Time-series analysis of high-resolution ebullition fluxes
from a stratified, freshwater lake
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Citation
Varadharajan, Charuleka, and Harold F. Hemond. “Time-series
Analysis of High-resolution Ebullition Fluxes from a Stratified,
Freshwater Lake.” Journal of Geophysical Research 117.G2
(2012). Copyright 2012 by the American Geophysical Union
As Published
http://dx.doi.org/10.1029/2011jg001866
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American Geophysical Union (AGU)
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Wed May 25 22:03:09 EDT 2016
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http://hdl.handle.net/1721.1/77913
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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, G02004, doi:10.1029/2011JG001866, 2012
Time-series analysis of high-resolution ebullition fluxes
from a stratified, freshwater lake
Charuleka Varadharajan1,2 and Harold F. Hemond2
Received 26 September 2011; revised 7 February 2012; accepted 11 February 2012; published 11 April 2012.
[1] 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.
Citation: Varadharajan, C., and H. F. Hemond (2012), Time-series analysis of high-resolution ebullition fluxes from a stratified,
freshwater lake, J. Geophys. Res., 117, G02004, doi:10.1029/2011JG001866.
1. Introduction
[2] Lakes are important natural sources of methane, and
ebullition is a key pathway by which methane from lake
sediments can be released to the atmosphere [Bastviken et al.,
2004; Walter et al., 2007]. Bubbling in aquatic ecosystems
tends to occur in episodes, and can be triggered by changes in
atmospheric pressure [Fechner-Levy and Hemond, 1996;
Mattson and Likens, 1990] or water level [Engle and Melack,
2000; Martens and Klump, 1980; Boles et al., 2001; Chanton
and Martens, 1988], as well as physical sediment disturbance
[Joyce and Jewell, 2003] and wind [Keller and Stallard,
1994]. Ebullition is also affected by the rate of in-sediment
methanogenesis as determined by temperature, oxygen level
and organic matter input [Christensen et al., 2003; Kelly and
Chynoweth, 1981; Liikanen et al., 2002].
[3] 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
1
Earth Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California, USA.
2
Department of Civil and Environmental Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts, USA.
Copyright 2012 by the American Geophysical Union.
0148-0227/12/2011JG001866
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.
[4] 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. [2001] and Leifer et al.
[2004] 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.
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Figure 1. Wavelet decomposition of a bubbling episode
for multiple time-scales. Events can be identified for each
time-scale using a selection criterion (e.g., thresholding).
Bubbling episodes comprise events that propagate across
several time-scales.
[5] 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:
[6] 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.
[7] 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
[8] 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.
[9] 3. The minimum sampling frequency needed to adequately characterize the temporal variability of bubbling in
this lake is on the order of minutes.
[10] 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, Fourierbased 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
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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.
[11] 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 (see auxiliary material, Text S1).1
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 prefilter data.
[12] We used the wavelet multiresolution analysis to
identify bubbling episodes that were important from a longterm 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.
[13] 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
1
Auxiliary materials are available with the HTML. doi:10.1029/
2011JG001866.
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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.
[15] 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 mM; dissolved methane concentrations were
in the range of 100–800 mM in the anoxic hypolimnion
between May and November [Varadharajan, 2009] and
were generally consistent with observations of Peterson
[2005]. 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.
Figure 2. Placement of bubble traps in the Upper Mystic
Lake in 2008; trap names reflect the approximate depth at
their location of deployment. Traps located within white circles were automated. Google Earth Imagery © 2008 Google
Inc. and Tele Atlas. Used with permission.
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. Methods
2.1. Study Site
[14] The Upper Mystic Lake (UML) is a eutrophic,
dimictic, kettlehole lake situated in eastern Massachusetts,
2.2. Measurement of Bubble Fluxes and Hydrostatic
Pressure
[16] Bubble fluxes were measured using automated traps
submerged at 1 m below the water surface as described in
Varadharajan et al. [2010]. 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.
[17] 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
Table 1. Summary of Bubbling at the Different Sites Where the Automated Traps Were Deployed
Trap
Period of Data
Collection (2008)a
Record
Length (h)
Peak Fluxb
(mL m 2 d 1)
9 m(A)
9 m(B)
13 m
19 m
22 m
23 m
25 m
6 m (Control)
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
5 Aug – 3 Sep
2832
1013
3482
1881
3164
3758
3022
693
630
396
313
87
250
172
248
20
Gas Released During
Episodes Using 75th
Percentile Threshold
(% Total Release)
Gas Released During
Episodes Using 90th
Percentile Threshold
(% Total Release)
82
73
75
91
81
76
85
69
35
51
57
56
50
56
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).
a
b
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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. [2010]. 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.
