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Wavelet Analysis of Water Quality Time Series
by
Albrecht Gnauck and Jean Duclos AlegueBrandenburg University of Technology at Cottbus
Dept. of Ecosystems and Environmental Informatics
Yearly Meeting of LTER - D
St. Oswald, March 26 - 28, 2007
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Content
1. Introduction
2. Some Remarks on Water Quality Indicators
3. Comments on Fourier Analysis of Havel River Data4. Comments on Wavelet Analysis of Havel River Data
5. MRA of
mov Reservoir Data6. Conclusions
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Introduction
The behaviour of water quality processes observed infreshwater ecosystems is an amalgam of components orprocesses operating in parallel manner not only at different
frequencies but as well at different time scales.
Answering scientific and management questions about the
processes represented by the measured data is ofteninherently linked to understanding their behaviour at differentfrequencies and time scales.
To extract as much information on climate changes aspossible from the signals, classical time series methods likecorrelation and spectral analysis or Fourier analysis as well
as modern methods like wavelet analysis can be used.
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The variables recorded can be denoted as stressors to the
system, ecological state variables or driving forces for the
freshwater ecosystem.
Driving force
Water
Temperature
Ecological
state
Chlorophyll-a
Stressor
Phosphate
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Freshwater
Issue
Indicator Indication Category
Eutrophication Chlorophyll-a
Phosphorus
Nitrogen
Secchi disktransparency
Biomass / Trophic state
Trophic state
Trophic state
Transparency / Trophicstate
Ecological state
Stressor
Stressor
Ecological state
Acidification pH Acidity, alkalinity Ecological state
Organic pollution BOD
DO
CO2
Pollution
Productivity
Respiration
Stressor
Ecological state
Stressor
Pathogenic
pollution
E. coli Fecal material Stressor
Freshwater quality problems and possible indicators
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Study Area and Indicators used(Dissolved oxygen, Chlorophyll-a, water temperature, pH)
0 10 km
Hv0120
Hv0200
Hv0190 Hk0030
Hv0170
345
TeK0030
SPK0020 SPK0010
Hv0110
Nu0120
Hv0180
Hv0130
Hv0140
Hv0160
flow direction
measuring point
POTSDAM
BRANDENBURG
Havel
Havel
Sacrow-ParetzerKanal
SchwielowSee
Templiner
See
Nuthe
Havelka
nal
Schlnitz-see
Jungfern-see
Wublitz
GroerZernsee
Trebelsee
Kleiner
ZernseeTeltow-kanal
Hv0170
Have
l
Fahrlander-see
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Time
CD
0 500 1000 1500
65
0
750
850
950
5 10 15 20 25
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Fourier analysis of water quality data
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Periodograms of water quality indicators
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Periodograms of water quality indicators
150
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0 200 400 600 800 1000 1200 1400 1600
0
50
100
150
Chlorophyll-a
Fourier 4
Fourier 4 model of chlorophyll-a
F(t) = 48.76 + 4.7cos(0.0087t)+0.47sin(0.0087t) 29.64cos(0.02t)
1.4sin(0.02t) 2.48cos(0.03t) + 1.3sin(0.03t) 20.35cos(0.04t) +
sin(0.04t)
Goodness of fitR-square: 0.68
Adjusted R-square: 0.67
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0 200 400 600 800 1000 1200 1400 1600
5
10
15
0
5
Water temperature
Fourier 3
Fourier 3 model of water temperature
F(t) = 13 + 0.39cos(0.009t) + 0.018sin(0.009t) 9.71cos(0.018t)
2.47sin(0.018t) 0.4018cos(0.027t) + 0.23sin(0.027t)
Goodness of fit
R-square: 0.95
Adjusted R-square: 0.95
H
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0 200 400 600 800 1000 1200 1400 1600
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
9.4
pH
Fourier 8
Fourier 8 model of pH
F(t) = 8.24 0.086cos(0.006t) + 0.044sin(0.006t) 0.10cos(0.012t)
0.009sin(0.012t) 0.15cos(0.018t) 0.24sin(0.018t) 0.005cos(0.024t) +
0.007sin(0.024t) + 0.05cos(0.03t) + 0.031sin(0.03t) 0.17cos(0.036t) 0.04sin(0.036t) 0.07cos(0.042t) 0.002sin(0.042t) 0.09cos(0.048t)
0.024sin(0.048t)
Goodness of fitR-square: 0.5
Adjusted R-square: 0.49
Di l d O
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0 200 400 600 800 1000 1200 1400 1600
4
6
8
10
12
14
16
Dissolved Oxygen
Fourier 7
Fourier 7 model of dissolved oxygen
F(t) = 10.23 0.13cos(0.006t) 0.75sin(0.006t) 0.41cos(0.012t)
0.52sin(0.012t) 0.26cos(0.018t) + 1.2sin(0.018t) +0.2cos(.024t) 0.27sin(0.24t) 0.38cos(.03t) 0.07sin(0.03t) -0.58cos(0.036t) +
0.53sin(0.036t) 0.16cos(0.042t) + 0.39(0.042t)
Goodness of fitR-square: 0.29
Adjusted R-square: 0.28
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Wavelet analysis of water quality data
Wavelets are mathematical tools that have proven quiteuseful for time scale based signal analysis in physicsand engineering.
