HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-20-3411-2016Technical note: Fourier approach for estimating the thermal attributes of streamsRyoMasahiromasahiroryo@gmail.comhttps://orcid.org/0000-0002-5271-3446LeysMarieRobinsonChristopher T.Department of Aquatic Ecology, EAWAG, 8600 Duebendorf, SwitzerlandDepartment of Civil Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, 152-0033 Tokyo, JapanInstitute of Integrative Biology, ETH-Zürich, 8092 Zürich, SwitzerlandMasahiro Ryo (masahiroryo@gmail.com)24August20162083411341815May201623May20163August20169August2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/20/3411/2016/hess-20-3411-2016.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/20/3411/2016/hess-20-3411-2016.pdf
Temperature models that directly predict
ecologically important thermal attributes across spatiotemporal scales are
still poorly developed. This study developed an analytical method based on
Fourier analysis to estimate seasonal and diel periodicities, as well as
irregularities in stream temperature, at data-poor sites. The method
extrapolates thermal attributes from highly resolved temperature data at a
reference site to the data-poor sites on the assumption of spatial
autocorrelation. We first quantified the thermal attributes of a glacier-fed
stream in the Swiss Alps using 2 years of hourly recorded temperature. Our
approach decomposed stream temperature into its average temperature of 3.8 ∘C,
a diel periodicity of 4.9 ∘C, seasonal periodicity
spanning 7.5 ∘C, and the remaining irregularity (variance) with
an average of 0.0 ∘C but spanning 9.7 ∘C. These
attributes were used to estimate thermal characteristics at upstream sites
where temperatures were measured monthly, and we found that a diel
periodicity and the variance strongly contributed to the variability at the
sites. We evaluated the performance of our predictive mechanism and found
that our approach can reasonably estimate periodic components and extremes.
We could also estimate the variability in irregularity, which cannot be
represented by other techniques that assume a linear relationship in
temperature variabilities between sites. The results confirm that spatially
extrapolating thermal attributes based on Fourier analysis can predict
thermal characteristics at a data-poor site. The R scripts used in this
study are available in the Supplement.
Introduction
Temperature is a fundamental determinant of physical and biogeochemical
patterns and processes in ecosystems. Organisms respond to temperature via
different adaptations and distinct life history strategies across various
spatial and temporal dimensions (Cossins and Bowler, 1987). More broadly,
global biodiversity has tracked external temperature transitions over
millennia (Mayhew et al., 2012). In the Anthropocene, unprecedented rapid
transitions in thermal regimes owing to changes in land use and climate are
a global concern (IPCC, 2014). Consequently, adequate characterization of
thermal regimes is an important prerequisite for understanding the role of
temperature in ecosystems and for ecological conservation.
Thermal attributes (seasonal and diel periodicity patterns and irregular
extremes), as representations of variability, have attracted a growing
interest in recent climate-relevant ecological studies (e.g., Thompson et
al., 2013). The combination of seasonal and diel periodicity cycles in
temperature promotes diverse behaviors and spatial distributions of
exothermic organisms according to respective life history strategies and
life cycle stages (e.g., Vannote et al., 1980). In contrast, irregular
extremes such as heatwaves can induce physiological exposure and
vulnerability (Paaijmans et al., 2013) as well as causing abrupt shifts in
biogeochemical processes (Frank et al., 2015) and ecological assemblages
(Thompson et al., 2013). Hence, the variability in thermal periodicities and
irregular extremes need to be distinguished from average temperatures.
