-water interface has been studied in detail for decades [Anderson, 2005], with most studies focused on using temperature to quantify GWSWE [e.g., Hatch et al., 2006]. Spatial patterns of exchange also can be inferred qualitatively from temperature variations at the groundwater/surface-water interface [e.g., Selker et al., 2006].
 The development of fiber-optic distributed temperature sensing (FO-DTS) technology [Selker et al., 2006] has helped to increase the spatial and temporal coverage of temperature measurements for hydrogeologic investigations. By providing measurements with high spatial (∼0.5–2 m) and temporal (∼10–60 s) resolution along cables several kilometers in length, FO-DTS has led to insights into local-scale hydrological processes that were previously impossible to resolve with traditional temperature-monitoring instruments. There is burgeoning interest in FO-DTS for studies of GWSWE, as evident in recent work [e.g., Selker et al., 2006; Henderson et al., 2009; Slater et al., 2010; Mwakanyamale et al., 2012; Briggs et al., 2012]. The utility of FO-DTS for studies of GWSWE derives from the temperature contrast between groundwater and surface water. By measuring streambed temperature in an environment with substantial differences in temperature between groundwater and surface-water, the intensity of temperature signals can be used as an indicator of GWSWE and flow directions [Anderson, 2005]. FO-DTS measurement quality depends on the DTS instrument, field-deployment configuration, and instrument calibration [Hausner et al., 2011]. The reader is referred to Selker et al.  and Hausner et al.  for details of the FO-DTS method, limitations, assumptions, and calibration.
 Previous efforts have demonstrated the information content of FO-DTS data for GWSWE, but relied on qualitative and subjective interpretation to map zones of focused GWSWE, defined here as those reaches of a stream where GWSWE is enhanced relative to other reaches as a result of high streambed or bank hydraulic conductivity and (or) large-scale hydraulic gradients. We recognize that hyporheic water has a buffered temperature signature, controlled by the GWSWE process [Gerecht et al., 2011]. In this study, we examine the stage-controlled exchange of groundwater and surface water, as explained in subsequent sections. A second, important limitation of some of the past efforts is that they lacked evaluations of uncertainty associated with the mapping of zones of GWSWE, as investigated here.
 We present a strategy for mapping zones of enhanced GWSWE and quantification of classification uncertainty. We combine and apply spectral analysis (SA) and discriminant analysis (DA) to FO-DTS data. DA is a widely applied statistical technique in hydrological and geophysical studies [e.g., Herrmann and Symader, 1976; Steinhorst and Williams, 1985; Daughney et. al., 2010]. SA of time series is also well established in hydrology, with applications ranging from catchment processes to fluvial processes [e.g., Gudmundsson and Sigbjarnarson, 1972].
 Using a combination of SA and DA, we demonstrate development of a statistical rule for identifying zones of enhanced GWSWE along a reach of a major river. Our objectives are to use a high-resolution (in space and time) FO-DTS data set to (1) objectively map GWSWE zones and (2) quantify uncertainty of these classifications. We also assess how the selection of SA frequency components, combination of different frequency components, and down-sampling of the training data set influences the effectiveness of the classification of zones of focused GWSWE. We note that our approach capitalizes on the signal associated with groundwater discharge, i.e., exchange in one direction; however, we use this signal to map zones where, with reversals in hydraulic gradient, focused GWSWE occurs in both directions over time.
2. Study Site and Experiment
 We acquired a FO-DTS data set on the western bank of the Columbia River, along a reach adjacent to the Hanford 300 Area in Richland, WA. Along this reach, groundwater discharge potentially carries radionuclide contamination from the Hanford site [Fritz and Arntzen, 2007]. Hydraulic conductivity ranges from ∼2000 m d−1 in the Hanford Formation (flood deposits) to 40–120 m d−1 in the Ringold Formation (fluvial deposits) [Williams et al., 2007]; hence, groundwater discharge focuses in zones where the Hanford Formation is present at the sediment/water interface [Peterson and Connelly, 2004; Fritz and Arntzen, 2007]. Rapid stage fluctuations and concomitant flow reversals occur in response to irregular operations of Priest Rapids Dam (upstream of the 300 Area) [Arntzen et al., 2006]. River water thus flows into the aquifer as the river stage increases, and groundwater discharges into the river as the river stage decreases, resulting in zones of focused GWSWE.
