18 January 2018

Stefano Galelli

people.sutd.edu.sg/~stefano_galelli/

Resilient Water Systems Group

REVEALS HISTORY OF REGIME SHIFTS

STREAMFLOW RECONSTRUCTION

IN NORTHERN THAILAND

Nguyen Tan Thai Hung

people.sutd.edu.sg/~ntthung/

A LINEAR DYNAMICAL SYSTEMS APPROACH

TO

The key to the future lies in the past.

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Paleohydrology

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Παλαιός = old, ancient

Paleohydrology

Proxy data

• Tree rings

• Ice core

• Corals

• …

Instrumental data

• Streamflow

• Precipitation

• Drought index

• …

Model

Paleoreconstructed

data

Παλαιός = old, ancient

Study site: Ping River

Monsoon Asia Drought Atlas (MADA)

Cook, E. R., Anchukaitis, K. J., Buckley, B. M., D’Arrigo, R. D., Jacoby, G. C., & Wright, W. E. (2010). Asian Monsoon Failure and Megadrought

During the Last Millennium. Science, 328(5977), 486–489. http://doi.org/10.1126/science.1185188

Figure 1B in Cook et al (2010)

Temporal resolution Annual

Spatial resolution 2.5o x 2.5o

Temporal range 1300 – 2005

Gridded time series of the Palmer’s Drought Severity Index

The conventional method

• How do we model catchment dynamics?

• Will a dynamic model be more accurate?

• What more insights can we gain with a

dynamic model?

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𝑦𝑡 = 𝛼 + 𝛽𝑢𝑡 + 𝜀𝑡

Linear dynamical systems

𝑥𝑡+1 = 𝐴𝑥𝑡 + 𝐵𝑢𝑡 +𝑤𝑡

𝑦𝑡 = 𝐶𝑥𝑡 + 𝐷𝑢𝑡 + 𝑣𝑡

𝑤𝑡 ∼ 𝒩 0,𝑄𝑣𝑡 ∼ 𝒩(0,𝑅)𝑥1 ∼ 𝒩(𝜇1, 𝑉1)

System𝑥

Input𝑢

Output𝑦

𝑥 ∈ ℝ𝑝 system state

𝑦 ∈ ℝ𝑞 system output

𝑢 ∈ ℝ𝑚 system input

𝐴 ∈ ℝ𝑝×𝑝 state transition matrix

𝐵 ∈ ℝ𝑝×𝑚 input-state matrix

𝐶 ∈ ℝ𝑝×𝑝 observation matrix

𝐷 ∈ ℝ𝑝×𝑝 input-observation matrix

𝑄 ∈ ℝ𝑝×𝑝 covariance matrix of the state noise

𝑅 ∈ ℝ𝑞×𝑞 covariance matrix of the observation noise

Learning: Expectation-Maximization

Shumway, R. H., & Stoffer, D. S. (1982). An Approach to The Time Series Smoothing and Forecasting Using the EM Algorithm. Journal of Time Series Analysis, 3(4), 253–264. https://doi.org/10.1111/j.1467-9892.1982.tb00349.x

Ghahramani, Z., & Hinton, G. E. (1996). Parameter Estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2. https://doi.org/10.1080/00207177208932224

Cheng, S., & Sabes, P. N. (2006). Modeling Sensorimotor Learning with Linear Dynamical Systems. Neural Computation, 18(4), 760–793. https://doi.org/10.1162/089976606775774651

E-Step

መ𝜃𝑘+1 = arg max ℒ 𝑌| 𝑋, መ𝜃𝑘

M-Step

𝑋 መ𝜃𝑘 = 𝔼 𝑋|𝑌, መ𝜃𝑘

Forward pass:

Kalman filter

Backward pass:

RTS recursion

Maximum

likelihood

ො𝑥𝑡|𝑡 = 𝔼 𝑥𝑡|𝑦1, … , 𝑦𝑡, መ𝜃𝑘

ො𝑥𝑡|𝑇 = 𝔼 𝑥𝑡|𝑦1, … , 𝑦𝑇 , መ𝜃𝑘

Kalman filter

Forward pass:

Kalman filter

Backward pass:

RTS recursion

Maximum

likelihood

ො𝑥𝑡|𝑡 = 𝔼 𝑥𝑡|𝑦1, … , 𝑦𝑡, መ𝜃𝑘

Kalman, R. E. (1960). A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82(1), 35. https://doi.org/10.1115/1.3662552

Faragher, R. (2012). Understanding the basis of the Kalman filter via a simple and intuitive derivation [lecture notes]. IEEE Signal Processing Magazine, 29(5), 128–132. https://doi.org/10.1109/MSP.2012.2203621

Figure 5 in Faragher (2012)

ො𝑥𝑡|𝑡−1 = 𝐴ො𝑥𝑡−1|𝑡−1 + 𝐵𝑢𝑡ො𝑦𝑡|𝑡−1 = 𝐶 ො𝑥𝑡|𝑡−1 +𝐷𝑢𝑡𝑉𝑡|𝑡−1 = 𝐴 𝑉𝑡−1|𝑡−1𝐴′ + 𝑄

𝐾𝑡 = 𝑉𝑡|𝑡−1𝐶′ 𝐶 𝑉𝑡|𝑡−1𝐶

′ + 𝑅−1

ො𝑥𝑡|𝑡 = ො𝑥𝑡|𝑡−1 + 𝐾𝑡 𝑦𝑡 − ො𝑦𝑡|𝑡−1𝑉𝑡|𝑡 = 𝐼 − 𝐾𝑡𝐶 𝑉𝑡|𝑡−1

For 𝑡 = 2,… , 𝑇

RTS recursion

Forward pass:

