Freshwater lakes can emit significant quantities of methane to the atmosphere by bubbling. The high spatial and temporal heterogeneity of ebullition, combined with a lack of high-resolution field measurements, has made it difficult to accurately estimate methane fluxes or determine the underlying mechanisms for bubble release. We use a high-temporal resolution data set of ebullitive fluxes from the eutrophic Upper Mystic Lake, Massachusetts to understand the triggers that lead to bubbling from submerged sediments. A wavelet approach is introduced to detect ebullition events for multiple time-scales, and is complemented with traditional statistical methods for data analyses. We show that bubble release from lake sediments occurred synchronously at several sites, and was closely associated with small, aperiodic drops in total hydrostatic pressure. Such results are essential to constrain mechanistic models and to design future measurement schemes, particularly with respect to the temporal scales that are needed to accurately observe and quantify ebullition in aquatic ecosystems.
 Accurate quantification of the total amount of ebullition occurring from lakes is not yet possible, primarily because the spatial distribution, magnitude, and timing of bubbling events is not adequately captured by existing measurements. Observations are often manually made at a limited number of locations in a lake, and commonly represent averages over durations of several days. By contrast, bubbling is spatially patchy as well as episodic. Large variations in bubble flux can occur over periods of minutes to hours, and the lack of data at high spatial and temporal resolution has made it difficult to objectively identify and characterize bubbling episodes. The mismatch between measured and actual temporal and spatial scales weakens what might otherwise be observable statistical relationships among ebullitive fluxes and their forcing mechanisms, relationships whose understanding could lead to better constraining of various mechanistic models of the overall ebullition process.
 The few reported high-temporal-resolution measurements of ebullition do support the contention that such data can contribute to understanding the process of ebullition. The data collected for this study have been used to constrain a physical model of methane release from soft sediments [Scandella et al., 2011]. Acoustic measurements in marine and freshwater settings have been used to estimate bubble sizes and rise velocities, and to observe spatial and temporal variability over periods of measurement of a few hours to days [e.g., Greinert and Nützel, 2004; Ostrovsky et al., 2008; Vagle et al., 2010]. Boles et al.  and Leifer et al.  used automated flowmeters to measure the gas captured in large tents near the seafloor of a marine hydrocarbon seep, revealing that ebullition was associated with tidal forcing. Automated chambers measuring total (diffusive plus ebullitive) surface fluxes [e.g., Goodrich et al., 2011; Mastepanov et al., 2008] and GPS-based measurements of surface deflections [Glaser et al., 2004] have been used to observe the effect of environmental variables such as atmospheric pressure, temperature, wind and water level on methane emissions in wetlands, while geophysical methods such as ground penetrating radar [Comas et al., 2007] and resistivity [Slater et al., 2007] have been used to determine the relationship between the free gas phase and methane fluxes in peatland soils.
 The present study experimentally addresses gaps in understanding the rates and timing of methane ebullition by measuring high-resolution bubbling fluxes in the eutrophic Upper Mystic Lake in Massachusetts. The study was designed to test the following hypotheses:
 1. Ebullition events can be defined in a statistically meaningful way by the identification of relatively short periods of time during which a disproportionate fraction of total bubbling in a lake occurs.
 2a. Ebullition, even from sediments subject to pressures corresponding to tens of meters of water depth, can be routinely triggered by relatively small, aperiodic fluctuations in hydrostatic pressure such as those that arise from changes in atmospheric pressure or lake water level. The response of sediments to these pressure fluctuations is sufficiently universal that bubbling events tend to be synchronous within the lake, or
 2b. Bubbling events tend to occur at identifiable frequencies, suggesting control more by local, in-sediment processes than by external forcing in the studied environment.
 3. The minimum sampling frequency needed to adequately characterize the temporal variability of bubbling in this lake is on the order of minutes.
 We used several approaches to data analysis, depending on the hypothesis being tested. The identification of bubbling events from methane flux time series data has typically been done using statistical methods such as a threshold-based selection [Kettunen et al., 2000; Greinert, 2008] or histogram-based frequency distribution studies [Harriss et al., 1985]. Regression and correlation analyses are commonly used to examine the relationships between methane fluxes and forcing mechanisms [e.g., Dise et al., 1993; Kettunen et al., 1996; Treat et al., 2007]. However, regression and correlation methods alone do not provide temporal information about a process. Alternatively, Fourier-based spectral analyses have been used to identify periodicity in ebullition fluxes [Boles et al., 2001; Greinert, 2008]. However, the short-term Fourier transform cannot resolve signals with high resolution in both the time and frequency domains, and is thus not helpful in identifying the precise timing of bubbling episodes. Furthermore, statistical and Fourier methods usually assume that the data are stationary, i.e., the statistical properties and frequencies of the signal and noise do not change with time. This assumption has not been demonstrated to be true for the seemingly sporadic natural process of ebullition.
