Operational Modal Analysis of Low Frequency Surface Waves in Lakes and Reservoirs

Abstract

Wind driven oscillations of water surfaces (surface seiches) are a key driver of ecological fluxes, such as of oxygen and nutrients, within lakes and reservoirs. These seiches exhibit low frequency oscillations with periods typically in the range from minutes to days, closely-spaced modes and a low signal to noise ratio, which makes modal analysis by simple spectral techniques challenging. Time domain operational modal analysis, specifically the Random Decrement Technique, was applied to extract modal parameters for the dominant surface seiches of Lake Geneva, Switzerland, and Lake Tahoe, USA. The results obtained show good agreement with existing literature on the periods and damping ratios of the seiches, historically obtained in this field by visual selection of free-decay sections of the time series or half-power peak picking. These analyses illustrate the potential for the use of techniques developed for operational modal analysis in areas outside the realm of mechanical and civil structures dynamics.

Full text

OPERATIONAL MODAL ANALYSIS OF LOW FREQUENCY
SURFACE WAVES IN LAKES AND RESERVOIRS
Zachariah Wynne 1, Thomas Reynolds 2, Damien Bouffard 3, Geoffrey Schladow 4, Danielle Wain 5
1PhD Student, University of Edinburgh, UK, Z.Wynne@sms.ed.ac.uk.
2Chancellor’s Fellow, University of Edinburgh, UK; Turing Fellow, The Alan Turing Institute, UK, T.Reynolds@ed.ac.uk.
3Researcher, Eawag, Swiss Federal Institute of Aquatic Science and Technology, Switzerland, Damien.Bouffard@eawag.ch.
4Professor, Tahoe Environmental Research Center, University of California Davis, USA, GSchladow@ucdavis.edu.
5Lake Science Director, 7 Lakes Alliance, USA, Danielle.Wain@7lakesalliance.org.
ABSTRACT
Wind driven oscillations of water surfaces (surface seiches) are a key driver of ecological fluxes, such
as of oxygen and nutrients, within lakes and reservoirs. These seiches exhibit low frequency oscilla-
tions with periods typically in the range from minutes to days, closely-spaced modes and a low signal
to noise ratio, which makes modal analysis by simple spectral techniques challenging. Time domain
operational modal analysis, specifically the Random Decrement Technique, was applied to extract modal
parameters for the dominant surface seiches of Lake Geneva, Switzerland, and Lake Tahoe, USA. The
results obtained show good agreement with existing literature on the periods and damping ratios of the
seiches, historically obtained in this field by visual selection of free-decay sections of the time series or
half-power peak picking. These analyses illustrate the potential for the use of techniques developed for
operational modal analysis in areas outside the realm of mechanical and civil structures dynamics.
Keywords: Random Decrement Technique, Low frequency oscillations, Novel applications
1. INTRODUCTION
When wind acts upon the surface of large bodies of water, such as lakes, reservoirs, lagoons, and semi-
enclosed coastal bays, for an extended period of time and with a consistent direction, the stress of the
wind on the water surface can lead to an upwelling of water at one end of the lake, as shown in Figure
1. Once the wind force undergoes a change of direction or reduction in magnitude, the water level seeks
to find an equilibrium and an oscillation, known as a surface or barotropic seiche, is induced within the
lake. The restoring force of this oscillation is gravity, with energy damped from the system through the
friction between the body of water and the lake bed (water-sediment interface) and friction between water