Kalman filter

Backward pass:

RTS recursion

Maximum

likelihood

ො𝑥𝑡|𝑇 = 𝔼 𝑥𝑡|𝑦1, … , 𝑦𝑇 , መ𝜃𝑘

𝐽𝑡 = 𝑉𝑡|𝑡𝐴 𝑉𝑡+1|𝑡−1

ො𝑥𝑡|𝑇 = ො𝑥𝑡|𝑡 + 𝐽𝑡 ො𝑥𝑡+1|𝑇 − ො𝑥𝑡+1|𝑡𝑉𝑡|𝑇 = 𝑉𝑡|𝑡 + 𝐽𝑡 𝑉𝑡+1|𝑇 − 𝑉𝑡+1|𝑡 𝐽𝑡

′

ො𝑦𝑡|𝑇 = 𝐶 ො𝑥𝑡|𝑇 + 𝐷𝑢𝑡

Rauch, H. E., Tung, F., & Striebel, C. T. (1965). Maximum likelihood estimates of linear dynamic systems. AIAA Journal, 3(8), 1445–1450. https://doi.org/10.2514/3.3166

For 𝑡 = 𝑇 − 1,… , 1

Maximum likelihood estimation

Forward pass:

Kalman filter

Backward pass:

RTS recursion

Maximum

likelihood

Quadratic terms only Analytical solutions

Algorithm 1: LDS-EM

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𝑡 = 𝑇,… , 1

Simultaneous learning–reconstruction

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𝑦𝑡 ← ො𝑦𝑡|𝑇

𝑦𝑡 ← ො𝑦𝑡|𝑡−1

Replace missing 𝑦𝑡 with its best available estimate

Forward pass

M-step

𝑥1

Rationale for SLR

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The substitution turns all terms related to missing 𝑦𝑡 into zero

ො𝑥𝑡|𝑡 = ො𝑥𝑡|𝑡−1 + 𝐾𝑡 𝑦𝑡 − ො𝑦𝑡|𝑡−1

E-Step

M-Step

Algorithm 2: SLR

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𝑡 = 𝑇, … , 1

Model performance

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𝑅𝐸 = 1 −

𝑡∈𝒱𝑦𝑡 − ො𝑦𝑡

2

σ𝑡∈𝒱 𝑦𝑡 − 𝑦𝑐

2

𝐶𝐸 = 1 −σ𝑡∈𝒱 𝑦𝑡 − ො𝑦𝑡

2

σ𝑡∈𝒱 𝑦𝑡 − 𝑦𝑣

2

Model performance

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Residual analysis

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A reconstructed history of the Ping

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Figure 2 in Cook et al (2010)

Stochastic replicates

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Conclusions

• Replacement for conventional method

Better model performance and desirable features

• A more conservative policy for the Bhumibol

There seems to be less water in the system

• Regional hydrological understanding (complementing the MADA)

History of regime shifts

• Direct application: regime-informed reservoir operation

Stochastic replicates of both streamflow and regime

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APPENDICES

Dendrochronologyδένδρον (tree limb) + χρόνος (time) = tree dating

Other reconstructions

Woodhouse et al,

2006

Gangopadhyay et al,

2009Devineni et al, 2013 Patskoski et al, 2015 Ho et al, 2016

Lo

cati

on

&D

ata

Colorado River

• 4 stations

• 62 chronologies

Colorado River at Lees

Ferry, Arizona

• 62 chronologies

Upper Delaware River

Basin

• 5 stations

• 8 chronologies

South-eastern US (NC,

SC, GA, FL)

• 8 stations

• 7 chronologies

Missouri River Basin

• 55 stations

• LBDA

Pe

rfo

rman

ce

• RE ~ 0.65 - 0.8

• nRMSE ~ 0.14

• adjusted R2

~ 0.7 -

0.8

• R2

= 0.76• RE ~ 0.2 - 0.5

• CE ~ 0.1 - 0.5

• Adjusted R2 ~ 0.1 -

0.4

• Normalized RMSE

~0.25 - 0.5

• NSE (positive /

negative, average

positive)

• Reduction of error

(mostly positive,

average around ~0.2

• Adjusted R2

~0.5 -

0.9

Search radius

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Spatial correlation

Site Distance r p-value

LS001 406.71438 -0.2225 0.0407

LS002 438.74650 -0.1447 0.1863

TH001 55.37224 0.2024 0.0632

TH002 354.28653 0.1293 0.2757

TH003 369.80428 -0.0365 0.7589

TH004 423.49371 0.1829 0.0919

TH006 85.10499 -0.0358 0.7464

MADA Tree rings

M-step solution

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Wavelet analysis

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Applications

• Drought adaptation planning

– Agriculture & Agri-Food Canada

– Prairie Provinces Water Board

– Denver Water Board

• Informing the public (Colorado River)

• Reliability of urban water supply

– Cities of Calgary and Edmonton

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