[18] 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
[19] 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.
[20] 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
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distribution with the maximum log likelihood value was
selected as the best fit for the data.
[21] 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.
[22] 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.
[23] 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 crosscovariance coefficients were calculated using the meanremoved values of the two signals, and were normalized to 1
at zero lag [Kettunen et al., 1996; Orfanidis, 1995].
2.4. Fourier Analysis
[24] 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
[25] 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
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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.
[26] 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 [2000], 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.
[27] 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 zerophase 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.
[28] 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.
[29] 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)
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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.
[30] 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. Results
3.1. Statistical Analyses
[31] 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 in Text S1), but had generally low
values (R2 ranging 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.
[32] 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].
[33] 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.
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Figure 3. Peak fluxes overlapped at the different sites, indicating that bubbling occurred synchronously
across the lake. Data were measured at 5–10 min resolutions between June and December 2008. “0” indicates the start of gas measurement using automated traps.
Figure 4. Histograms of trap fluxes indicate a wide range of ebullition fluxes at all sites. The solid green
line shows a lognormal fit, and the dashed red line represents a gamma distribution fit. Fluxes shown here
were computed using 24-h time bins.
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Figure 5. High fluxes only occurred during 5 to 25% of the period when the traps were deployed (see
Table 1). Fluxes shown were computed using 6 and 24-h time bins; similar results were observed for other
bin widths.
[34] 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
[35] 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).
[36] 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
[37] 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)
[38] 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 S1 for 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 in Figure 9, Sep 29th – Oct 3rd). In
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VARADHARAJAN AND HEMOND: ANALYZING HIGH-RESOLUTION BUBBLING DATA
Figure 6. Autocorrelations computed for different time-bins over 48 h suggest that once bubbling is initiated, future events will most likely occur within the first day. The red, filled-in markers represent significant coefficients (p < 0.05).
Figure 7. Significant negative correlations (p < 0.05) were observed between the logarithms of trap
fluxes (mL m 2 d 1) and relative hydrostatic pressure (cm of water) at all sites other than the control;
the weakest correlation was at the 19 m trap that had low fluxes for most the season. Fluxes shown were
computed using 24-h time bins.
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Figure 8. Changes in total hydrostatic pressure at the Upper Mystic Lake were caused in approximately
equal part by variations in atmospheric pressure and in water level.
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).
[39] 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 in
Figure 9), did not propagate up to the 1–2 day time-scales.
Figure 9. Identification of bubbling episodes and events at the 9 m(A) trap using a wavelet MRA. The
thick red dots highlight significant events at each scale. Bubbling episodes often consisted of events that
could be detected across multiple time-scales.
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Table 2a. Episodes Identified at the 9 m(A) Site Using Waveletsa
Start Date
End Date
Gas Captured in Trap
(% Total Release)
1-Sep
11-Sep
20-Sep
23-Sep
28-Sep
7-Oct
12-Oct
19-Oct
24-Oct
6-Nov
13-Nov
3-Sep
14-Sep
22-Sep
25-Sep
3-Oct
10-Oct
17-Oct
22-Oct
1-Nov
10-Nov
17-Nov
2
4
2
1
28
4
3
4
18
5
19
Total
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trends and most of the energy was concentrated at the higher
scales (e.g., Figure A12 in Text 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. Discussion
89
a
See Figure 10.
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 in Figure 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.
[40] 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
[41] Bubbling events typically occurred during periods
when the hydrostatic pressures were low (e.g., Figure 11a
for 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 sitedependent 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.
3.6. Wavelet Variance
[42] The wavelet variance was computed for time-scales
ranging from 5 min to 170 h; all sites exhibited similar
4.1. Characterization of Bubbling Fluxes
[43] 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).