 Thus, we present the use of wavelet transforms as a novel method of identifying the timing and length of bubbling episodes. Wavelets can be used to simultaneously analyze signals in both the time and frequency domains, as well as to identify local variations in non-stationary time series data. The mathematical background, concepts and implementation of wavelets for various applications are described in numerous texts and papers [e.g.,Addison, 2002; Graps, 1995; Mallat, 1999], and there are several instances when it has been used for time series analysis in geoscientific studies [e.g., Glaser et al., 2004; Grossmann and Morlet, 1984; Torrence and Compo, 1998; Kumar and Foufoula-Georgiou, 1997]. A wavelet multiresolution analysis (MRA) can be used to represent a signal as a series of decompositions at different time-scales (or conversely, frequencies) with high time-resolution at smaller time-scales, and high frequency resolution at coarser time-scales. Thus an MRA is particularly suitable for detecting events in signals that contain both high-frequency events occurring over short periods as well as low-frequency components that occur over longer durations (seeauxiliary material, Text S1). The MRA is also useful for identifying features of interest in noisy data, since the progressive smoothing of the signal with increasing time-scales can eliminate the need to pre-filter data.
 We used the wavelet multiresolution analysis to identify bubbling episodes that were important from a long-term seasonal perspective, as well as to characterize the short-term events that comprised the episodes. Our wavelet event detection scheme was motivated by manual observations showing that bubbling episodes at the lake tended to occur over periods lasting several hours to days and were interspersed between long periods (several days) of low methane emissions. The data from the automated traps revealed that such episodes typically comprised several, abrupt bubble release events of shorter durations, on the order of minutes to hours.
 For this study, wavelet transform coefficients were calculated for several time-scales that spanned resolutions ranging from minutes to days. Potential episodes of interest were identified at higher time-scales; at these scales, noise is minimized by using larger window lengths, thus providing a ‘big picture’ view. Local characteristics were then found by examining the corresponding time points of interest in the finer time-scales. For each time-scale, ‘events’ were identified as periods when the bubbling was distinguishable from the rest of the signal using a scale-dependent denoising threshold. ‘Bubbling episodes’ were then defined as concurrent time periods when events were detected across multiple time-scales (Figure 1). While the time-scales chosen for our analyses were particularly suitable for detecting bubbling events at the Upper Mystic Lake using a 6-month record of high-resolution fluxes, the method can be extended to detect events and long-term trends in time series data sets of longer durations, even if they have lower sampling resolutions.
2.1. Study Site
 The Upper Mystic Lake (UML) is a eutrophic, dimictic, kettlehole lake situated in eastern Massachusetts, north of Boston. It has a surface area of 0.58 km2 and a volume of 6.8 million m3. It is 25 m deep at the center, and has generally steep bottom slopes near the shore. A spillway at the southern end serves as a control on water level. The lake typically stratifies in spring, with the thermocline deepening through the summer and fall. Overturn commonly occurs in November or December [Aurilio et al., 1994; Spliethoff, 1995] and the lake surface typically freezes during the winter. Ice melt followed by spring overturn generally occurs in March.
 During the period of this study, the temperature near the deepest sediments was constant at 4°C throughout the year, and the water column was anoxic below a depth of 15 m from April to December. The concentration of methane in the upper mixed layer was measured on several dates and was less than 1 μM; dissolved methane concentrations were in the range of 100–800 μM in the anoxic hypolimnion between May and November [Varadharajan, 2009] and were generally consistent with observations of Peterson . Total methane concentrations in sediment pore water, as measured in a 1 m-long freeze core taken in September 2008, were almost uniform at 4 mM below 25 cm sediment depth, and were at ∼70% saturation for the temperature (4°C) and absolute pressure (∼3 atm) conditions at the site. Bubble patches persisting for time periods ranging from 1 to 10 min have been observed at the surface on several occasions.
2.2. Measurement of Bubble Fluxes and Hydrostatic Pressure
 Bubble fluxes were measured using automated traps submerged at ∼1 m below the water surface as described in Varadharajan et al. . Briefly, these traps were inverted funnels with an attached collection chamber in which bubbles rising through the water column accumulated. Gas volumes were determined using the pressure difference between the gas in the trap and the water pressure outside the trap that was measured by a sensor. Gas volumes were normalized to 1 atm and 20°C.