In freshwaters, ecological responses to thermal regime shifts may be less
understood than in marine and terrestrial ecosystems (Thompson et al.,
2013), even though freshwater biodiversity has experienced major decreases
in recent decades (WWF, 2014). Progress in understanding response patterns
has been delayed partially because the quantification of thermal attributes
is difficult for running waters. Physical disturbance in streams can be
considerable due to periodic flow pulses (Poff et al., 1997). Particularly
where flashy flows carry debris such as driftwoods or mobilize gravel
riverbeds, e.g., braided rivers, the installation of data loggers for
long-term time series monitoring can be logistically difficult. For natural
floodplains acting as biodiversity hot spots (Tockner and Stanford, 2002),
frequent time series temperature data (daily or sub-daily) recorded
throughout a year or longer are difficult to acquire because the floodplain
mosaic changes frequently and dramatically (Van der Nat et al., 2003). In
these cases, researchers often rely on spot measures of temperature or the
use of reference temperature time series at stations along the streamline.
Both approaches have caveats (a lack of time series information or likely
bias in the data) when estimating the thermal attributes at a data-poor
site, thereby limiting understanding of the ecological consequences in
freshwaters. Estimating thermal stream attributes using both
spot measurements at study sites and time series measures at the nearest
hydrological station would likely be a more robust approach.
There is still much room for improvement of statistical models for stream
temperature (see review in Benyahya et al., 2007). Regression models employ
correlative relationships with air temperature (e.g., Pilgrim et al., 1998)
and streamflow (Webb et al., 2003), but correlative approaches considering
water temperature at a nearby hydrological station along the streamline have
not been implemented to date. Autoregressive models take into account the
autocorrelation structure within water temperature time series data and also
the correlation with external variables (Kothandaraman, 1971; Cluis, 1972;
Long, 1972). To deal with seasonally changing parameters in regression and
autoregressive models, periodic autoregressive models were introduced
(Benyahya et al., 2007). However, temperature patterns at multiple temporal
scales, particularly the combination of seasonal and diel periodicity
patterns, are still rarely considered (Steel and Lange, 2007). Moreover,
temperature models that directly predict ecologically important thermal
attributes as response variables do not exist presently, as many models
focus solely on temperature numeric values at a given time. Considerable
error and bias in ecologically relevant attributes can arise if they are
calculated using a modeled time series environmental condition (e.g.,
hydrologic indices calculated from a simulated river discharge; Ryo et al.,
2015). Directly estimating ecologically relevant thermal attributes,
therefore, is required for reliably predicting associated ecological
responses.
Fourier analysis (Fourier, 1878) is well suited for analyzing combined
multi-temporal patterns and predicting thermal attributes. Unfortunately,
the use of Fourier analysis for assessing stream temperature patterns has
slowed since its early emergence in 1970 (Kothandaraman, 1971; Cluis, 1972;
Long, 1972; but see Maheu et al., 2015). This is surprising given the high
potential for Fourier techniques to detect and describe periodicity at
multiple scales in time series data, and water temperature data in
particular. Here, we investigate the application of Fourier analysis to the
assessment of thermal patterns in running waters.
We developed an analytical method to estimating average, seasonal, and diel
periodicities in stream temperature, as well as irregular extremes at
data-poor sites (i.e., spot measures) using Fourier analysis. We first
quantified these thermal attributes with 2 years of hourly recorded
time series temperature data from an Alpine glacial-fed stream in Val Roseg,
Switzerland (see Ward and Uehlinger (2003) for a synthesis of research
conducted in this catchment). Using those results, we predicted thermal
patterns at sites along the same stream, where monthly spot measures of
temperature were taken during the same 2 years. We compared the performance
of the method with that of a linear regression model to underscore the
advantages.
MethodsCompositional variables in stream temperature
We assume that hourly stream temperature T(t) at a given time t is composed of its
long-term average value T¯, a seasonal periodicity pattern S(t), and diel
periodicity pattern D(t). The seasonal and diel periodicity patterns are driven
by meteorological (e.g., solar radiation and precipitation) and hydrological
(e.g., discharge and snowmelt/ice melt) conditions. The remaining unexplained
component of variance in the temperature time series results from multiple
external factors such as sub-daily changes in weather conditions and a
week-long heat waves: we call this component an irregularity ε(t). A distinctive high/low value in the irregularity indicates thermal
extremes that strongly disturb the periodicity patterns in temperature.
