0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Runoff

Baseflow

Seas

onali

ty0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

BFIH

OST

The dependence structure of daily UK hydrological time seriesNicholas Howden1, Tim Burt1, Fred Worrall2, François Birgand1, Sebastian Gnann1, Ross Woods1

Introduction

The presence of dependence structure in hydrological time series is bothproblematic (it prevents the use of standard statistical tools for analysis)and useful (as it indicates a pattern in the data that may be predicted bya model. The dependence structure determines several keycharacteristics of the time series all of which relate to the way in whichthe structure contributes to the variance as a function of theautocorrelation function:

We use an analytical model to represent the dependence structure (non-randomness) in hydrological time series from the UK. We can then identify the variance of these series to mainly comprise four distinct components: quick, slow, seasonal and random noise.

This analytical model allows us to:

1. Project the distribution of totals at larger timescales• forecast stochastic averages

2. Identify change-points to a level of probability• sensitive trend detection and attribution

3. Identify the information content of observations• define a time-domain alternative to the power spectrum

4. Demonstrate a robust flow decomposition technique

0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

monthly rainfall total [mm]

prob

abilit

y de

nsity

[−]

400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

x 10−3

annual rainfall total [mm]

prob

abilit

y de

nsity

[−]

annual rainfallannual distribution

months1 2 3 4 5 6 7 8 9 10 12 15 18 24 30 36 48 60 120 240

prec

ipita

tion

tota

l as

a pe

rcen

tage

of t

he e

xpec

ted

valu

e - [

%]

0

20

40

60

80

100

120

140

160

180

20095%90%75%25%10%5%

Empirical probability distribution ofprecipitation total in n consecutivemonths based on the period 1888 to1966 and on precipitation amountsassessed over the Oxford area. Thecurves show the probability of notreaching the amounts specified by theordinates.

Theoretical probability distribution ofprecipitation total in n consecutivemonths based on the dependencestructure model for precipitationamounts assessed over the Oxford area.

Example: Oxford (UK) Daily Precipitation 1827 - 2016

Analytical model of the dependence structureProject monthly and annualdistributions of P total

Forecast long-term P total distributions

Test for non-stationarity (trend) in the daily record

We can plot the cumulative deviations from mean daily rainfall (pre-1850), and look at theprobability that this has changed:

CI%⇥X̄⇤

µ⇡ ��1

⇥1� ↵

⇤⇢ �̂

µ̂

�sF⇥n⇤

n

Results show that, although there has been change, there is only an 80% probability of thisbeing significant, less than required by standard statistical tests for change.

The dependence structure tends to inflate the variance, as individualmeasurements don’t provide unique information. This is represented bythe variance inflation factor:

The variance inflation factor calculated from:

where:

Standard experimental design

Treatment vs Control

Randomized bloc design

The value of integrated hydrological and biogeochemical indicators

Experiments are designed to find which treatments lead to statistically significant differences in outcomes

Example: Wheat Yield

These experiments are monitored using integrated indicators of responses to the treatments:

Net

pro

duct

ivity

.dt

Integrated indicators tend to smooth out variabilityand amplify the mean. In the presence of shortmemory, they also lead to the variance in theintegrated indicator being normally-distributedwhich makes classical statistics more readilyapplicable.

Simple numerical tests can show howhydrological processes can be considered asintegrated indicators, which may enablemore sensitive measures of trend detection.

Example: River Thames Streamflow 1883 to 2015

Analytical model fitted to the autocorrelation of the Thamesstreamflow data:

The contribution of different component frequencies to thevariance is usually determined using the power spectral density.Here we can use our data to produce this, and produce a versionpredicted by our analytical model:

And we can do the equivalent in the time domain using our newapproach:

Application for flow decomposition

slow flow

quick flow

seas

onal

ity

1 2

The three flow componentsin UK river systems…

analytical model

empirical PSD (FFT)