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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.
3
Paleohydrology
4
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?
8
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
14
𝑡 = 𝑇,… , 1
Simultaneous learning–reconstruction
15
𝑦𝑡 ← ො𝑦𝑡|𝑇
𝑦𝑡 ← ො𝑦𝑡|𝑡−1
Replace missing 𝑦𝑡 with its best available estimate
Forward pass
M-step
𝑥1
Rationale for SLR
16
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
17
𝑡 = 𝑇, … , 1
Model performance
18
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
19
Residual analysis
20
A reconstructed history of the Ping
21
Figure 2 in Cook et al (2010)
Stochastic replicates
22
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
23
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
27
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
29
Wavelet analysis
30
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
31