Consequently, hourly stream temperature is expressed as
Tt=T¯+St+Dt+ε(t).
Fourier analysis for temperature decomposition
Fourier transformation converts a function of time T(t) into a function of
frequency G(f) by transforming time series data into a sum of trigonometric
curves. Both forward (Eq. 2) and backward transformations (Eq. 3) are identically
reversible:
Gf=∫-∞∞T(t)e-i2πftdt,Tt=∫-∞∞G(f)ei2πftdf.
As measured stream temperature is a discrete variable, we used the fast
Fourier transformation algorithm to perform the transforms. The algorithm
searches for a single solution to identically explain a time series variable
T(t) by summing the trigonometric curves (ei2πft) of different frequency
f amplified by a corresponding spectral intensity G(f):
Gf=∑t=0N-1T(t)e-i2πft,Tt=1N∑t=0N-1G(f)ei2πft,
where N is the length of the time series (rounded to the nearest power of 2)
and equivalent to the number of summed curves. As seen in Eq. (5), the
time series variable, hourly stream temperature T(t), is formulated with summed
trigonometric curves (with frequencies f=0,1,2,…,N-1) with
magnitudes amplified by a corresponding spectral intensity G(f). Frequencies
with high spectral intensities are the most important contributors to
variability of the time series. Stable patterns often dominate the variance
of the time series and result in strong spectral intensities at frequencies
close to 24 h and 1 year.
As the time series temperature T(t) is formulated by Eqs. (1) and (5), we
consider that all of the terms in Eq. (1) result from a sum of a subset of
the terms in Eq. (5). Although our intention is to explain time series
temperature with the four components above, Fourier analysis decomposes it
into N components based on a series of trigonometric curves. Terms in Eq. (5),
therefore, need to be “classified” as belonging to the terms in Eq. (1).
The mismatch in the number of components requires summing some of the
trigonometric curves to best represent seasonal and diel periodicity
patterns. However, it is unknown which curves at what frequencies and
spectral intensities are required to sufficiently express these periodicity
patterns. For selecting curves to be summed, we considered the condition
that the seasonal and diel periodicities consist of their period lengths
(inverse of frequency, 1/f) within the ranges of 1–365 days and 1–24 h,
respectively. Moreover, we introduced a threshold value in spectral
intensity in order to select only the dominant components as well as to avoid
mixing noise with the periodicity patterns. The irregular component accounts
for all frequencies with a spectral intensity below the (a) threshold value.
Consequently, the shape of functions S(t) and D(t) depends on a spectral threshold
value. We set the threshold value at 0.1 ∘C, a minimal unit of
temperature measurement in the case study system (see Sect. 2.4). The
analysis was performed using the fast Fourier transform (fft) function of the
“stats” library in R 3.1.2 (R Core Team, 2014).
Extrapolation to spot-measured temperature data
Assuming that thermal attributes are autocorrelated in space along the
river continuum, we can extrapolate time series data from reference
locations to spot-measured data at sites along the same stream. For
simplicity, we do not include external information (e.g., discharge and air
temperature). After decomposing the time series temperature T0 as Eq. (1)
at a reference site, the decomposed factors are used to estimate
temperature Ta at spot-measured site A along the same stream network. We
assume that the four temperature components in Eq. (1) are linearly
correlated between sites when other factors (e.g., major tributary or
groundwater inputs) affecting stream temperature along the network are low
enough to maintain the spatial-autocorrelation temperature patterns between
sites. The temperature at site A is formulated as
Tat=T¯a+Sat+Dat+εa(t)=β1T¯0+β2S0t+β3D0t+β4ε0(t)+β5,
where coefficients β1-4 are the weighting parameters for each
component (> 0) and β5 is a parameter to
adjust systematic bias. The following procedures are performed to estimate
each parameter. First, linear regression between Ta and T0
corresponding to temperatures at the measurement time of Ta is conducted
to estimate β1 (slope) and β5 (intercept). Second,
β2 and β3 are estimated by minimizing the mean
square error value based on the linear regression between an estimated
β4ε0=Ta-β1T¯0+β2S0t+β3D0t+β5 and εa at the corresponding measurement times.