 The traps were deployed at several locations having different water column depths (Figure 2). The periods of automated data collection for all sites are listed in Table 1. The sampling interval was 5 min until 23 October 2008, and 10 min from then onwards. Manual measurements of total ebullition were also conducted roughly once or twice a week at several sites from July to November 2007, and April to November 2008. An automated control trap was deployed at a location that had no observed bubbling (6 m site), based on ebullition flux measurements from 2007. Volume data obtained from the bubble traps were corrected for effects of sensor drift and temperature variation as outlined in Varadharajan et al. . The uncertainty in the volume measurements was of the order of 3 to 6 mL depending on the diameter of the trap collection chamber. Gas volumes for periods during which data was missing or had to be rejected due to leaks or loose connections were linearly interpolated. Missing data comprised between 1 and 15% of the length of the entire data set collected at seven of the eight automated traps; three traps (at the 13 m, 19 m and 23 m sites) had longer periods of 3–4 weeks when the sensors were not functional (Table 1). The volumes collected at 10-min resolution were linearly interpolated to 5-min intervals.
Table 1. Summary of Bubbling at the Different Sites Where the Automated Traps Were Deployed
Gas Released During Episodes Using 75th Percentile Threshold (% Total Release)
Gas Released During Episodes Using 90th Percentile Threshold (% Total Release)
Short periods when some traps were not functional (less than ∼1 week) are not listed separately.
Peak fluxes were computed by dividing the maximum volume of gas collected in the traps over a day by their cross section area (0.2 m2).
5 Aug – 1 Dec
9 Oct – 20 Nov
12 Jun – 29 Oct, 14 Nov – 20 Nov
2 Jul – 25 Aug, 13 Sep – 8 Oct
10 Jul – 1 Dec
30 May – 12 Jun, 10 Jul – 1 Dec
16 Jul – 1 Dec
6 m (Control)
5 Aug – 3 Sep
 Total hydrostatic pressure and atmospheric pressure were each measured every 10 min using commercial sensors (Model 3001 Levelogger Gold, Solinst) from 5 August 2008 to 4 December 2008. Water level measurements were also obtained using a modified version of our automated bubble trap pressure logger [Varadharajan et al., 2010]; the water level data from this device agreed with the readings from the commercial sensor within 1%. Manual measurements from a nearby staff gage were used to periodically verify the automated water level measurements; the difference between the manual and automated water levels was always less than 2 cm of water. The uncertainty in automated water level readings was typically ±0.5 cm due to sensor drift and noise caused by wave action, while the error in the manual readings was approximately ±1 cm. The standard deviation in lake water level measured over the entire period of measurement was 10.5 cm. Water level and atmospheric pressure data were linearly interpolated to 5-min resolution for comparison with the trap data. For calculation of total hydrostatic pressures, all water level data were adjusted relative to the lake level on 17 March 2008, and atmospheric pressures were converted to units of cm of water.
2.3. Statistical Analysis
 Ebullitive gas fluxes were calculated by dividing smoothed cumulative gas volumes by the trap cross-section area (0.2 m2) and a time-bin width (2, 6, 12, 24 or 48 h), starting with the first data point of each signal. Data from the control trap were used to estimate the magnitude of noise present in the automated trap measurements; fluctuations in apparent gas volumes caused by wind/wave-induced buoy motion were found to be the dominant source of noise [Varadharajan et al., 2010]. Smoothed gas volumes were calculated by applying a 12-point moving average filter to data from all the traps, as this was the minimum length that visually achieved adequate noise reduction in the control trap measurements (from ±3 mL to approximately ±0.5 mL). This corresponds to smoothing over a 1-h interval for data measured at 5-min resolution. Small negative fluxes, which occurred about 15% of the time due to the 3–6 mL error in recorded volumes, were treated as zero fluxes.
 Histograms normalized to unit area were computed for fluxes using Freedman-Diaconis bin widths [Freedman and Diaconis, 1981]. Maximum likelihood parameters were estimated for different probability distributions; the distribution with the maximum log likelihood value was selected as the best fit for the data.
 The existence of synchronicity in bubbling was tested using correlations between the logarithms of site fluxes that were computed with the pairwise intersection of their data records. Since the traps were deployed on different dates during the season, fluxes for each trap pair were calculated from the first data point of the later-deployed trap. Correlations were considered significant if the p-values were lower than 0.05 and were not significantly affected by changes in binning start times.
 The hypothesized relationship between the logarithms of bubbling fluxes and corresponding hydrostatic pressures was tested using correlations that were similarly obtained from the pairwise intersection of the signals. Hydrostatic pressure data were pre-smoothed using a 1-h moving average filter and resampled to match the time-bins for which the corresponding flux data were calculated.
