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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


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)



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)




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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)



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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