molecules [1]. The damping of surface seiches leads to a reduction in the amplitude of each subsequent
oscillation and a slight increase of the period of oscillation in the absence of further forcing of the system
[2, 3].
Wind
Oscillations of lake surface
Figure 1: Surface seiche oscillation
Surface seiche motion can lead to flooding of areas around the perimeter of the lake and damage to
boats and other structures within or close to it [4, 5, 6, 7]. They are of interest to ecologists as they
provide a mechanism through which currents and nutrient fluxes are induced within the lake. Due to
the introduction of additional nutrients into lakes from fertilizer run-off, this area of study is becoming
increasingly important to prevent eutrophication, the build up of excessive nutrients within a water body,
which may destroy fragile ecosystems and make the water unfit for human use or consumption due to
algal and bacterial blooms [8].
There are a number of factors which make seiches difficult to analyse in practice, including the long
period of the oscillation, which may range from minutes to days per cycle, the low signal to noise
ratio and the excitation of multiple vibration modes within the lake, associated with different axis of
oscillations. For lakes with multiple axes of similar lengths and bathymetry (underwater topography),
these modes are close-modes of similar frequency.
The frequencies of surface seiches are well established for many lakes but few measurements of their
associated damping behaviour were found in an extensive literature review. A widely cited [2, 9, 10, 11,
12, 13] but unverified study published in 1934 by A. Endr¨
os [1], with the results reproduced in English
in A. Defant’s Physical Oceanography Volume II [2, Table 25. Pg. 187], used visual inspection of water
elevation records for 35 lakes. Surface seiche periods and damping ratios were obtained through visual
identification of 5 consecutive oscillations with decaying amplitudes which were deemed by Endr¨
os to
be free from further forcing.
The damping ratio of the lakes was expressed by Endr¨
os through the logarithmic decrement, λ, and a
damping constant or factor of friction, β. For the mth and nth peaks in each oscillation the logarithmic
decrement is given by:
λ=log (Am)−log (An)
m−n=β.T
2(1)
Where T is the period of the oscillation in minutes and A is the amplitude of the peak. The amplitude of
the nth wave is therefore given by:
An=A1.e−β. n.T
2(2)
For the sake of clarity, in this paper, all damping is presented as a percentage of energy lost between
oscillations, expressed by:
Energy loss (%) = 100 1−1
e2π.δ (3)

where δis the damping ratio given by:
δ=β
2π/T (4)
Hence:
Energy loss (%) = 100 1−1
eβ/T (5)
Other studies of surface seiches have used frequency-domain spectral methods including power spectral
density, coherence and least-squares harmonic analyses [14] to identify natural frequencies using spectral
peaks. The associated damping ratios are often extracted using the peak-amplitude method, also known
as the half-power or quality factor method [15]. These techniques are highly dependent on the filtering
and windowing of the data during the frequency domain transformation and may struggle to extract
accurate damping estimates for lakes with close-modes. A drawback of frequency domain techniques
for use in the study of surface seiches is that they often appear intimidating to the user and there are
numerous considerations in the selection of appropriate analysis parameters, improper selection of which
may invalidate the modal parameters obtained.
In this paper we demonstrate the application of the random decrement method to extract the dominant
seiches for Lake Geneva, Switzerland, and Lake Tahoe, USA. The results obtained for the frequencies
of both seiches offer good agreement with previous measurements calculated through frequency domain
analysis. The damping ratios obtained for Lake Geneva are of a similar magnitude to the results obtained
by Endr¨
os [1], while the damping ratio obtained for the Lake Tahoe dominant surface seiche is the
first measurement of its kind. As this work represented the first application of the random decrement
technique to the study of seiches, extensive sensitivity analyses were carried out on the input parameters
for the random decrement method.
As there may be modes of vibration which are rarely excited but which result in lower frequencies
of oscillation, the term ’dominant’ seiche has been used throughout this paper to signify the lowest
frequency seiche which is observable within the data analysed. The dominant seiche observed for both
lakes within this study correlates with the lowest frequency surface seiche oscillation recorded within the
literature. Due to the low frequency of seiche oscillations all results are presented as seiche periods in
either cycles per hour or minutes as appropriate.
2. METHODOLOGY
2.1. Formation of the Random Decrement Model for Seiche Analysis
The random decrement model was formed of three components:
1. Data was filtered using a band-pass filter to remove low frequency oscillations from the dataset.
These low frequency motions are associated with diurnal tides, the Coriolis effect and seasonal
variations in water level. In all cases the periods of these motions are in excess of six hours and
are easily filtered.
2. The data was analysed using a level-crossing Random Decrement Technique (RDT) [16, 17] with
amplitude trigger levels specified as multiples of the standard deviation of the filtered dataset.
3. The Random Decrement Signature (RDS) obtained was analysed in the time domain using the
Matrix-Pencil Method [18], a time-domain curve-fitting technique which uses the identification of
response function poles as a solution to the generalised eigenvalue problem. The RDS is decom-
posed into a series of complex exponentials, from which the damping ratios and seiche periods are
calculated.

