��Space-Time Modeling of Hydroclimate Extremes for Risk and Resiliency� �Balaji Rajagopalan�Department of Civil, Environmental and Architectural Engineering�&�CIRES�University of Colorado, Boulder, USA��&�Will Kleiber�Applied Math����

NSF Workshop on Extremes

Nov 5 – 8, 2024

Colorado School of Mines

Golden, CO

Contact:

balajir@Colorado.edu

University of Colorado, Boulder, USA

• Cameron Bracken (PNNL)
• Andrew Verdin (Stantec)
• Will Kleiber
• Sean Horvath
• Alvaro Ossandon (Santa Maria Univ., Valporaiso, Chile)
• Prasad Thota
• David Woodson
• Javi Gual (visiting student from Spain)
• Manuela Brunner (NCAR now Switzerland)
• Upmanu Lall (Columbia Univ./Arizona State Univ.)
• Vimal Mishra (IIT Gandhinagar)
• Satish Regonda (IIT Hyderabad)

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• Rajagopalan et al. (2002)
• Bracken et al. (2016a, b, 2018)
• Ossandon et al., 2021a,b, 2022a,b, 2023a,b)

WRR, HESS, Earths Future, J. Hydrology, Climate Dynamics, Monthly Weather Review

Collaborators

• Space-time Hydroclimate Extremes have major socio-economic impacts
• Extreme precipitation
• Daily streamflow (floods)
• Droughts

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Motivation

SwissRe Sigma Report 2020

Motivation

CEEW 2022 Report

Increasing frequency of

• Floods and droughts

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Build India’s resilience through improved public early warnings

Nov 28 – Dec 4, 2015

Dec 2 – Dec 4, 2023

• Increase in precipitation Extremes
• Increase Flood risk

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• Future Variability of ENSO in a warmer

climate

SST Sep-Nov 2023 – El Nino

(Mis)-Management of Water Resources�Leading to Massive Floods

Nov 9 – 16, 2015

Pakistan Floods 2022

-Extensive Flooding

-Major crop loss

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Kerala Floods 2018

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• Heavy Precipitation

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• Population centers along the foothills of Western Ghats
• Management of dams and reservoirs (lack thereof)

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• Geography + Meteorology + Human Factors

Mitigate the impact of extreme events (e.g., rainfall, streamflow) - need to

• Understand
• Moisture sources / pathways that produce extremes
• Role of Meteorology vs Role of Land surface
• Interplay between the two

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• Model/Forecast
• Space-time variability of extremes
• General Framework
• Develop skillful probabilistic forecasts at meaningful lead times (hrs ~ days ~ season) for risk-based decision making

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Building Resiliency

Ossandon et al. (2021, J. Hydrology)

Ossandon et al. (2021, WRR, 2022, Earth’s Future)

Lakshmi et al. (2019, j. Atm. Sol. Terr. Phys.)

Nanditha et al. (2022, Clim. Dyn.)

IVT direction (vectors) and intensity (kg m⁻¹ s⁻¹; color shading) fields at (a) 0000, (b) 0600, (c) 1200, (d) 1800 UTC 26 Jul 2005, of mean daily composite on (e) 27 Jul 2005, (f) 28 Jul 2005, (g) 29 Jul 2005 and IMD daily rainfall (mm day⁻¹; colors over land) distribution for the above period [(h)–(k)] during Mumbai floods.

Statistical Learning - Bayesian Paradigm

Model m

Rev. Thomas Bayes (1701-1761) a

f

Statistical Learning - Bayesian Paradigm

Hydroclimate problems are fraught with uncertainty in

• Data
• Model
• Expert knowledge

• Markov Chain Monte Carlo (MCMC)

Sample from the posterior PDF

• Computationally intensive – possible

Akin to Generalized Linear Model (GLM) Framework using MLE

Statistical Learning - Bayesian Hierarchical Models directed acyclic graphs (DAGs)

randomForest R package^*

• Individual decision trees generated by sampling a subset of the training data then performing classification or regression of variables
• Feed individual trees with new data to generate ensemble forecast

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*Liaw, A., & Wiener, M. (2002)

Ho, T.K. (1995)