 Taking the Hanford Formation thickness as a strong indicator of focused GWSWE, Slater et al.  used waterborne electrical surveys to determine spatial variability in the depth to the Hanford-Ringold contact along the river corridor adjacent to the 300 Area; they also report on temperature anomalies from FO-DTS along this reach. Mwakanyamale et al.  demonstrated how SA, in this case based on an S-transform, better discriminated zones of focused GWSWE, relative to temperature anomalies alone. GWSWE was found to have a distinct spectral signature in FO-DTS data and to coincide with locations along the river where the Hanford-Ringold contact is predicted to be locally deeper (i.e., thicker Hanford Formation) in electrical images [Slater et al., 2010; Mwakanyamale et al., 2012]. Dam releases govern the power spectrum of the stage time series, and this spectral signature is imprinted on the streambed temperature spectra where enhanced GWSWE occurs. Strong amplitudes (≥ 0.4) at longer periods (2–16 d) found in the stage power spectrum are also apparent in FO-DTS power spectra in locations of enhanced GWSWE [Mwakanyamale et al., 2012]. The 2- to 4-d periods are interpreted as the original high-amplitude periods driving river stage fluctuations, with periods at multiples of these assumed to be harmonics. In this study, we further capitalize on the distinct spectral signature of GWSWE.
 A 1.6-km ruggedized SensorNet EnviroFlex fiber-optic cable with two 50-μm multimode fibers was installed on the streambed at ∼2 m from the river bank and about 0.15–0.76 m below the water surface. The cable was secured to the streambed (underwater) using cobbles, stakes, and/or concrete blocks. The riverbed temperature was recorded by a Sensortran 8-channel Gemini control unit at 5-min intervals for every 0.51 m along the cable length (for a total of 2871 measurement locations) using single-ended measurements. The FO-DTS system acquired data from November 2008 to January 2012 with occasional downtime used for system maintenance and recalibration. The river-stage data were collected at 1-h intervals by a probe stationed upstream of 300 Area. We analyze three uninterrupted parts of the data set acquired during winter (38 days), summer (48 days), and fall (22 days). The data sets have variable durations because of interruptions for maintenance and recalibration of the FO-DTS system.
3. Combined Spectral and DA
 We used time-frequency analysis (the S-transform) in previous work [Mwakanyamale et al., 2012] examining the FO-DTS time series. Alternatives to the S-transform include wavelets [e.g., Henderson et al., 2009] and empirical orthogonal functions (EOF) [e.g., Ghil et al., 2002]. In this study, we independently analyze short intervals of data, for which stationarity can be assumed, thus allowing a simple demonstration combining an SA based on the discrete Fourier transform (DFT) and DA. We stress, however, that DA also could be combined with EOF, S-transform, or wavelet-based time-frequency SA, but such a demonstration would be more complicated than required here. We implement the DFT using a fast Fourier transform (FFT) algorithm [Cooley and Tukey, 1965]. The FFT decomposes the temperature time-series signals into components of different frequencies/periods, yielding amplitude and phase information for each component.