Third, β4 is numerically estimated similarly to the second
step based on Ta and the estimated Ta. Note that this approach will
require a reasonably high density of spot measurements, covering the diel
range – ideally including minimum and maximum – in different seasons.
To highlight the benefits of the extrapolation method, we compared the
component extrapolation approach to a linear regression model that simply
extrapolates time series temperature based on the linear regression between
Ta and T0. Importantly, if the coefficients β1-4
have the same value as the slope of the linear regression and β5 is the intercept, the Fourier approach is equivalent to the
linear regression model. The R scripts used in this study are available in
the Supplement.
Location of the Val Roseg catchment in Switzerland. Study
and reference (i.e., Pontresina hydrological station) sites are indicated by
black dots and labeled A and B, and R, respectively. Glaciers are shown as
shaded areas with the pro-glacial lake shown below the Roseg Glacier
(modified from Uehlinger et al., 2003).
The case study: an Alpine glacier-fed stream (Val Roseg,
Switzerland)
The method was applied in the Roseg catchment, an Alpine valley located in
the Bernina Massif of the Swiss Alps (Fig. 1). The catchment area is 66.5 km2 and ca. 30 %
glaciated (Swiss National Hydrological and
Geological Survey (OFEV); year of record 2010). Elevations range from 1766
to 4049 m a.s.l. The Roseg River is fed by meltwaters of the Tschierva and
Roseg glaciers. The Roseg glacial runoff first drains into the pro-glacial
lake Roseg before merging with the flow from the Tschierva Glacier (Fig. 1).
The thermal attributes of the glacial meltwaters and runoff from the lake
strongly influence the seasonal and diel periodic thermal patterns in this
river. Mean annual discharge at the end of the catchment was 2.8 m3 s-1
(discharge record averaged for 1955–2013), and daily discharge
ranged from 3.3 to 19.6 m3 s-1 in July and August and from 0.2 to
2.0 m3 s-1 between November and March 2013 (OFEV). The study
system comprises (i) a long proglacial reach below the Tschierva Glacier,
exhibiting extremely low temperatures due to glacial runoff (kryal); (ii) a
single-thread channel downstream of the confluence of the proglacial reach
and the Roseg lake outlet; (iii) a complex braided floodplain; and (iv) a
constrained reach extending to the end of the catchment where the Pontresina
hydrological station is located (see Tockner et al. (1997) and Uehlinger et al.
(2003) for a detailed description).
Hourly time series temperature was recorded at a reference site, the
Pontresina hydrological station (site R in Fig. 1: 46∘29′23.6′′ N,
09∘53′53.3′′ E; 1766 m a.s.l.) in 2012 and 2013 (Fig. 2a: provided
by OFEV). Spot-measured water temperature was taken monthly at two sites,
one located within the proglacial reach (site A: 46∘24′38.3′′ N,
09∘51′31.2′′ E; 2106 m a.s.l.) and one below the lake outlet
confluence reach (site B: 46∘25′05.9′′ N, 09∘51′27.1′′ E;
2054 m a.s.l.) (Fig. 1). For both sites, stream temperature was measured
using a conductivity meter (WTW LF323, Weilheim, Germany) at different daily
times on each visit from April to October 2012 and 2013 (in total 14 times).
Hourly time series temperature at the reference site
decomposed using Fourier decomposition analysis. (a) The observed record
Ta in 2012–2013 is decomposed into an average value (a), seasonal
periodicity pattern Sa(b), diel periodicity pattern Da(c), and
the irregularity εa(d). The comparison between the
observed and measured temperature excluding εa at the
reference site (e).