Breiman, L. (2001)

Statistical Learning – Random Forest

• Woodson et al. (2020, WRR)
• Multi-year streamflow projection

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• Woodson et al. (2024, ASCE, JWRPM)
• 0 ~ 24 month streamflow forecast

• Bracken, C., B. Rajagopalan, W. Kleiber and L. Cheng and S. Gangopadhyay, Efficient Bayesian hierarchical modeling of spatial precipitation extremes over a large Domain, Water Resources Research, 52(8), 6643-6655, 2016

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• Bracken, C., B. Rajagopalan, and C. Woodhouse, Bayesian hierarchical nonhomogeneous hidden Markov model for multisite streamflow reconstructions, Water Resources Research, 52(10), 7837-7850, 2016

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• Bracken, C., K. Holman, B. Rajagopalan and H. Moradkhani, A Bayesian hierarchical approach to multivariate nonstationary hydrologic frequency analysis, Water Resources Research, 54 (1),243-255, 2018

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• Verdin, A., B. Rajagopalan, W. Kleiber, G. Podesta and F. Bert, BayGEN: A Bayesian space-time stochastic weather generator, Water Resources Research, 55(4), 2900-2915, 2019

Recent Bayesian/Statistical Learning Research

• Horvath, S., J. Stroeve, B. Rajagopalan and W. Kleiber, A Bayesian logistic regression for probabilistic forecasts of the minimum September Arctic sea ice cover, Earth and Space Science, 7(10, e2020EA001176, 2020

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• Ossandon, A., B. Rajagopalan and W. Kleiber, Spatial-Temporal Multivariate Semi-Bayesian Hierarchical Framework for Extreme Precipitation Frequency Analysis, Journal of Hydrology, 600(126429), 2021

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• Ossandon, A., B. Rajagopalan, U. Lall, J. Nanditha and V. Mishra, A Bayesian hierarchical network model for daily streamflow ensemble forecasting, Water Resources Research, WR029920, 2021

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• Ossandon, A., M. Brunner, B. Rajopalan and W. Kleiber, A space-time Bayesian hierarchical modeling framework for projection of seasonal streamflow extremes, HESS, 2022

Recent Bayesian Statistical Learning Research

• Verdin, A., B. Rajagopalan, W. Kleiber,and C. Funk (2015), A Bayesian krigingapproach for blending satellite andground precipitation observations,Water Resour. Res., 51, 908–921,doi:10.1002/2014WR015963

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• B., Suchetana, B. Rajagopalan and, J. Silverstein, Investigating regime shifts and the factors controlling Total Inorganic Nitrogen concentrations in treated wastewater using non-homogeneous Hidden Markov and multinomial logistic regression models, Science of The Total Environment, 646(625-633), 2019

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• Tixier, A., M. Hallowell, B. Rajagopalan and D. Bowman, Application of machine learning to construction injury prediction, Automation in Construction, 69(102-114), 2016.

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• May-Ostendorp, G. Henze, B. Rajagopala and C. Corbin, Extraction of supervisory building control rules from model predictive control of windows in a mixed mode building, J. Buld. Perfor. Sim., 2013

Recent Bayesian Statistical Learning Research

‘Semi-Bayesian Blending’

Gauge Data

• Direct measurement of field
• Low uncertainty
• Coarse resolution

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Satellite-derived Estimates

• High spatial resolution
• Near-real time observation
• Indirect measurement of field
• Areal average
• Potential over/under estimation
• Higher uncertainty

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Solution: Blending information from multiple sources for accurate estimation of the underlying field with uncertainty

Verdin et al., 2015, WRR)

Data Sources

“Bayesian Blending” – Bayesian Hierarchical Model

July 2009

July 2009

• Enhanced blended estimate at Higher elevations and over regions of With more observations
• Satellite underestimation improved

Uncertainty:

1. 5^th percentile
2. 95^th percentile
3. Standard error

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• Lower errors over regions with observations

July 2009

Precipitation

• Daily observed precipitation - Global Historical Climatology Network �(GHCN) dataset (https://www1.ncdc.noaa.gov/pub/data/ghcn/daily/)
• Years: 1964-2018, no. of stations 73.
• Seasonal total and 3-day maximum precipitation were computed from daily precipitation.