 DA [Fisher, 1936] is a powerful tool for multivariate classification problems. The method generates a set of classification functions based on a combination of input variables (training data) to predict the category (class) to which an observation belongs [Fisher, 1936]. Each observation is classified into the category for which the probability of belonging is greatest. The effectiveness of the computed functions is quantified as the posterior probability . Following Hung et al. , this is the probability that a predicted sample (x) belongs to a prior defined class (j) where wj denotes the fact that x is a member of j. There are a number of DA approaches in the literature, including linear, quadratic, logistic, and nonparametric. In this study, we use quadratic discriminant analysis (QDA). Unlike linear discriminant analysis (LDA), QDA does not require identical covariance matrices for each class. QDA assumes (1) multivariate normality, although QDA can be effective even for non-normal data sets [Arevalillo and Navarro, 2011], and (2) no correlation between variables. We note that linear, logistic, or nonparametric approaches could also be used. We selected QDA because it was the simplest approach that proved an effective classifier for our data set. LDA also was investigated, but covariance is not expected to be homogenous across classes, and results for LDA were inferior to those from QDA.
 Discriminant functions are functions of the posterior probabilities. The discriminant function for the QDA, used here, is:
where, and are covariance matrices for groups i and j, respectively; μi and μj are the mean vectors for groups i and j, respectively; and P(wi) and P(wj) denote prior probabilities for groups i and j, respectively, calculated from the training data. Using a 0–1 cost, DA will assign sample x to group i if gij(x) > 0 and to group j if gij(x) < 0. The relation between prior and post observations is described in the confusion matrix. In this matrix, the number of correctly classified observations lies on the diagonal of the matrix, whereas the number of erroneously classified observations lies on off-diagonals [Cooley and Lohnes, 1971]. We summarize overall classification success as a hardness (H) value [Deutsch and Journel, 1998], which for a binary classification problem is the probability of assignment to class 1 given an observation belongs to class 1 (correct positive classification) minus the probability of assignment to class 1 given an observation belongs to class 0 (false positive classification). H = 1 would thus correspond to the case of perfect classification of all observation locations.
 We used as DA input the SA amplitudes for different periods. For DA training data, we used the thickness of the Hanford Formation as previously determined by boat-towed electrical surveys [Slater et al., 2010]. DA was performed for different combinations of input data (different seasons and periods) and training data (100, 10, and 5% of training data). A 5-m cutoff was used to classify all the spatial points along the river corridor as zones of focused GWSWE (≥5 m) or no GWSWE (<5 m). This threshold was selected based on consistent hardness values for DA results of different sampling periods (seasons). More direct information (e.g., seepage measurements) would be preferable as training data, but such information is available only at ∼10 locations sparsely distributed along the reach [Fritz et al., 2007] because the streambed is armored and exceedingly difficult to work in—hence, the motivation for alternative, noninvasive approaches such as FO-DTS and, in turn, our proposed SA/DA framework. Furthermore, access to spatially exhaustive training information from the electrical surveys allows us to evaluate how much training information is required to develop robust discriminant functions. Although we note that the geophysics-derived thickness of the Hanford Formation is not ideal training information, the results shown subsequently provide a strong indication that it is an appropriate choice for the Hanford 300 Area reach.
4. Results and Discussion
 The combination of DA and SA proved an effective strategy to classify exchange vs. non-exchange sites using FO-DTS time-series data and training data in the form of Hanford Formation thickness. Comparing results from different analyses yields insights into the effectiveness of the SA/DA approach for combining information across frequencies and seasons. Amplitudes of selected periods (0.5, 1, 3, 4, 6, and 8 d) obtained from SA were used as inputs in the DA. Periods were chosen based on high amplitude (≥0.4) in the sampling seasons [Mwakanyamale et. al., 2012]. Figure 1 shows riverbed water temperature and river stage of three subsets of the FO-DTS data collected in January to March 2011, June to August 2011, and November 2010. The temperature time series exhibit evidence of temperature anomalies in winter (Figure 1a) and fall (Figure 1c) that occur with decreases in river stage [Mwakanyamale et al., 2012]. The river stage is characteristically high in the summer (Figure 1b) [Lindberg and Chou, 2001], which results in suppression of temperature anomalies. The SA results exhibit distinct spatial and temporal patterns in different sampling seasons (Figure 2). Higher amplitudes (≥0.4) at longer periods (3, 4, 6 and 8 d) are seen for some locations along the cable (dashed gray box). Long periods (3, 4, and 6 d) exhibit strong amplitude in the winter data (Figures 2c, 2d, and 2e, blue lines), and these strong amplitudes coincide with the locations of thicker Hanford Formation (≥5 m) (Figure 2g), supporting our use of Hanford Formation thickness as training data. Short periods (0.5 and 1 d) (Figures 2a and 2b, blue lines) show weaker amplitudes in the zones of focused GWSWE while exhibiting similar amplitudes to the longer periods at the other locations along the cable, further supporting our use of Hanford Formation thickness.