 Autocorrelations were used to determine the memory in a signal, i.e., the effect that a data point has on future values of a signal. Autocorrelations can yield information about the characteristic time duration of bubbling episodes when computed using trap fluxes, and are also necessary to interpret cross-correlations between fluxes and their possible trigger mechanisms. Cross-correlations indicate the delayed effect that one process has over the other, and can be calculated for different time lags. Auto-covariance and cross-covariance coefficients were calculated using the mean-removed values of the two signals, and were normalized to 1 at zero lag [Kettunen et al., 1996; Orfanidis, 1995].
2.4. Fourier Analysis
 Spectral analysis of fluxes was carried out using both raw and filtered trap volumes to determine whether bubbling episodes occurred periodically. Fluxes were computed using several time-bins (5 min to 24 h for unfiltered data; 1 to 24 h for filtered data), since the periodicity with which bubbling occurred was not known a priori. Periodograms were generated by the Welch method using different windows (rectangular, Hamming and Hann) with varying degrees of overlap (0–75%) chosen for each signal. Averaged periodograms generated by the short-term Fourier transform with overlapping windows were used in order to reduce the noise in the spectrum [Stearns, 2002]. Window sizes were varied from ∼10 to 50% of the signal length to optimize the visual tradeoff between frequency peak resolution and peak detection; 95% confidence intervals were used as cutoffs for identification of significant peaks.
2.5. Wavelet Analysis
 A stationary wavelet transform (SWT) using the Haar wavelet was applied to the cumulative volume data from the traps at dyadic (powers of 2) time-scales. The SWT is a variation of the classical discrete wavelet transform (DWT), and has been presented in literature under several names such as MODWT, undecimated wavelet transform or translation-invariant DWT [e.g.,Nason and Silverman, 1995; Percival and Walden, 2000]. The SWT was preferred over the classical DWT since it is shift-invariant, i.e., the magnitudes of the wavelet coefficients are independent of the start point for analysis. The Haar wavelet was chosen since the signal was assumed to be approximately piecewise constant, given the predominant step edges in cumulative gas volume data from the traps. The transform was directly performed on the cumulative volumes without prior filtering.
 The wavelet analysis was carried out using the WMTSA toolkit developed at the University of Washington (http://www.atmos.washington.edu/∼wmtsa/). The software implements the MODWT, which is described in detail in Percival and Walden , and has been used in previous geoscientific analyses [e.g., Kallache et al., 2005; Percival, 2008; Whitcher et al., 2000]. The MODWT produces two sets of decompositions – (1) a multiresolution analysis (MRA) that can be used to precisely identify events of interest and (2) a wavelet variance that can be used to determine the dominant time-scales in the signal and hence identify periodicities or trends.
 We identified bubbling episodes using an MRA comprising 10 dyadic time-scales. The MRA generates ‘detail coefficients’ and ‘smoothing coefficients’ at each of the different time-scales, which correspond to outputs from zero-phase high-pass and low-pass filters, respectively. These are calculated by averaging the wavelet coefficients for all possible start point shifts at each time-scale, and represent an additive decomposition that can be summed up to perfectly reconstruct the original signal. Events are detected by analyzing the detail coefficients, which are properly aligned with features in the original time series, and represent variations caused by successively smoothing the signal. The relative magnitudes of the detail coefficients across the time series are indicative of the extent of changes in the signal for each time-scale. The detail coefficients at the 10th scale correspond to a physical time period of ∼2 days (42.5 h), beyond which the time-scale begins to approach the typical manual sampling interval.
 For each time-scale, bubbling ‘events’ are identified by using the time points when the absolute modulus of the detail coefficients exceed their denoising threshold computed using the Stein's Unbiased Risk Estimate (SURE) [Coifman and Donoho, 1995; Donoho et al., 1995]. The SURE method minimizes the mean squared error associated with the selection of thresholds, and assumes that oscillations in the recorded data due to wind and wave effects, as well as small random bubble release over the 6-month deployment period, can be represented as stationary Gaussian white noise. However, since bubbling fluxes can exhibit significant seasonal and interannual variability, this assumption might not hold for longer studies; in such cases different thresholds can be computed by dividing the time series into locally stationary segments. Two other common denoising thresholds, the minimax and the universal thresholds [Donoho, 1995; Jansen, 2001], were found to be too selective; use of these thresholds resulted in missing events that would be identified as significant from visual inspection of the data. Signals were padded at the beginning with zeroes corresponding to the length of the filter at each scale to handle initial boundary effects. Signals were also reflected at the end, and the MODWT coefficients were subsequently truncated to the original signal length.