[44] 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
[45] The results from both the statistical and wavelet
analyses showed that bubbling episodes tend to occur lakewide, 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
Table 2b. Episodes Identified at the 25 m Site Using Wavelets
Start Date
End Date
Gas Captured in Trap
(% Total Release)
20-Aug
23-Aug
29-Aug
14-Sep
30-Sep
8-Oct
15-Oct
20-Oct
25-Oct
7-Nov
14-Nov
19-Nov
23-Nov
22-Aug
26-Aug
5-Sep
15-Sep
3-Oct
10-Oct
18-Oct
23-Oct
30-Oct
10-Nov
17-Nov
21-Nov
26-Nov
1
7
21
1
5
3
9
6
8
3
6
1
2
Total
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Table 3. Detailed Structure of the Episodes That Were Identified at the 1.3 h Scale in the Wavelet MRA at the 9 m(A) site
Volume Gas Collected
in Trap During
Event (mL)
Relative Hydrostatic
Pressure at Start
(cm of Water)
Relative Hydrostatic
Pressure at End
(cm of Water)
61.8
71.4
69.6
69.6
68.3
76.6
80.7
77.5
75.0
72.0
63.6
63.0
67.0
62.3
61.4
75.1
61.8
53.6
51.2
47.4
63.3
67.5
57.9
49.8
45.0
62.0
69.6
68.6
69.3
67.9
76.6
79.7
76.4
72.3
70.7
62.0
63.0
67.0
61.4
62.0
75.5
56.1
53.9
48.4
49.4
65.5
67.1
51.9
48.7
44.1
Event Start Time (LT)
Event End Time (LT)
Event
Length (h)
Sep-02 04:20
Sep-12 16:25
Sep-12 20:44
Sep-20 20:44
Sep-21 12:04
Sep-24 10:20
Sep-29 15:49
Sep-30 03:04
Sep-30 12:49
Sep-30 16:54
Oct-01 10:14
Oct-01 15:24
Oct-08 18:09
Oct-09 05:05
Oct-09 08:35
Oct-13 21:15
Oct-25 18:35
Oct-26 02:39
Oct-28 12:54
Oct-28 21:29
Nov-09 16:30
Nov-14 12:15
Nov-15 10:59
Nov-15 19:34
Nov-16 06:40
Sep-02 07:39
Sep-12 20:15
Sep-12 23:19
Sep-20 22:45
Sep-21 15:04
Sep-24 13:05
Sep-29 19:50
Sep-30 07:05
Sep-30 16:20
Sep-30 19:30
Oct-01 13:30
Oct-01 19:20
Oct-08 21:15
Oct-09 08:10
Oct-09 11:50
Oct-14 00:09
Oct-25 23:19
Oct-26 04:25
Oct-28 15:20
Oct-29 00:00
Nov-09 19:09
Nov-14 14:39
Nov-15 15:40
Nov-15 23:35
Nov-16 10:45
3.3
3.8
2.6
2.0
3.0
2.8
4.0
4.0
3.5
2.6
3.3
3.9
3.1
3.1
3.2
2.9
4.7
1.8
2.4
2.5
2.7
2.4
4.7
4.0
4.1
13
29
10
5
9
8
28
30
62
52
15
33
11
13
19
8
118
4
9
8
9
14
100
5
19
80.3
630a
Total
Sixty-three percent of the total volume of gas captured by this trap (1000 mL) occurred in 3% of the total deployment time (2832 h).
a
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.
[46] 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-
Figure 10. Events identified at the 9 m(A) and 25 m traps using wavelet coefficients from the
21.3-h time-scale (indicated by thick dark lines). The green circles mark the onset of bubbling events, while
the red squares mark the end of the event.
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Figure 11. Comparison of wavelet detail coefficients (red line) for the 9 m(A) trap at, e.g., (a) 11.6-h and
(b) 1.3-h time-scales with the hydrostatic pressure signal (blue line). Bubbling typically occurred during
periods of low hydrostatic pressures (indicated by markers and thick lines).
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
Trap Name
Mean Hydrostatic
Pressure at Onset
of Episodes
Standard Deviation
of Onset Pressures
Mean Hydrostatic Pressure
When Bubbling Ceases
Standard Deviation
of Cessation Pressures
9 m(A)
13 m
22 m
23 m
25 m
68
67
66
62
67
10
7
10
5
7
64
63
67
61
67
7
9
10
5
7
a
Mean hydrostatic pressure for the entire deployment period = 74 cm of water and standard deviation = 10 cm of water.
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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.
[47] 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 in McGinnis et al. [2006]).
[48] 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.
[49] 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-tosite responses to the hydrostatic pressure forcing. Thus,
while correlations and regression have often been used to
understand the causal forcings that lead to certain
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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 nonstationary. 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
[50] In any sensor-based measurement scheme, the
choice of an appropriate sampling interval is critical. Undersampling 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.
[51] 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
[52] 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
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with any particular periodicity and did not have a characteristic duration, which would make it impossible to plan
manual measurement campaigns in advance.
[53] 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.
[54] Acknowledgments. 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.
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