MATLAB was used for the application of the Random Decrement model to both lakes. The use of
the Matrix-Pencil Method is based on the assumption that the free-response of the lake oscillation is
comprised exclusively of sinusoidal motion with a constant damping ratio, an assumption supported by
prior observations [2, 19, 12, 7, 5].
3. CASE STUDIES
Figure 2: (a) Bathymetry of Lake Tahoe showing location of sampling site, (b) Bathymetry of Lake Geneva
showing location of sampling sites, (c) Filtered 30-day sample of water level data for Lake Tahoe. (d) Filtered
30-day sample of water level data for Lake Geneva,
3.1. Study Site 1: Lake Geneva
Situated on the border of Switzerland and France, Lake Geneva is the largest freshwater lake in Western
Europe with an average depth of 154m, a maximum depth of 309m and a surface area of 580km2. The
lake has a maximum width of 14km, with the crescent shape of the lake basin leading to northern and
southern shorelines of length 95km and 72km respectively.
The dominant surface seiches in Lake Geneva both run along the east-west axis of the lake and have
observed oscillation periods of 73.5 minutes (first dominant mode) and 36.7 minutes (second dominant
mode). These seiches are associated with the prevailing north-easterly and south-westerly winds across
the lake basin, with a small degree of seasonal variation observed in the dominant wind direction. At the
shallower and narrower western end of the lake the mean seiche amplitude is 17.5cm for the first mode
surface seiche, compared to a mean amplitude of 4cm at the eastern end of the basin [20].

Water elevation data was collected at ten minute intervals at three locations, referred to as locations 2026,
2027 and 2028 and shown on Figure 2b, between 00:00 on the 1st of January, 1974, and 23:50 on the 7th
of January, 2013.
3.2. Study Site 2: Lake Tahoe
Lake Tahoe is the largest alpine lake in North America and is known for its exceptional water clarity. It
has an average depth of 300m, a maximum recorded depth of 501m, a surface area of 490km2[8], and is
situated on the border between Nevada and California in the western United States of America. The lake
has an approximate length of 19km and 35km along its east-west and north-south axis respectively, as
shown in Figure 2a. The prevailing south-westerly winds in the lake basin lead to four dominant seiche
modes, with the first and second dominant surface seiche modes occurring along the north-south axis of
the lake and the third and fourth dominant surface seiche modes occurring along the east-west axis. The
mean surface seiche amplitude for the first dominant mode is between 2.5cm and 5cm at northern and
southern extremities of the lake [21, pp 6.9.].
The water elevation datasets for Lake Tahoe were recorded at 30 second intervals at a thermistor chain
located at Homewood, marked on Figure 2a, between 30 July 2013 00:00:00 and 6 December 2013
23:59:30, 1 January 2014 00:00:00 and 10 May 2014 23:59:30 and, 6 January 2015 00:00:00 and 15
May 2015 23:59:30. These datasets were analysed individually and as a combined dataset to allow for
the validation of stationarity assumptions across the datasets.
3.3. Results of Random Decrement Technique Analysis
Where there are multiple channels of simultaneous data available to measure a system response there
is benefit in the adoption of a linked-triggering approach for the RDT. The RDT is applied to a single
channel of data referred to as the trigger channel, with the sampling of all other data channels occurring
each time the triggering condition is met for the trigger channel [22]. For optimal identification of modal
parameters, the trigger channel should have a high signal to noise ratio at the frequency of interest [23].
The power spectral densities for each of the three Lake Geneva datasets are provided in Figure 3. The
dataset noise has been quantified based on the mean magnitude of the power spectral density at 1.85
cycles/hour and 0.79 cycles/hour, an area assessed through visual inspection of Figure 3 to fall outside
the spectral peaks associated with the seiche oscillations. The ratios for the magnitude of the power
spectral density at the first dominant seiche frequency and the mean noise levels is quantified in Table 1
for each of the datasets. Location 2028, presented in green, has the highest signal to noise ratio and has
been used as the trigger channel for the linked-triggering RDT analysis.
Table 1: Comparison of signal to noise ratios for power spectral densities for Lake Geneva, Locations 2026 to
2028. Peak signal value corresponds to obtained dominant seiche frequency. Higher estimate of noise level based
on noise level at 1.85 cycles per hour. Lower estimate of noise level based on noise level obtained at 0.79 cycles
per hour.
Background noise level Peak value- Peak Signal:
Location Higher estimate Lower estimate Dominant Seiche Noise ratio -
Mode Peak 1 - Mean
2026 1.65E-05 8.11E-06 5.65E-05 5.20
2027 1.04E-05 3.67E-06 3.39E-05 6.23
2028 3.65E-05 2.16E-05 4.51E-04 16.58
The results of the RDT analysis with a triggering value of 4 standard deviations of the location 2028
dataset is presented in Table 2 for each location. Consistent damping estimation for the seiche oscillation
at locations 2026 and 2027 was not possible due to the smaller amplitude of the seiche and the greater
levels of noise at these locations. The larger amplitude of the seiche oscillation at Location 2028 allowed
consistent results for the energy loss associated with the dominant seiche to be obtained and are provided