The wavelet transform is a lens for decomposing asignal into components at different resolution and time
scale.
,dt)t()t(f)s,u(W s,u =
=
s
ut
s
1)t(s,u
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Multiresolutiondecomposition (MRD)
reveals the variations at
different scales denoted
by d.
Multiresolution analysis
(MRA) filters information
in the signal at differentscales represented by
a.
MRA
Scale
MRD
Scale
Freq
a1 d1 1
a2 d2 2
a3 d3 4
a4 d4 8
a5 d5 16
a6 d632
a7 d7 64
a8 d8 128
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Questions to be answered
by wavelet methods
Are the statistical variations in a given ecologicalindicator homogenous across time ? What are thetime dependent variations such as the presence oftrends ?
What is the dominant scale of variation influencingthe long term variation of the indicator ? Are thevariations from one day to the next more prominentthan the variations from one week to the next ?
How are two indicators related on a scale by scalebasis? How do they covary ? Are the time lags from
one scale to the other significantly different ?
DO-10 minutes
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DO-10 minutes
DO hourly
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DO-hourly
DO daily
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DO-daily
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Dissolved oxygen
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 04
6
8
1 0
1 2
1 4
1 6
1 8
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DO Variance
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DO-Variance
W l t i f DO CHA
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Wavelet covariance of DO-CHA
Wavelet cross-correlation of DO-CHA
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Wavelet covariance of DO WT
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Wavelet covariance of DO-WT
Wavelet cross-correlation of DO-WT
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0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 00
5 0
1 0 0
1 5 0
Chlorophyll-a
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W a v e le t v a ria n c e
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**
*
**
*
0
50
100
150
200
Wavelet Scale
L L LL L
LU
U
U
U
U
1 2 4 8 16 32
W a v e le t v a ria n c e
Wavelet covariance of CHA-pH
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Wavelet covariance of CHA pH
Wavelet covariance of CHA-DO
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Wavelet covariance of CHA DO
Wavelet covariance of CHA Water temperature
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Wavelet covariance of CHA-Water temperature
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Water temperature
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Water temperature
0 200 400 600 800 1000 1200 1400 1600 18000
5
10
15
20
25
30
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U
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* **
**
*
0.0
0.5
1.0
1.5
2.0
2.5
Wavelet Scale
L L L LL
L
U UU
U
U
U
1 2 4 8 16 32
Wavelet analysis ofmov reservoir data
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Data for water temperature, chlorophyll-a, DO,
COD and pH from 1984-2004
MRA
Scale
MRD
Scale
Freq/days
a1 d1 21
a2 d2 42 days (1.4 months)
a3 d3 84 days (2.8 months)
a4 d4 168 days (5.6 months)
Water temperature (1984-2004)
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Chlorophyll-a
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Dissolved oxygen
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COD-Cr
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pH
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Total phosphorus
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Conclusions
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Conclusions
Time series analysis methods are helpful toolsto investigate long-term ecological observations.They serve as means to understand the timeand frequency structure of measurements.
Fourier analysis gives us suitableapproximations for physical water qualityindicators but cannot follow the disturbeddynamic ecological processes which take placein freshwater bodies.
Wavelet analysis helps us to interpret thefrequency dependent long-term changes of timeseries due to environmental (or climate)
changes.
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