ResultsThermal attributes at the reference site
By converting the time series temperature data at the reference site R (Fig. 2a)
to the frequency domain, we identified frequency ranges with high
spectral power (Fig. 3). Nine trigonometric curves exceeded the threshold of
0.1 ∘C (Table 1). Based on period length (Table 1), these curves
were allocated to seasonal and diel components: three curves for seasonal
periodicity patterns (Fig. 2b) and six curves for diel periodicity patterns
(Fig. 2c). The time series temperature at the reference site R (Fig. 2a) was
thus decomposed into an average of 3.8 ∘C, a seasonal cycle
spanning 7.5 ∘C (Fig. 2b), a diel cycle spanning 4.9 ∘C
(Fig. 2c), and an irregularity spanning 9.7 ∘C with an average
of 0.0 ∘C and standard deviation of 0.92 ∘C (Fig. 2d).
Hourly stream temperatures excluding the irregularity (i.e.,
T¯+St+Dt; Fig. 2e) explained 92 % of
the data variability (r2), indicating a successful decomposition of the
time series data and a high reliability of the approach to characterize
stream temperature components at the reference site. The irregularity had a
normal distribution (no inferred bias), indicating that the seasonal and
diel periodicity patterns were extracted accurately from the original
time series data.
Frequency and spectral intensity of trigonometric curves
converted from the 2 years of temperature data at the reference site.
Trigonometric curves whose spectral intensity was higher than the threshold
value are found in Table 1.
Frequency f, period 1/f, and spectral intensity G(f) of
trigonometric curves (threshold of G(f) > 0.1) composing the following
thermal attributes: average
T¯, seasonal periodicity pattern S(t), and diel periodicity pattern D(t).
Attributef1/fG(f)[Hz][h][days][∘C]T¯0–3.78S(t)287603652.01443801830.11629201220.19D(t)72824.110.277302410.5973223.910.2873423.910.10145812.00.50.121460120.50.21Estimating time series temperature for spot-measured stream
sites
Temperature at site A was approximately twofold lower than at site B during
the study period. Temperature at site A had an average of 2.3 ∘C
and spanned 0.4–4.8 ∘C; temperature at site B had an average of
5.4 ∘C and spanned 0.8–9.2 ∘C. Temperatures at site A
and B were linearly correlated to temperature at the reference site R
(r2=0.60 and 0.92, respectively; Fig. 4). This result indicates that
spatial autocorrelation of the thermal attributes exists between them, even
though a thermal influence from the pro-glacial lake differentiates the
thermal patterns between sites A and B (Fig. 1).
Linear regression of stream temperature at site A and B
with the reference site temperature R.
For Eq. (7), the proposed method estimated that the weighting parameters at
site A (β1-4={0.32,0.32,0.50,0.70}) showed higher heterogeneity in the relative
contributions of the components than those at site B (β1-4={0.97,0.97,0.99,0.93}). For site A, the
relative contribution of diel variability and irregularity to the estimated
temperature (β3=0.50, β4=0.70) was
higher than the average and seasonal variability (β1=β2=0.32). This result indicates that our approach can
accurately estimate periodic components and extremes, including the
variability in irregularity that cannot be represented by linear regression
focusing on an average estimate. At site B, the parameter composition was
less variable (0.93–0.99), and therefore the performance of the
decomposition approach was similar to the performance of the linear model.
Using the developed method, the temperature at site A was estimated better
than the estimated temperature using linear regression (r2=0.66
and 0.60, respectively). For site B, both approaches performed equivalently
(r2=0.92). These differences between sites A and B indicate that the
thermal attributes at site A are more different from the reference site R
than those at site B. The thermal attributes at site A are not affected by a
thermal effluence from the pro-glacial lake (Fig. 1).