10-3 10-2 10-1 100
Frequency (Cycle/hour)
10-5
100
Power Spectral Density
(m2/cycle/hour)
1st dominant seiche frequency
2nd dominant seiche frequency
Location 2026
Location 2026 5%-95% CI
Location 2027
Location 2027 5%-95% CI
Location 2028
Location 2028 5%-95% CI
Figure 3: Power spectral density plot of filtered elevation data for Lake Geneva. Power spectra filtered through
band-averaging of the power spectral density with a rectangular window of length 120 samples. Confidence inter-
vals calculated using a Chi-square distribution with 120 degrees of freedom [24]
in Table 2. The seiche period obtained show strong agreement with the seiche period obtained through
spectral methods by Graf [19] of 73.5 to 74.2 minutes and the 73.5 minute period obtained by Endr¨
os [1]
through visual inspection of unprocessed time-series data collected by Forel [25].
Endr¨
os [1] reported an energy loss per oscillation of 2.96%, lower than the 4.72% obtained in this study
but of a similar magnitude.
Table 2: Dominant seiche period obtained for Locations 2026 to 2028 on Lake Geneva through use of the linked
triggering RDT analysis and energy loss for dominant seiche period for Location 2028. The triggering channel is
defined as the channel for which when the triggering condition is met data sections are taken for all three locations
(2026 to 2028) for the location specific RDSs. The triggering location used is Location 2028 with a triggering
values of four standard deviations.
Triggering Channel = Location 2028
Location 2026 Location 2027 - Location 2028 - Location 2028 - Number of Sections
Dominant Seiche - Dominant Seiche - Dominant Seiche - Location 2028 - of data included
Period (Minutes) Period (Minutes) Period (Minutes) Period (Minutes) in RDS per Location
73.63 73.80 73.78 4.72% 8,126
Presented in Figures 4a) and b) are the variation in the measured seiche period and energy loss respec-
tively with varying level-crossing trigger values. In building vibration it has been suggested that this
variation may be due to an amplitude dependant variation in the modal parameters [26, 27]. The varia-
tion in the parameters obtained at the lower levels of the triggering condition are likely due to triggering
by measurement noise, at the higher levels of the triggering condition insufficient data sections are in-
cluded within the RDS resulting in a poor estimation of the autocorrelation function. Based on Figure
4a) and b) a RDT trigger level of one standard deviation was found to be a reasonable lower bound for
consistent modal parameter estimation for both lakes, with an upper bound set as when the number of
data sections within the RDS fell below 2,000, a limit recommended by Tamura et al [28] for the use
of the RDT with building vibration measurements. Within these limits the seiche period varied by be-
tween 0.01% and 0.21% of the mean period for the three locations across the trigger level range. The
energy loss per oscillation for location 2028 varied between 4% and 4.9% across the trigger level range,
suggesting a potential increase in energy dissipation with amplitude.
Presented in Figure 5 is the RDS for Locations 2028 with a triggering value of 4 standard deviations and