 Events identified from the bubble trap data at the different time-scales were compared directly with the hydrostatic pressure signal, by matching their horizontal time-axes. For this study, the events appearing at time-scales that were on the order of 1–2 days (i.e., 21.3 and 42.5-h time-scales) were considered to be bubbling events of interest from a long-term seasonal perspective. The detailed structure of these events, and comparisons with the hydrostatic pressure signal, were then studied by examining the corresponding events identified in the minute to hour time-scales.
 The second MODWT decomposition yields a set of ‘wavelet’ and ‘scaling’ coefficients that are effectively the coefficients obtained from a classical DWT, but are computed for all data points in a signal without any downsampling. The wavelet coefficients obtained using the Haar wavelet are proportional to the bubbling fluxes computed at each signal data point (section 2.3); the time-scales in the wavelet analysis correspond to the time-bins in the flux calculations. These coefficients are used to generate a wavelet power spectrum in order to detect possible periodicity in ebullition, and provide a comparison with results of the Fourier analysis. The time-scales with the highest variance correspond to the dominant frequencies at which bubbling occurs, since the sum of the variances equals the total energy of the signal. The wavelet variance was calculated for the sum of all details obtained from a 12-level MODWT decomposition, corresponding to a 170-h time-scale. Cumulative volumes could not be used as a signal for wavelet variance analysis, since the “trend” corresponding to the volume increase with time would have caused the larger time-scales to have higher energies.
3.1. Statistical Analyses
 Peak fluxes at different sites overlapped to an extent (Figure 3), visually indicating that there were periods, of the order of a day in length, during which bubbling episodes occurred simultaneously throughout the lake. The pair-wise correlation coefficients of the logarithms of site fluxes were significantly related (p < =0.05) for 17 of the 20 trap combinations (Table A1 inText S1), but had generally low values (R2ranging from 0.2 to 0.5). The best correlation coefficients were observed for the 24-h time-bins, with less significant relationships observed for the smaller time-bins, which is likely the result of minor differences in the exact timing of bubbling events.
 However, the magnitude of ebullition fluxes varied considerably over the period of deployment from site to site (Figure 4). The histograms show that ebullition at most sites, except the ones with low fluxes, consistently follows either a lognormal or a gamma distribution. The log likelihoods for the lognormal distribution were marginally better than the gamma distribution values for the 12, 24 and 48 h time-bins, but the case was reversed for the 2 and 6 h time-bins. The lognormal nature of the frequency distribution of methane fluxes has also been observed in other freshwater ecosystems [Harriss et al., 1985; von Fischer et al., 2010].
 Bubbling episodes can be identified using an approximate, site-dependent threshold determined from a sorted distribution of fluxes (Figure 5 and Table 1). Episodes identified in this manner were typically periods when fluxes were in the top 10th to 25th percentile for all time-bins; thus the exact choice of this threshold is subjective and can sometimes vary with the time-bin chosen for flux calculations. The results were not affected by changes in binning start times.
 Autocorrelation coefficients computed using 24-h fluxes over 10 days were not significant for periods longer than a day at all sites, indicating that the effects of ebullition events are relatively short-lived. The autocorrelation coefficients calculated for 2, 6 and 12-h unfiltered fluxes over 48 h also decayed rapidly, with most significant coefficients occurring within 30 h (Figure 6). Autocorrelation results for the control trap data showed that fluxes computed using time-bins smaller than 2 h were affected by wind noise. Some wind-induced noise in the data was expected, because the traps respond to the difference between external hydrostatic pressure and internal gas pressure, both of which can be affected by small water pressure fluctuations and trap motions induced by wind-driven surface waves.
3.2. Comparison of Fluxes With Hydrostatic Pressure
 Significant negative correlations were observed between the logarithms of trap fluxes and hydrostatic pressure for all time-bins (e.g.,Figure 7), although the low R values suggest that changes in hydrostatic pressure only account for a part of the fluxes or that their relationship is not linear. A few outliers were observed in the correlations computed using the small time-bins (2 and 6 h) where fluxes were low despite low hydrostatic pressures. This could have been due to several reasons, such as short lags between a pressure drop and bubble initiation, or a decrease in the sediment gas inventory following a bubbling episode. The contributions of variations in atmospheric pressure and water level to the magnitude of total hydrostatic pressure at the UML were similar; the standard deviations of both variables were approximately 10 cm of water through the season (Figure 8).
 The autocorrelation coefficients of the hydrostatic pressure signal slowly decayed to zero over approximately 100 h. Atmospheric pressure and water level data were similarly autoregressive indicating that the effects of their variations were persistent for about four days. Thus, cross correlations could not be used to determine the time lag between changes in hydrostatic pressure and corresponding changes in ebullition fluxes, since the effects of hydrostatic pressure variations typically lasted for several days.