Based on the obtained parameters, time series temperatures at sites A and B
were inferred (Fig. 5). At site A, due to relatively high contributions of
diel cycles and irregularities (β3=0.50, β4=0.70),
higher hourly variability was estimated by the proposed
approach than the linear regression (Fig. 5a). The maximum values estimated
by the proposed approach and linear regression were clearly distinct at 5.9 and
4.0 ∘C, respectively, while an average of 1.1 ∘C was equivalent for both estimates. The linear model clearly
underestimated (< 4.0 ∘C) the high temperatures recorded
at 4.4 ∘C on 30 May 2012 and 4.8 ∘C on 5 June 2013. In
contrast, the proposed approach inferred a possibility of temperature
reaching 5.9 ∘C, having a 97.5 % percentile value of 4.0 ∘C.
This difference between approaches indicates the ability of
the proposed approach to estimate extreme thermal pulses and their
occurrence probability.
Hourly stream temperatures estimated using Fourier
decomposition (red) and linear regression (purple) at site A and B (plots
(a) and (b), respectively) and temperature at the reference site (grey).
Horizontal red line represents the maximum temperature recorded from the
monthly spot measures (4.8 ∘C on 5 June 2013 at site A, and 7.9 ∘C
on 9 July 2013 at site B).
Discussion
Currently, no stream temperature models explicitly predict ecologically
important thermal attributes (seasonal and diel periodicity patterns and
irregular extremes) because of the difficulty in capturing the combined
patterns at multiple temporal scales in stream environments (Benyahya et
al., 2007). This study developed a regression approach that predicts these
thermal attributes at data-poor sites based on the pre-analysis of
time series temperature data at a data-rich reference site along the stream.
The method merges a Fourier transformation technique into a linear
regression model to better represent periodic patterns at multiple temporal
scales. The approach could estimate the relative composition of thermal
attributes from a limited number of spot-measured data (see Eq. 7), while
linear regression weighted the composition equally. The results emphasized
the significance of developing further ecologically based thermal prediction
models, aiming at deeper understandings in ecological responses to thermal
attributes (Paaijmans et al., 2013; Thompson et al., 2013; Frank et al.,
2015).
The developed prediction method confirmed its potential to evaluate the
relative contribution of thermal attributes at a data-poor site. The method
is somewhat comparable with multiple linear regressions (Y=β0+β1X1+…) in terms of the
assumption that the thermal attributes are independent of each other and
autocorrelated between sites, and their relative contributions change
according to location. The unique difference is that the developed method
can directly use the periodicity patterns and irregularities for prediction
as Eq. (7). This type of model was not addressed in a recent review on
temperature models (Benyahya et al., 2007). As the statistical expression of
our approach is linear (Eqs. 1 and 6), it can be easily coupled with
approaches in the review; i.e., using other regression models employing air
temperature (e.g., Pilgrim et al., 1998) and streamflow (Webb et al., 2003).
Adding such information has the potential to increase accuracy, especially
if these factors contain unique information that is unexplained by the
spatial correlation of water temperature between sites. For example, if
discharge represents a volume of snowmelt water that can influence the
correlative relationship of water temperature between sites, inclusion of
discharge into the model's structure would increase accuracy.
The advantage of Fourier decomposition analysis at multiple temporal scales
is the ability to evaluate the possible range maxima for each thermal
attribute of time series data (Fig. 2). Fourier analysis clearly detected
strong periodicity patterns in time series temperature data (Fig. 3) and
represented them with a small number of trigonometric curves (Table 1). By
doing so, we found that the irregularity had a broader range (9.7 ∘C)
than the spanning ranges of seasonal and diel periodicity patterns (7.5
and 4.9 ∘C, respectively) at the study stream. This
high irregularity may be an important thermal attribute in glacier-fed
streams. Further, our results clearly showed that seasonal trends existed in
the amplitudes of diel periodicities and irregularities, both having lower
variability in winter and greater variability in spring (Fig. 2c, d)
(Hopkins, 1971). Importantly, these characteristics are difficult to detect
without coupling the periodicity patterns at multiple temporal scales. By
decomposing the time series temperature data at the reference site as a
pre-analysis for prediction, this decomposition method (see Sects. 2.1 and
2.2) can be useful for quantifying thermal attributes in ecological studies.