with Location 2028 used as the triggering channel. In agreement with the observed increase in seiche
amplitude at the shallower western end of the lake basin [20], the seiche amplitude obtained for Location
2028 through the linked-triggering RDT is an order of magnitude larger than that for locations 2026 and
2027, illustrated in Figure 7.
Figure 4: a) Variation in dominant seiche period and b) Variation in energy loss per seiche oscillation with varying
triggering value for Location 2028. The resulting number of data sections averaged to produce the RDS is also
plotted.
0 10 20 30 40 50 60
Time (hours)
-1
-0.5
0
0.5
1
Variation in water depth
scaled based on maximum
seiche amplitude
Location 2028
Figure 5: RDS for Location 2028, where Location 2028 is utilized as the triggering channel with a triggering
value of four standard deviations. Amplitude scaled based on maximum seiche amplitude for Location 2028.
The seiche periods and damping ratio obtained for the dominant surface seiche at Lake Tahoe are pro-
vided in Table 3 alongside values obtained from the literature. No previous calculation of the damping
ratio for this seiche was found in an extensive literature review. The variation in the seiche period
obtained for various level crossing values of the RDT are presented in Figure 6. Consistent modal pa-
rameters were obtained for triggering values between one standard deviation of the dataset and when the
number of sections in the RDS fell below 2,000. A slightly greater variation is observed in the energy
loss per oscillation. Given the small magnitude of the seiche itself it seems unlikely that the physical
damping mechanism would be greatly affected by even relatively large changes in amplitude. A percent-
age variation for the 2013 to 2015 combined dataset of 3.24% for the seiche period and 14.07% for the
energy loss per oscillation is observed as the amplitude of the RDT triggering is varied from one standard
deviation to when the number of segments within the RDS falls below 2,000.

Table 3: Dominant period and associated energy loss obtained through RDT analysis for Lake Tahoe for individ-
ual annual datasets (2013 to 2015) and for all data combined sequentially into a single dataset alongside values
obtained from existing literature. The triggering value for the collection of data sections for the RDS was 1.5
standard deviations from the mean of the filtered data.
Source
Tahoe
Ichinose Environmental RDT RDT RDT RDT
et al [30] Research Center 2013 2014 2015 2013 - 2015
[21, pp 6.9.]
Period (Minutes) 11.22 11.7 11.22 11.21 11.22 11.21
Energy loss N/A N/A 1.09% 1.95% 1.79% 1.62%
Number of data sections N/A N/A 3,250 5,210 5,152 15,104
included within RDS
Figure 6: Comparison of seiche period (crosses and squares) extracted for Lake Tahoe, 2013 to 2015 water level
data, using RDT, and number of data sections in the RDT analysis (circles) for varying trigger values. Dashed line
corresponds to limit of 2,000 data sections proposed by Tamura et al [28] for the extraction of meaningful results.
Once the number of data sections included within the RDS falls below 2,000, the obtained damping ratio begins to
display random behaviour. Note: The following data points fall outside the plotted data range - 2013: 0.25 and 2
to 4 standard deviations used as trigger. 2014: 4 standard deviations used as trigger. 2015: 0.25, 2.25, 2.5 and 3.25
standard deviations used as trigger.
3.4. Lake Geneva mode shape
An advantage of the use of the linked triggering RDT for the Lake Geneva analysis is that the mode shape
of the seiche can be approximated, illustrated in Figure 7. The phase difference and relative magnitude
of the seiches is retained in the RDSs produced for all locations, as shown in Figure 8. At each time
step a straight line is used to connect the surface elevation at each location, producing an estimate of the
mode shape. This illustrates the greater seiche amplitude at the narrow, shallow western end of the lake
basin, with a node located between location 2028 and 2027. As it is not expected that the mode shape is
longitudinally symmetric, due to the variation of the lake bathymetry, this allows us to ascertain that the
node location lies much closer to location 2027 than 2028, with locations 2026 and 2027 being in-phase
with each other and in perfect anti-phase with location 2028.
4. DISCUSSION
4.1. Seiche Period
As the water level of Lake Geneva has been artificially maintained since 1885 [29] it would be expected
that there would be little variation in seiche observations, as supported by the agreement in the seiche
periods obtained in this study and in existing literature, presented in Table 2.
The seiche periods obtained in this study for the separate yearly data sets for Lake Tahoe give a mean pe-