 The distribution of amplitudes in the summer SA analysis results (Figure 2, red lines) differs from those observed in the winter data (Figure 2, blue lines). Only one long period shows strong amplitudes in all the locations projected to be groundwater discharge zones based on the Hanford Formation thickness (Figure 2e, black line). All long periods (3–8 d) (Figures 2c–2f, red lines) show high amplitudes at cable length 220–380 m. This location is identified as a zone of focused GWSWE. Short periods (0.5 and 1 d) in the summer data (Figures 2a and 2b, red lines) show stronger amplitudes than observed in winter results. Strong amplitudes in the short periods are interpreted as being caused by diurnal variations. The SA results from the fall season (not shown here for brevity) exhibit amplitudes that are in contrast to all other sampling seasons (winter and summer) with strong amplitudes along most of the cable for all periods (0.5–8 d). We note that in the fall, the surface and groundwater temperatures converge, rendering fall (or spring, presumably) data less informative than summer or winter, when temperature contrasts are optimal for detection of groundwater discharge.
 The DA classification results and performance (expressed as H) are presented in Figures 2h and 2i. DA input was tested for correlation between variables (frequency components). From this testing, 10 of 19 variables showed correlation coefficients less than 0.7, 16 of 19 less than 0.8, and all less than 0.95. The identified zones of focused GWSWE occur where higher amplitude is seen at longer periods and generally align with the thicker Hanford Formation. The use of spatially exhaustive training data allows us to evaluate the data requirements for our SA/DA approach. Application of the DA to the full data set (100%), including all periods from all sampling seasons, provided the best classification (Figure 2i), with H of ∼75% (0.75). Down-sampling to consider only 5% of available training data (distributed randomly along the cable) produces a discriminant function that results in a classification with H = 0.67 (Figure 2h).
 The DA results are displayed as maps in Figures 1d and 1e. The locations of mapped zones of focused GWSWE correspond well with Hanford Formation thickness and include previously identified seep locations [Peterson and Connelly, 2004]. Classification uncertainty is shown in Figures 1d and 1e as the probability of correct classification (P). Considering all data for all periods from all sampling seasons (Figure 1e), the SA/DA approach classifies 98% of the sites with >90% confidence. Using only 5% of training data for all periods from all sampling seasons, 90% of the sites were classified with certainty of >90% (Figure 1d). These results indicate that, at least for our study area and data set, the SA/DA approach produces a classification that is robust when developed based on sparse data.
 Table 1 presents all DA classification results for seasonal and combined data sets. Considering only single frequency (4-d period) and fall data, the SA/DA produces classifications with low confidence, such that only ∼20% of the sites are classified with 90–100% certainty. The use of all periods in winter and summer and combining two different seasons produced better results than the use of single frequency (and all periods in fall). Less than 50 sites out of ∼1700 were erroneously classified as zones of focused GWSWE when using all periods in winter and summer and combining two different seasons. The winter classification (all periods) is similar to the summer classification (all periods) performance with H of 0.66 and 0.63, respectively. Combining winter and summer data (all periods) performed better than combining either of the seasons with the fall data. Down-sampling the training data to 5% reduced H by ∼10% compared to using 10% of the training data. Retaining 10% of training data and then discriminating everywhere results in a classification almost identical to that developed with the full data set, with only negligible change to H.
