probabilistic downscaling to detect regional present and

Probabilistic downscaling to detect regional present andfuture climate hazards

Sherman Lo, Ritabrata Dutta, Peter Dueben, Peter Watson

University of Warwick, ECMWF, University of Bristol

28th April 2020

Sherman Lo, Ritabrata Dutta, Peter Dueben, Peter Watson (University of Warwick, ECMWF, University of Bristol)Probabilistic downscaling to detect regional present and future climate hazards28th April 2020 1 / 55

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Funding

University of Warwick

Alan Turing Institute

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Introduction

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Introduction

Figure: Precipitation

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Introduction

We have computer simulations of the weather/climate, they are calledmodel fields.

Temperature

Precipitable water content

Humidity

Geopotential

Wind speed and velocity

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Introduction

We have computer simulations of the weather/climate, they are calledmodel fields.

Temperature

Precipitable water content

Humidity

Geopotential

Wind speed and velocity

Page 7: Probabilistic downscaling to detect regional present and

Introduction

Figure: Air temperature

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Introduction

(a) Model fields (b) Precipitation

Figure: Sample from data

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Introduction

40 years of data (1979 - 2019)

Observed precipitation (∼ 10 km)

Simulated model fields (∼ 80 km)

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Introduction

(a) Model fields (b) Precipitation

Figure: Sample from data

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Introduction

Can we use the model fields on a coarse grid to forecast the precipitationon the fine grid? This task is known as downscaling.

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Introduction

1989-01 1989-03 1989-05 1989-07 1989-09 1989-11 1990-01Time

0

5

10

15

20

25

30

35ra

infa

ll (m

m)

Figure: Forecast (orange) and observed (blue)

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Introduction

1.2± 1.5 mm

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Statistical model

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Statistical model

Occurrence of precipitation

Quantity of precipitation

Autocorrelation of precipitation

Affected by the model fields

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Statistical model

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Statistical model

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Statistical model

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Statistical model

0 5 10 15 20lag (day)

0.0

0.2

0.4

0.6

0.8

1.0au

toco

rrela

tion

London: autocorrelation of rain

Figure: Autocorrelation of precipitation

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Statistical model

We are dealing with a zero-inflated random variable.

Sometimes, theprecipitation is exactly 0 mm, sometimes it is a positive number.

We introduce the compound-Poisson.

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Statistical model

We are dealing with a zero-inflated random variable. Sometimes, theprecipitation is exactly 0 mm,

sometimes it is a positive number.

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Statistical model

We are dealing with a zero-inflated random variable. Sometimes, theprecipitation is exactly 0 mm, sometimes it is a positive number.

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Statistical model

We are dealing with a zero-inflated random variable. Sometimes, theprecipitation is exactly 0 mm, sometimes it is a positive number.

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Statistical model

Zt = number of times it rains at day t

R(i)t = amount of precipitation in a rain event at day t

Yt = amount of precipitation at day t

Yt |Zt =Zt∑i=1

R(i)t

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Statistical model

Yt |Zt =Zt∑i=1

R(i)t

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Statistical model

Yt |Zt =Zt∑i=1

R(i)t

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Statistical model

Yt |Zt =Zt∑i=1

R(i)t

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Statistical model

Zt ∼ Poisson(λt)

R(i)t ∼ Gamma(1/ωt , 1/(ωtµt))

Yt |Zt =Zt∑i=1

R(i)t ∼ Gamma(Zt/ωt , 1/(ωtµt))

Yt ∼ Compound-Poisson

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Statistical model

Zt ∼ Poisson(λt)

Yt |Zt =Zt∑i=1

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Statistical model

Zt ∼ Poisson(λt)

Yt |Zt =Zt∑i=1

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Statistical model

Zt ∼ Poisson(λt)

Yt |Zt =Zt∑i=1

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Statistical model

We make λt , µt , and ωt a function of the model fields xt.

We make Zt and Yt |Zt undergo an autoregressive and movingaverage process.