0 20 40 60
Distance from western end of lake (km)
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Amplitude (m)
Figure 7: Linear approximation of Lake Geneva
mode shape created using one oscillation (80 minutes)
of the RDSs for locations 2026 (71km), 2027 (49km)
and 2028 (4km). Locations of data collection marked
in red. All distances reported based on centre line of
lake.
0 5 10 15 20
Time (hrs)
-1
-0.5
0
0.5
1
Normalised Amplitude
[RDS/Max RDS]
Location 2026
Location 2027
Location 2028
Figure 8: Normalised RDSs for the three Lake
Geneva datasets (2026, 2027 and 2028) illustrating
phase relationship.
riod of 11.29 minutes, with a coefficient of cariation of 1.06% across the yearly datasets. Values of 11.22
minutes and 11.7 minutes were recorded by Ichinose et al [30] and the Tahoe Environmental Research
Center [21] respectively showing an excellent agreement between the methods used, with prior results
falling within the range of values obtained through the RDT. Coefficients of variation of 0.13%, 0.01%
and 3.19% for the 2013 to 2015 datasets respectively were recorded showing low levels of sensitivity to
the seiche period obtained for triggering conditions between 1 standard deviation and when the number
of data sections included within the RDS falls below 2,000.
4.2. Damping Ratios
The damping ratios obtained for both lakes are more variable across both the range of triggering values
and the individual datasets than the seiche periods. The energy loss for the dominant seiche obtained from
the Lake Tahoe combined dataset for a triggering value of 1.5 standard deviations was 1.96%, with values
for the individual annual datasets for the years 2013 to 2015 being 1.09%, 1.95% and 1.79% respectively.
These values of energy loss obtained for the individual yearly datasets are highly sensitive to the initial
triggering condition specified, due to the relatively short length of the datasets. As fewer data sections are
included within the RDS for the individual yearly datasets the estimation of the autocorrelation function
is likely to be poorer. For the combined dataset greater averaging has occurred and the resulting RDS is
a better approximation of the autocorrelation function allowing more accurate damping estimation. This
is illustrated by the reduced variation in the energy loss estimation for a wider range of initial triggering
conditions for the combined (longer) dataset. It is expected that this variation would continue to decrease
if a longer dataset were available and further data sections are included within the RDS.
Endr¨
os [1] produced damping estimates for several lakes through visual inspection of water elevation
records, provided in Figure 9a) and b) for the energy loss per oscillation versus the maximum and mean
lake depth respectively. Comparison of the results for Lakes Geneva and Tahoe, presented as red circles
for Lake Tahoe and red triangles for Lake Geneva, supports the overall trend of lakes with greater depths
having lower energy loss than shallower lakes.

Figure 9: a) Maximum lake depth versus energy lost to damping per oscillation. b) Mean lake depth versus energy
lost to damping per oscillation. RDT results for Lake Tahoe (Table 2) and Lake Geneva (Table 3) presented as
a red circle and triangle respectively. All other values taken from Endr¨
os [1]. Error bars for Lake Tahoe energy
loss show the range of results obtained across the three annual datasets used within the RDT analysis. Error bars
for Lake Geneva energy loss show the range of results obtained across the three locations used within the RDT
analysis.
4.3. Sensitivity Analyses
As this study represents the application of the RDT to a new area which is quite different from that which
it was originally developed for, extensive sensitivity analyses of the results were carried out. Alongside
providing verification on the suitability of the RDT for the study of seiches this has a wider use as
providing credence to some of the assumptions which underlies its use in mechanical and structural
operational modal analysis.
As would be expected the filter cut-off values used in the bandpass filtering of the results prior to ap-
plication of the RDT had little effect on the results obtained while the low and high pass filter cut-off
values were higher and lower respectively than the seiche period of interest. The length of the RDS may
be selected by an iterative process; if the RDS is too short the damping estimates produced are more
sensitive to the trigger level selected. When the RDS length becomes too great the modal parameter
extraction is dominated by the background noise in the latter parts of the signal, leading to spurious
damping estimates. It is the trigger value which has the greatest impact on the results obtained. For both
lakes the limit of 2,000 data segments within the RDS recommended by Tamura et al [28] and Chang
[31] provides a good upper bound for the maximum amplitude which may be used as a trigger level. The
lower limit of the trigger value for both lakes was close to one standard deviation of the mean of the
dataset; RDSs triggered at amplitudes lower than this value were found to contain greater number of data
segments triggered by noise.
5. CHALLENGES AND LIMITATIONS OF THE RANDOM DECREMENT TECHNIQUE
A key drawback of the the use of the RDT for seiche analysis is the requirement for a large dataset. The
size of this dataset is dependant on the frequency with which seiche oscillations are induced in the lake.