The proposed method is highly promising towards evaluating potential changes
in thermal variability due to climate change or anthropogenic thermal
effluents in rivers and streams (Caissie, 2006; Webb et al., 2008). For
instance, environmental change could modify each component of stream
temperature with a different degree of severity such as increasing the
average, shifting the peak timing in seasonality, reducing diel variability,
or amplifying the irregularity. In addition, as thermal regimes play a key
role in aquatic ecosystem structure and functioning (e.g., environmental
niches determining community composition, biogeochemical cycles; Vannote et
al., 1980), each component of stream temperature can be used to better
understand ecosystem dynamics and responses (Thompson et al., 2013).
While demonstrating the broad applicability of this approach, we caution
over some of its limitations. First, the analysis connecting Eqs. (1) and
(7) is threshold dependent. We first estimated an appropriate range of
threshold value visually so as to capture a handful of trigonometric curves
(i.e., the three major peaks were shown in Fig. 3). Then, we compared the
patterns modeled based on some threshold values. Too-low threshold value
results in high sensitivity to noise, while too-high threshold value results
in high insensitivity to periodic patterns. We compared the performance of
the models based on a set of threshold values (0.05, 0.1, 0.2 ∘C, etc.)
and determined the threshold graphically as the value 0.1 ∘C clearly separated
periodicity patterns and irregularities (Fig. S1;
Supplement). Therefore, a threshold value must be carefully chosen, and it needs
to be evaluated whether the irregularity attribute is unbiased. Second,
although the concept is feasible, complete validation was limited due to the
low number of sample sites. To obtain more robust estimation, the model
needs additional validation as well as calibration. Third, if a systematic
shift in thermal pattern is observed during a target period, the method
needs adjustment. Our target 2-year period was short enough to neglect
long-term trends in temperature. For example, some studies using long-term
records (> 20 years) detected an interannual increasing trend in
mean temperature due to anthropogenic thermal discharge and land-use change
(Beschta and Taylor, 1988; Hostetler, 1991). In such cases, the proposed
equation in Eq. (1) must be modified because it assumes constant average
temperature over a target period for analysis; i.e., T¯ should be a
function of time as T¯(t). Lastly, the assumption of linearly related
components between the reference site and other sites is not always met in
all cases. For instance, glacier-fed and groundwater-fed streams can be
composed of different seasonal and diel periodicity patterns (Brown and
Hannah, 2008). In our target system, flow from the Roseg pro-glacial lake
differentiated thermal patterns of sites A and B (Fig. 1). Even though our
analysis confirmed that the linear assumption was apparent in this system
(r2=0.60; Fig. 4), incorporation of the thermal pattern at the lake
outlet into the analysis would certainly increase estimation accuracy.
The Supplement related to this article is available online at doi:10.5194/hess-20-3411-2016-supplement.
Acknowledgements
The study was done as part of a fellowship in the
“Young Researchers Exchange Program between Japan and Switzerland” under
the “Japanese-Swiss Science and Technology Programme” (EG 11-2015) and
supported by the Japan Society for the Promotion of Science (26-11771). The
data used in this study were collected for a Swiss National Science
Foundation project (31003A_152815). We thank the Swiss
National Hydrological and Geological Survey from the Federal Office for the
Environment (OFEV) for kindly providing water temperature data and S. Blaser
for field assistance. We are indebted to the anonymous reviewers for their
constructive reviews and comments. We thank India Mansour for proofing the
language.
Edited by: S. Thompson
Reviewed by: two anonymous referees
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