For example

λt = exp

[βλxt + φλ(lnλt−1 − kλ) + θλ

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

[βµxt + φµ(lnµt−1 − kµ) + θµ

yt−1 − Zt−1µt−1

µt−1√

Zt−1ωt−1+ kµ

]

ωt = exp [βωxt + kω]

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Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

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Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

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Statistical model

For example

λt = exp

[βλxt

+ φλ(lnλt−1 − kλ) + θλZt−1 − λt−1√

λt−1+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

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Statistical model

For example

λt = exp

[βλxt + φλ(lnλt−1 − kλ)

+ θλZt−1 − λt−1√

λt−1+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

Page 37: Probabilistic downscaling to detect regional present and

Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

Page 38: Probabilistic downscaling to detect regional present and

Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

Page 39: Probabilistic downscaling to detect regional present and

Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

Page 40: Probabilistic downscaling to detect regional present and

Statistical model

For example

λt = exp

Zt−1 − λt−1√λt−1

+ kλ

]

µt = exp

µt−1√

Zt−1ωt−1+ kµ

]

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Statistical model

Figure: Graphical model

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Statistical model

1980 1982 1984 1986 1988 1990time

0

10

20

30

40

50

60

70

rain

fall

(mm

)

(a) Time series

0 10 20 30 40 50 60 70rainfall (mm)

0

500

1000

1500

2000

2500

3000

3500

4000

cum

ulat

ive

frequ

ency

(b) Cumulative

Figure: Simulated data

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Statistical model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20lag (day)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

auto

corre

latio

n

Figure: Simulated data autocorrelation ARMA(5,5)

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Markov chain Monte Carlo

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Markov chains Monte Carlo

This model is wrong so it is important to quantify how wrong themodel is.

The problem of modelling fitting can be done using Bayesianinference to get uncertainity quantification.

This was done using Markov chains Monte Carlo (MCMC).

We use Monte Carlo which uses random numbers to solve a modelfitting problem. Flucations in solutions, (samples), reflect theuncertainity.

We use a Markov chain to draw dependent samples in such a waythey converge to the right answer.

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In Bayesian inference, we are interested in studying the posteriordistribution

posterior ∝ likelihood× prior

π(β, z1:T |x1:T , y1:T )︸︷︷︸posterior

∝ p(y1:T , z1:T |x1:T , β)︸︷︷︸likelihood

× π(β)︸︷︷︸prior

Page 51: Probabilistic downscaling to detect regional present and

In Bayesian inference, we are interested in studying the posteriordistribution

posterior ∝ likelihood× prior

π(β, z1:T |x1:T , y1:T )︸︷︷︸posterior

∝ p(y1:T , z1:T |x1:T , β)︸︷︷︸likelihood

× π(β)︸︷︷︸prior

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p(data|parameters)︸︷︷︸likelihood

p(parameters|data)︸︷︷︸posterior

∝ p(data|parameters)︸︷︷︸likelihood

× p(parameters)︸︷︷︸prior

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But the prior is subjective!

1980 1982 1984 1986 1988 1990time

0

5

10

15

20

25

30

35

rain

fall

(mm

)

(a) Sensible

1980 1982 1984 1986 1988 1990time

0

50

100

150

200

250

300

350

rain

fall

(mm

)

(b) Not sensible

Page 56: Probabilistic downscaling to detect regional present and

But the prior is subjective!

1980 1982 1984 1986 1988 1990time

0

5

10

15

20

25

30

35

rain

fall

(mm

)

(a) Sensible

1980 1982 1984 1986 1988 1990time

0

50

100

150

200

250

300

350

rain

fall

(mm

)

(b) Not sensible

Page 57: Probabilistic downscaling to detect regional present and

We use Gibbs sampling to split the posterior into two parts

Sample Zt |β,Z1:T\t , y1:T , x1:T

Sample β|y1:T ,Z1:T , x1:T

We use slice sampling and elliptical slice sampling.

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We use slice sampling to sample Zt |β,Z1:T\t , y1:T , x1:T .

yt = 0⇒ Zt = 0

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We use slice sampling to sample Zt |β,Z1:T\t , y1:T , x1:T .

yt = 0⇒ Zt = 0

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Slice sampling:

s ∼ Uniform(0, π(z(i)))

z(i+1) ∼ Uniform(z : π(z) > s)

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We use elliptical slice sampling to sample β.

It can sample a posterior witha Normal prior. For the technical:

Chain at β(i)

ν ∼ prior

s ∼ Uniform(0, posterior(β(i)))

θ ∼ Uniform(0, 2π)

β(i+1) = β(i) cos θ + ν sin θ if posterior(β(i+1)) > s, else try another θwith a smaller support containing zero

Page 64: Probabilistic downscaling to detect regional present and

We use elliptical slice sampling to sample β. It can sample a posterior witha Normal prior.