There have been conflicting reports on how many data sections are required to be averaged to form the
RDS for accurate damping and frequency estimation, Yang et al [32] set a minimum of between 400 and
500, while Tamura et al [28] set the lower limit as 2,000 [33] for frequency and damping estimation.
Chang [31] recommended a lower limit of 2,000 data sections for frequency and damping estimation
and 1,000 for frequency estimation only. The sensitivity analyses carried out for this study support the
limit of 2,000 data sections per RDS for damping estimation of surface seiches, however as illustrated by
Figure 10, based on data from Location 2028, Lake Geneva, the estimation of the frequency of surface
seiches may be carried out with far fewer data segments. The RDS visibly deviates from the constant
exponential decay as the number of data sections included within the RDS reduced due to the presence
of single data sections dominating the RDS and insufficient averaging of the system forcing after the
triggering condition. Kareem and Gurley [33] has presented a full analysis of the effect of the number of
data sections included in the RDS.
0 10 20 30 40 50 60
Time (hours)
-0.1
-0.05
0
0.05
0.1
Variation in water depth (Metres)
Standard deviation trigger = 14
Number of data sections = 18
Standard deviation trigger = 10
Number of data sections = 138
Standard deviation trigger = 4
Number of data sections = 8,126
Figure 10: Comparison of signal clarity of RDS with varying triggering values and associated numbers of data
sections included within the RDS extracted for Lake Geneva, Location 2028. Triggering values defined as multiple
of standard deviation of filtered elevation data.
For both lakes analysed within this study, the periods of other surface seiche modes were extracted.
These results were more highly sensitive to the input conditions and the lower order seiches occurred
less frequently within the dataset than the dominant seiche modes. The periods of these seiches may be
extracted with little difficulty but their associated damping ratios are more difficult to obtain due to the
higher amplitude, lower frequency seiche mode dominating the RDS. Potential methods for extracting the
damping ratios for these lower order modes could be decomposing the signal into its modal components
by using a signal decomposition approach such as that presented by Chen and Wang [34], and excluding
all but the mode of interest, before application of the RDT, or through defining a secondary stage to
the RDT model by which the power spectra of each data section is analysed to identify the dominant
frequency and excluding it from the RDS if this corresponds to the frequency of the dominant surface
seiche. Similar approaches could be used in larger bodies of water where the seiche frequency approaches
the frequency of the diurnal-tides. However, as the diurnal tides impact on the water surface elevation is
highly consistent and predictable, and has low levels of damping, this could be easily identified during
the modal analysis of the RDS and the RDS reconstructed without this oscillation for further analysis.
It should be noted that for the application of the RDT to semi-enclosed water bodies such as bay, fjords
and harbours, the damping ratio extracted will not represent the damping of the seiche itself but the
summation of energy added and lost at the open-boundary (seaward boundary) and the friction damping
of the seiche itself.

6. CONCLUSION
Techniques developed for operational modal analysis may bring benefit to a wide range of fields beyond
mechanical and civil dynamics due to the prevalence of oscillatory motion in nature. Despite the rapid
advances in operational modal analysis techniques there has been little uptake of some methods in other
fields. Time-domain modal analysis techniques may be of special interest in this regard due to the
perception of complexity which persists with those unfamiliar with frequency domain analysis. This
study has applied the random decrement technique to a new field and proved its validity and suitability
for long-term data analysis and with extremely low frequency oscillations. The results for the dominant
seiches of both Lake Geneva and Lake Tahoe have shown excellent agreement with published literature,
with the damping ratio for Lake Tahoe representing the first measurement of its kind and the damping
ratio for Lake Geneva being of a similar magnitude to that obtained by Endr¨
os [1] through pain-staking
visual analysis of water elevation data. Both sets of results are in line with the general trend observed
by Endr¨
os [1] for decreasing damping ratios with water depth, with the RDT offering a method through
which modal analysis of existing water elevation data may be rapidly carried out.
ACKNOWLEDGEMENTS
The authors would like to thank the Swiss Federal Office for the Environment for the raw Lake Geneva
data and Heather Sprague of University of California at Davis for providing the raw Lake Tahoe data. The
processed data used here for the RDT analysis can be found at https://doi.org/10.15125/BATH-00504.
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