3.3. Power Spectrum Analysis
 No significant peaks (based on 95% confidence intervals) could be distinguished from the noise in the power spectra of unfiltered fluxes, for any of the window choices as well as variations in window length or overlap. The noise in unfiltered fluxes results from the effects of wind on the traps, which occur at relatively high frequencies (corresponding to 1–2 h time-scales or less). However, significant peaks were absent in the spectra of filtered fluxes for all window variations. These results were not affected by changes in bin lengths or start times. Thus, no particular periodicity could be identified using Fourier analysis in the UML ebullition fluxes.
3.4. Wavelet Multiresolution Analysis (MRA)
 Bubbling events could be identified at several different time-scales in the wavelet MRAs (Figure 9 for trap 9 m(A), Figures A1–A6 in Text S1for other traps). These were used to determine if bubble events on the shorter (minute to hour) time-scales eventually evolved into a bubbling episode of the order of several days' duration (e.g., Episode 1 inFigure 9, Sep 29th – Oct 3rd). In general, bubbling that appeared as one large event on the 21.3-h (or greater) time-scales comprised a series of shorter events that could be distinguished in the minute to hour scales (e.g.,Figure 9 and Tables 2a, 2b, and 3). For example, ∼55% of the gas bubbled between Nov 13th and Nov 17th at the 9 m(A) site (Episode 2 in Figure 9), was released during a few hours, namely from 11:00 to 15:40 local time (LT) on Nov 15th (Event 2(a) in Figure 9 and Table 3).
Table 2a. Episodes Identified at the 9 m(A) Site Using Waveletsa
 The progressive smoothing in the wavelet transform ensured that noise due to wind, which could appear as spurious events on the shorter time-scales (e.g., Event 3 inFigure 9), did not propagate up to the 1–2 day time-scales. This avoided the need to preprocess the signal using filters that could potentially eliminate events of interest on smaller scales. The MRA could also detect episodes that involved relatively steady gas release over several hours rather than shorter bursts of ebullition. For example, gas release from Nov 6th–10th turned out to be a significant episode for the season (Event 4 inFigure 9) and was recognized as such at the longer time-scales, even though this particular episode did not include any notable events at the 5 to 20-min time-scales.
 Most of the ebullition during the season deployed happened over a very short length of time (e.g., Figure 10). For example, about 63% of the total gas collected by the 9 m(A) automated trap was found to occur during 3% of the deployment period (Table 3), which is consistent with the analysis using statistical thresholds (Table 1). Several bubbling episodes that were important from a seasonal perspective involved at least one significant event at the 5 to 10 min time-scales, indicating that a substantial amount of gas can be released over extremely short periods. Bubbling episodes also overlapped at different traps (e.g.,Tables 2a and 2b) showing that bubble releases tend to occur at similar times across the lake.
3.5. Comparison of Wavelet Coefficients With Hydrostatic Pressure
 Bubbling events typically occurred during periods when the hydrostatic pressures were low (e.g., Figure 11afor trap 9 m(A) and Figures A7–A11 in Text S1 for other traps). The comparison of the wavelet detail coefficients from the smaller time-scales (Figure 11b) against the pressure signal revealed that the timing of the biggest bubble releases usually occurred at local minima of the hydrostatic pressure record. Bubbling events generally commenced when the hydrostatic pressure dropped below a site-dependent threshold, and stopped within a few hours of a pressure rise (Table 4). The hydrostatic pressures at which ebullition stopped were typically similar to the onset pressures, although there were a few instances when ebullition continued despite rising hydrostatic pressure.
Table 4. Relative Hydrostatic Pressures (cm of Water) at Different Sites During the Start and End of Bubbling Events Identified at the 11.6-h Scalea
Mean Hydrostatic Pressure at Onset of Episodes
Standard Deviation of Onset Pressures
Mean Hydrostatic Pressure When Bubbling Ceases
Standard Deviation of Cessation Pressures
Mean hydrostatic pressure for the entire deployment period = 74 cm of water and standard deviation = 10 cm of water.
3.6. Wavelet Variance
 The wavelet variance was computed for time-scales ranging from 5 min to 170 h; all sites exhibited similar trends and most of the energy was concentrated at the higher scales (e.g., Figure A12 inText S1). The results indicate that the variability in ebullition fluxes during the deployment period was not dominated by any particular time-scale that was less than 28 days, and is consistent with the results from the Fourier analysis.