Table 1. Classification Performance of Discriminant Analysis for Groundwater Discharge and Nondischarge Zonesa
All data represent classification performance results when using all amplitudes from all sampling seasons, and 10 and 5% represent classification performance results when using down-sampled training data and amplitudes from all sampling seasons. Winter, summer, and fall represent classification performance results when using all periods (0.5–8 days). Winter, summer, and fall 4 day periods, represent classification performance results when using 4 day periods only.
Fall 4 day period
Summer 4 day period
Winter 4 day period
Summer and Fall
Winter and Fall
Winter and Summer
5% of all data
10% of all data
 Classification based on amplitude at a single period is less effective than classification based on multiple periods. The SA/DA approach thus uses combinations of frequency components to give a superior classification than possible using only a single frequency component. The SA/DA approach effectively combines data across frequency and time and is not adversely affected by inclusion of low-quality data from seasons with unfavorable conditions (e.g., fall). Although QDA was used here, other DA approaches (e.g., logistic or nonparametric) may prove necessary in the presence of stronger correlation between variables, non-normality, or noisier data.
 Extracting information from large FO-DTS data sets is challenging, and new approaches are needed to objectively map zones of enhanced GWSWE while providing assessments of classification uncertainty. Although past work [Henderson et al., 2009; Mwakanyamale et al., 2012] relied on subjective selection of a single spectral component as an indicator of groundwater discharge, in this study, we have combined DA and SA to capitalize more fully and objectively on the information content of FO-DTS data; this combination proved an effective strategy to discriminate zones of enhanced GWSWE using FO-DTS time-series data and training data in the form of bed thickness. Although the SA/DA approach is effective in reporting the uncertainty associated with classification using FO-DTS, it lacks information on uncertainty associated with training data. Although we used the Hanford Formation thickness as the training data in this study, more concrete training data, e.g., seepage flux measurements, could be used if available. The results obtained by DA show a clear distinction between zones of focused GWSWE and other locations along the reach. Our results show that classification combining information from multiple frequencies over multiple seasons provided results superior to those derived from a single frequency component of the FO-DTS power spectrum from a single season (Table 1). Although use of fall data alone provided poor results, as expected given the low groundwater/surface-water temperature contrast in fall, the inclusion of fall data did not adversely affect the results from SA/DA analysis of multiseason data. This reinforces the point that a clear contrast in temperature between groundwater and surface water is critical for FO-DTS to effectively map GWSWE. Indeed, the SA/DA approach proved to be most effective when considering amplitudes from all periods at all sampling seasons. In addition to providing superior classifications compared to past work, our approach also provides a quantitative measure of classification uncertainty. The SA/DA approach was shown to be robust when developed from down-sampled training data, at least for the Hanford 300 Area. This finding underscores the effectiveness of the SA/DA approach with sparse training data, and thus points to the potential consideration of direct seepage measurements with sparse spatial distribution for training. We conclude that the SA/DA approach will be useful in settings where geology plays a major control on GWSWE. Although our classification is binary, DA could be used with multiple categories. Regression could be used instead of classification if a more gradational or continuous population of exchange is expected.
 This research was supported by the U.S. Department of Energy under grant number DOE - DE - FG02 - 08ER64561 with additional support from the U.S. Geological Survey (USGS) Toxic Substances Hydrology Program and Groundwater Resources Program. We thank Andy Ward, Jason Greenwood, and Christopher Strickland (Pacific Northwest National Laboratory), and Carole Johnson (USGS) for field assistance. We are also grateful to Dimitrios Ntarlagiannis, Roelof Versteeg, and Timothy Johnson. We also thank the three anonymous reviewers for their suggestions and Rory Henderson (AECOM) for a colleague review of the draft manuscript. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.