Sampling Design Avoiding Pitfalls in Environmental Sampling Part 1
Sampling design optimization for rainfall prediction …/file/... · Sampling design optimisation...
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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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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 s
Standardized random error
Samplingdesign
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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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#
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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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
gauge merging
Wadoux et al.,2016
Introduction
Model
Material
Results
Final remarks
Model calibration
Example, February 11th, 2010...
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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
optimisation forradar-rain
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
optimisation forradar-rain
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
optimisation forradar-rain
gauge merging
Wadoux et al.,2016
Introduction
Model
Material
Results
Final remarks
Sampling design optimization
400000
450000
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0 40km
Initial
400000
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500000
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0 40km
Optimized
Samplingdesign
optimisation forradar-rain
gauge merging
Wadoux et al.,2016
Introduction
Model
Material
Results
Final remarks
Sampling design optimization
400000
450000
500000
350000 400000 450000 500000
0 40km
Distance fromradar
Elevation
beamblockage
Samplingdesign
optimisation forradar-rain
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
optimisation forradar-rain
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.