4.1. Characterization of Bubbling Fluxes
 Bubbling episodes were identified using two different methods, namely conventional statistical thresholds based on histograms of flux distributions, and the alignment of detail coefficients in a wavelet multiresolution analysis. Both supported our first hypothesis, and showed that sporadic bubbling episodes that occurred about 10–25% of the time represented the periods during which most methane release took place in the Upper Mystic Lake. About 50–80% of the total gas bubbled from July to November 2008 was emitted during these episodes. However, while the statistical method requires somewhat subjective threshold and episode selections, the wavelet analysis based on a scale-dependent denoising threshold presents a means to define bubbling events and episodes more precisely (e.g.,Tables 2a and 2b).
 The multiresolution wavelet analysis also reveals that most bubbling episodes comprise several short bubble releases that occur on scales of 5 to 10 min or less, although episodes could occasionally result from a gradual, continuous bubble release lasting for several hours. We could identify no single characteristic duration of ebullition for the UML; episode lengths could range anywhere between a few hours and several days. However, the autocorrelations suggest that the highest probability of gas release following an initial bubbling event occurs within the first day.
4.2. Synchronous Bubbling Episodes and Apparent Forcing Mechanisms
 The results from both the statistical and wavelet analyses showed that bubbling episodes tend to occur lake-wide, coincident with periods of low hydrostatic pressure, supporting hypothesis 2a that ebullition can be routinely triggered by relatively small, aperiodic fluctuations in hydrostatic pressure. The Fourier and wavelet variance results both indicate that no periodicity was present in ebullition fluxes at the UML for time-scales less than 28 days; these results do not support hypothesis 2b, but are consistent with the theory that initiation of bubbling is dominated by a lake-wide aperiodic forcing.
 Perhaps somewhat counterintuitively, it was seen that hydrostatic pressure decreases as small as a few cm of water can apparently cause episodes of bubbling even at sites where the water column depth is as much as 25 m. At most sites, the largest bubble releases typically occurred at times when the hydrostatic pressure decreased below a site-dependent threshold, and continued until the hydrostatic pressure rose above its initial triggering value (Table 4). Relatively larger increases (∼10 cm of water) in hydrostatic pressure were always associated with an immediate cessation of bubbling at this lake. This contrasts with marine systems, where ebullition fluxes have been observed to be correlated with tidal variations that are approximately an order of magnitude larger than the fluctuations observed at the UML [Boles et al., 2001]. The close relationship between the hydrostatic pressure forcing and ebullition fluxes may explain the lack of a single characteristic episode length as well as the absence of periodicity at time-scales less than 28 days, since atmospheric pressure and lake water levels can vary considerably over a few hours, and tend to be aperiodic at these time-scales.
 However, the observation that gas bubbling typically occurs below a site-dependent threshold hydrostatic pressure combined with the instances when delayed ebullition occurred while the hydrostatic pressure was rising suggests that in situ sediment mechanics plays a role in determining the exact timing and magnitude of gas that is released at each site. There are also several instances when little or no ebullition was observed even though hydrostatic pressure was well below the site threshold. These could have been the result of insufficient sediment gas inventory following an earlier bubbling episode, although it must be cautioned that the traps might have failed to capture some of the bubbles, due to movement around their watch circles, at some of the times of low hydrostatic pressure. Another possibility is that bubble release from the sediments was triggered; but that some of the bubbles may not have reached the surface waters (as shown inMcGinnis et al. ).
 A 1-D conduit dilation model developed using this time series data further explores the causal effect of changes in hydrostatic pressure on ebullition and sediment mechanics [Scandella et al., 2011]. The model is based on the theory that dynamic conduits in the sediment will dilate and release gas when hydrostatic pressure decreases cause the effective stress in the sediment to drop below its tensile limit. The model numerically calculates the evolution of gas pressures and saturation in response to changes in hydrostatic pressure, and was able to predict the magnitude and timing of fluxes at the lake based on our hydrostatic pressure record.
