Sampling design optimization for rainfall prediction …/file/... · Sampling design optimisation...

Samplingdesign

optimisation forradar-rain

gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Sampling design optimization forrainfall prediction using a

non-stationary geostatistical model

Alexandre Wadoux1 Dick Brus2 Miguel Rico-Ramirez3

Gerard Heuvelink1

1Environmental sciences, Soil Geography and Landscape groupUniversity of Wageningen, Netherlands

2Environmental Sciences, Soil, Water and Landuse groupAlterra, Netherlands

3Civil engineering, Water and Environment Management groupUniversity of Bristol, United Kingdom

July 5, 2016

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Overview

1 Introduction

2 Model

3 Material

4 Results

5 Final remarks

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Introduction

Conventional geostatistical models assume that theproperty being monitored is the realisation of asecond-order stationary random process

Z (s) = µ + ε(s)

µ = constant

Cov(ε(s), ε(s + h)) = C(h)

if h = 0 => Cov(ε(s), ε(s)) = Var(ε(s)) = C(0)

But this is often an invalid assumption=> can be checked with exploratory analysis of

the observed data

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Expectation

Objectives...1 Account for non-stationarity in the mean and

variance of rainfall2 Optimize the sampling locations of rain gauges

for mapping rainfall over time

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Introduction

Simple solutions exist for non-stationarity

In the meanZ (s) = m(s) + ε(s)

and in the variance

Z (s) = m(s) + σ(s) · ε(s)

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Nonstationary variance model

Mean rainfall at location s

Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Multiplier for error at location sStandardized random error

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks



Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Multiplier for error at location s

Standardized random error

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks



Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Multiplier for error at location sStandardized random error

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Universal kriging for merging

In matrix notation

Z = Fβ + Gκ · ε︸︷︷︸C = diag{Gκ} · R · diag{Gκ}T is the variance-covariance matrix

Predictions at new location

z(s0) = f(s0)T β + g(s0)

T κ · ε(s0)

Prediction error variance at new location

σ2(s0) = c(0)− cT0 C−1c0︸︷︷︸

prediction error variance of the residuals

+ (f 0 − FT C−1c0)T (FT C−1F)−1f 0 − FT C−1c0)︸︷︷︸error variance of the trend

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Parameter estimation

With exponential correlogram,

r(h) = c0 + (1− c0){exp(−3h

a)}

We need to estimate Φ = [κi , c0,a], and βi

Independant of βi , Restricted loglikelihood:

`(Φ|z) = Constant − 12

ln|C| − 12

ln|XT C−1X|

− 12

yT C−1(I−Q)z

βi are estimated with GLS using REML estimates ofkappa, c0 and a.

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Case study

Illustration with a simple case, daily rainfall mappingwith radar and rain-gauge

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#

#

#

0°0'0"

0°0'0"

1°0'0"W

1°0'0"W

2°0'0"W

2°0'0"W

3°0'0"W

3°0'0"W

54°0'0"N 54°0'0"N

53°0'0"N 53°0'0"N0 5025Km

¯# Radar E Rain gauge

Study area

0°0'0"5°0'0"W

55°0'0"N

50°0'0"N

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Covariates

Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Rainfall fromradar

Distance fromradar

Previouspredicted

rainfall

fk +

Elevation

Distance fromradar

beamblockage

gl ·0 50000 100000 150000 200000 250000

0.0

0.2

0.4

0.6

0.8

1.0

h [meters]

r

Correlogram

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Covariates

Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Rainfall fromradar

Distance fromradar

Previouspredicted

rainfall

fk

+

Elevation

Distance fromradar

beamblockage

gl ·0 50000 100000 150000 200000 250000

0.0

0.2

0.4

0.6

0.8

1.0

h [meters]

r

Correlogram

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Covariates

Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Rainfall fromradar

Distance fromradar

Previouspredicted

rainfall

fk +

Elevation

Distance fromradar

beamblockage

gl ·

0 50000 100000 150000 200000 250000

0.0

0.2

0.4

0.6

0.8

1.0

h [meters]

r

Correlogram

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Covariates

Z (s) =K∑

k=0

βk fk (s) +L∑

l=0

κlgl(s) · ε(s)

Rainfall fromradar

Distance fromradar

Previouspredicted

rainfall

fk +

Elevation

Distance fromradar

beamblockage

gl ·0 50000 100000 150000 200000 250000

0.0

0.2

0.4

0.6

0.8

1.0

h [meters]

r

Correlogram

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Model calibration

Example, February 11th, 2010...

350000 400000 450000 500000

4000

0045

0000

5000

00

Mean

0

1

2

3

4

350000 400000 450000 500000

4000

0045

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Standard deviation

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

350000 400000 450000 500000

4000

0045

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5000

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Kriging prediction

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350000 400000 450000 500000

4000

0045

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Kriging variance

0.5

1.0

1.5

2.0

2.5

3.0

Parameter Estimated value Associated to

c0 0.0001278 nuggeta1 8914 range [meters]β1 -0.02205 interceptβ2 -0.1141 radar imageβ3 1.967e-05 distance from radar*radar imageβ4 0.1771 previous estimated rainfall mapκ1 0.3699 interceptκ2 4.555e-11 elevation*radar imageκ3 6.445e-06 distance from radar*radar imageκ4 1.35e-10 beam blockage*radar image

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Sampling design optimization

Minimizing the variance criterion by a random searchcalled Spatial Simulated Annealing (SSA)1.

Criterion =1T

∫ T

t=0

1|A|

∫s∈A

Var(Z (s)− Z (s))dsdt (1)

0 2000 4000 6000

5.4

5.5

5.6

5.7

Simulated annealing iterations

Crit

erio

n

1Van Groenigen, J. W., Siderius, W., and Stein, A. (1999). Constrained optimisation of soil sampling forminimisation of the kriging variance.Geoderma, 87(3):239–259

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks


400000

450000

500000

350000 400000 450000 500000

0 40km

Initial

400000

450000

500000

350000 400000 450000 500000

0 40km

Optimized

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks


400000

450000

500000

350000 400000 450000 500000

0 40km

Distance fromradar

Elevation

beamblockage

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Final remarks

Decrease of the rainfall prediction error variance isobtained by the optimized rain-gauge network

1 It pays off to place rain-gauges at locationswhere the radar imagery is inaccurate

2 Uniform distribution of rain-gauge over the studyarea is also important

Samplingdesign


gauge merging

Wadoux et al.,2016

Introduction

Model

Material

Results

Final remarks

Thank you for your attention

This project has received funding from the European Unions Seventh Framework Programme for research,technological development and demonstration under grant agreement no 607000.

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