For the technical:

Chain at β(i)

ν ∼ prior

Page 65: Probabilistic downscaling to detect regional present and

We use elliptical slice sampling to sample β. It can sample a posterior witha Normal prior. For the technical:

Chain at β(i)

ν ∼ prior

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Even more technical:

Initial: [θmin, θmax] = [θ − 2π, θ]

If: θ < 0, θmin = θ

Else: θmax = θ

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0 2000 4000 6000 8000 10000Sample number

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0Po

isson

Rate

_geo

pote

ntia

l

Figure: Example of a chain

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Forecasting

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Forecasting

Simulate the future using different samples from the posterior (MCMCchain). Any variation reflect the uncertainity.

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Forecasting

1989-01 1989-03 1989-05 1989-07 1989-09 1989-11 1990-01Time

0

5

10

15

20

25

30

35ra

infa

ll (m

m)

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Forecasting

Figure: Probability precipitation > 15 mm

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Forecasting

1989-01 1989-03 1989-05 1989-07 1989-09 1989-11 1990-01time

0.0

0.1

0.2

0.3

0.4

0.5

0.6fo

reca

sted

pro

babi

lity

of >

15

mm

of r

ain

Figure: Probability precipitation > 15 mm

Page 73: Probabilistic downscaling to detect regional present and

Forecasting

0.0 0.2 0.4 0.6 0.8 1.0false positive rate

0.0

0.2

0.4

0.6

0.8

1.0tru

e po

sitiv

e ra

te

0 mm5 mm10 mm15 mm

Figure: ROC curve

Downscaling

We interpolate the model fields for now

(a) Coarse grid (b) Fine grid

Figure: Interpolation

Downscaling

We impose a prior on β where neighbouring locations with similartopography have similar values

β1β2...βN

∼ N(0, τ−1K

)

where [K]i ,j = exp[−ν

2(wi − wj)

2]

and wi is the topography

Downscaling

We impose a prior on β where neighbouring locations with similartopography have similar values

β1β2...βN

∼ N(0, τ−1K

)

where [K]i ,j = exp[−ν

2(wi − wj)

2]

and wi is the topography

Downscaling

1_poisson_rate

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Figure: Simulation from prior

Downscaling

0 2000 4000 6000 8000 10000sample number

0.7

0.6

0.5

0.4

0_0_

Poiss

onRa

te_c

onst

Figure: MCMC chains

Downscaling

Figure: Forecast at Wales

Downscaling

Instead of interpolating the model fields, we can learn the model fields onthe coarse grid. We use a Gaussian process.

Downscaling

x ′i ,t = model fields at time t, location i on coarse grid

xj ,t = model fields at time t, location j on fine gridx ′1,tx ′2,t

...x ′M,t

∼ N(0, τ−1K)

Downscaling

We can sample the model fields on the fine grid given:

model fields on the coarse grid

precipitation & Z on the fine grid

x1,tx2,t

...xN,t

∣∣∣∣∣∣∣∣∣

x ′1,tx ′2,t

...x ′M,t

, {Y1:T} , {Z1:T}

Downscaling

We can sample the model fields on the fine grid given:

model fields on the coarse grid

precipitation & Z on the fine gridx1,tx2,t

...xN,t

∣∣∣∣∣∣∣∣∣

x ′1,tx ′2,t

...x ′M,t

, {Y1:T} , {Z1:T}

Page 86: Probabilistic downscaling to detect regional present and

Conclusion

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Conclusion

We used the model fields on a coarse grid to forecast the precipitationon the fine grid.

We have quantified uncertainity.

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Conclusion

We used the model fields on a coarse grid to forecast the precipitationon the fine grid.

We have quantified uncertainity.

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Conclusion

1989-01 1989-03 1989-05 1989-07 1989-09 1989-11 1990-01Time

0

5

10

15

20

25

30

35ra

infa

ll (m

m)

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Conclusion

The task of downscaling is computational expensive.

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Conclusion

To transistion from weather forecasting to climate prediction, we needto forecast many years into the future.

Page 92: Probabilistic downscaling to detect regional present and

Conclusion

Thank you

probabilistic downscaling to detect regional present and

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