 It must be noted that although our model was able to accurately predict methane ebullition caused by drops in hydrostatic pressure, the statistical correlations (not shown) between changes in hydrostatic pressure and fluxes were poor, and in many cases insignificant. Several factors in such a dynamic system – for example, a delayed response to pressure drops or a decrease in the sediment gas content following an ebullition episode – could have resulted in insignificant correlations, even when there was a real relationship between the signals. While the methods used in this paper clearly show that ebullition typically occurred during periods of low hydrostatic pressure, the biggest bubbling episodes identified in the wavelet analysis were not necessarily associated with the largest pressure drops. The relatively weak pairwise trap correlation coefficients may also be a result of minor differences (less than a day) in site-to-site responses to the hydrostatic pressure forcing. Thus, while correlations and regression have often been used to understand the causal forcings that lead to certain observations, the results from such analyses should be treated with caution in studies of methane ebullition for several reasons. In some situations, true relationships can be missed due to minor differences in the timing of the forcing and flux signals. However, in other cases, spurious correlations may be observed when there is no relationship between the variables because ebullition flux data are likely to be non-stationary. Furthermore, bubbling is a process where the system history could play an important role in determining future behavior, especially with regard to the sediment gas content. Correlations and regressions cannot fully describe the behavior of such a system, because they impose algebraic relationships in lieu of an evolution equation. However, correlations may provide a simple means to identify relationships between different processes, which can then be subjected to further analyses that incorporate physical explanations for the relationships.
4.3. Selection of Sampling Intervals for Future Measurements
 In any sensor-based measurement scheme, the choice of an appropriate sampling interval is critical. Under-sampling can lead to important events being missed, whereas over-sampling leads to large volumes of data that may be unnecessarily difficult to store and process. At the UML, we hypothesized that the sampling resolution needed to be on the order of minutes, to capture the essential temporal characteristics of ebullition. The final choice of 1 sample per 5 or 10 min was a trade-off between the second to minute time-scales at which bubbling events had been previously observed [Greinert, 2008; Walter et al., 2006; C. Varadharajan, personal observations, 2007] and storage limitations of the commercial data logger (HOBO H8, Onset Systems). However, in some of the data post-processing, the effects of wind on the trap data had to be reduced by smoothing the signal using a moving average filter of about 12 sample points, which yielded an effective sampling resolution of approximately one hour. Fortunately the one sample per hour rate was adequate for determining both total ebullitive fluxes as well as the dependencies of flux on external forcing mechanisms such as hydrostatic pressure.
 However, significant bubbling events could be defined at the time-scale of 5 min, which is a result that may be important for understanding the details of in-sediment processes. Thus, studies that seek to understand the coupling between sediment mechanics and methane venting will probably need very high-temporal resolution data (order of seconds to minutes). However, in such cases we also recommend that traps be placed as close to the sediments as possible, to minimize the time lag between bubble release and capture within the traps as well as to minimize the effects of trap movement about their mooring point.
5. Summary and Recommendations
 The temporal variability observed in the automated trap data at the Upper Mystic Lake illustrates the need for high-resolution, continuous long-term monitoring to adequately characterize bubbling in aquatic ecosystems. Most of the ebullition during the six-month deployment period occurred during short episodes that were triggered by drops in hydrostatic pressure. These episodes did not take place with any particular periodicity and did not have a characteristic duration, which would make it impossible to plan manual measurement campaigns in advance.
 We also present the use of the stationary wavelet transform as a new tool to analyze high-temporal resolution ebullition data sets. The wavelet analyses were consistent with results obtained using conventional statistical methods, with the added advantage of being able to precisely identify the timing and characteristics of bubbling episodes. The ability to retain temporal information is especially desirable for an analysis of methane ebullition, where dynamic relationships with hydrostatic pressure and other triggering mechanisms are expected. In particular, the application of a multilevel decomposition to detect abrupt, high-frequency events at shorter time-scales, as well as low-frequency trends on the longer time-scales, could be useful in understanding methane ebullition data collected in environments where multiple time-scales are involved, such as in eddy covariance towers or chamber measurements in peatlands. For example, an analysis using wavelet variances where the energies are preserved across scales may be able to distinguish between seasonal variations in fluxes, from the changes induced by short-term forcings such as temperature and hydrostatic pressure. Wavelets are also useful for analyzing noisy data, since the signal is smoothed with increasing time-scales, thus retaining potentially important information in the finer time-scales. It may also be possible to extend the use of wavelets to detect ebullition in high-resolution spatial data, given that wavelets have been used extensively for event and edge detection in image processing. In all cases, an appropriate choice for the type of wavelet transform (continuous, discrete or stationary), the time (or space) scales for analysis and the mother wavelet function should be made based on the variability of the processes being studied.
 This work was supported by NSF doctoral dissertation research grant 0726806, NSF EAR 0330272, a GSA graduate student research grant and MIT Martin, Linden and Ippen fellowships. Alexandra Patricia Tcaciuc and Emanuel Borja were funded by the MIT and Martin UROP programs and assisted with the fabrication and testing of equipment, and with collection of field data. We thank Phil Gschwend, Sudarshan Raghunathan and Steve Lerman for discussions about the data analysis, and Ruben Juanes and Ben Scandella for discussions regarding the role of sediment mechanics in ebullition.