Covariates

• Elevation data - NASA Land Data Assimilation Systems (NLDAS) (https://ldas.gsfc.nasa.gov/nldas/elevation)
• ENSO and PDO climate indices (https://www.esrl.noaa.gov/psd/data/climateindices/list/)
• Spatial average of seasonal total precipitation.

Climatology of the extreme seasonal precipitation for 73 precipitation stations and 0.5-degree elevation grid in (m) of the study area. The red square corresponds to the site of interest, considered in Section 4.

Space-Time Field of Extremes

Ossandon et al. (2021, J. Hydrology)

Connections to Large Scale Climate

��Data layer: “True” values of the GEV regression coefficients

• We Consider that extreme precipitation at each location follows a GEV distribution
• We considered only location and scale parameters nonstationary
• Models were fitted using Maximum likelihood
• The best model were selected based on the lowest total BIC value
• The best model only consider the standardized spatial average of seasonal total precipitation (SASP) as covariate

Model Structure – Semi-Bayesian Hierarchical Model (SBHM)

• Building on Bracken et al. (2016)
• Ossandon et al. (2021)

Model Structure

Priors

• Multivariate Normal

on regression parameters

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• Inverse Wishart for variance

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Model Structure

Priors

• Multivariate Normal

on regression parameters

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• Inverse Wishart for variance

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• The parameters of the nonstationary GEV parameters at each location were estimated via maximum likelihood.

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• The spatial Bayesian multivariate model was fit using a Markov Chain Monte Carlo (MCMC) method

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• One chain of length 7500 was run, with the first 2500 iterations discarded as warmup, resulting in 5000 samples for each parameter

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• To assess convergence, trace plots were visually inspected

Implementation and Fitting

Caption. a) Q-Q plot , (b) PDF of the GEV distribution fitting, and (c) time series of nonstationary of the MLE return levels for different return periods for extreme summer precipitation at Pasamonte station, NM. In panel (b), the data are first transformed to an appropriate standardized GEV scale. The empirical PDF is obtained by a kernel density estimator. In panel (c), the black line corresponds to the observed.

Model Fit

Model Fits – Posterior �distributions of�Model Parameters

Fitting

Spatial-Time Variability of Return Levels 2-year

Return Levels

Conditional posterior median (2-year return level) of extreme summer precipitation and the corresponding 95% confidence interval width for the

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• whole record (a and b)
• a wet year (1997, c and d),
• a dry year (2001, e and f).

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• Points in panels correspond to the median of the observed.

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• Historical extremes are very well captured

Spatial Variability of Return Levels – 100-year

• Median 100-year return level of extreme summer precipitation along with the median of the observed (left) and median 100-year return level of 95\% confidence interval width of extreme summer precipitation (right).

Fitting

• Nonstationary 100-year return levels from (a) the semi-Bayesian univariate model (b) the semi-Bayesian multivariate model of extreme summer precipitation at Pasamonte station, NM. Red line corresponds to the nonstationary 100-year return levels from the MLE estimates of the GEV regression coefficients.

Temporal Variability

stations dropped (7) for the cross-validation (red circles) and the subset data (66 stations) using to fit the model (gray circles).

Cross Validation

Predicted extreme summer precipitation

• for a dry (2001, left) year
• wet (1997, right) year
• 10% random leave-out stations

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Semi-BHM - Summary

• Semi-BHM offers a computationally efficient approach to space-time modeling of extremes over a large domain

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• Robust estimates of return levels of extremes in space-time
• Helpful in risk-based decision making
• For flood hazard mitigation

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• Full Bayesian extensions with moisture tracks

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Outline

• Bayesian Hierarchical Model for annual Streamflow
• BHNHMM
• Reconstruct past streamflow

1473 ~ 1997

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Streamflow

• Annual(Oct-Sep) streamflow at 20 gauges in the Colorado River Basin
• Period 1906 – 1997
• Tree ring data from 50 sites
• Tree rings are an excellent integrator of climate and hydrology. Thus, capture the streamflow variability very well

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Traditional Methods

• Individual linear regression on streamflow at each gauges as a function of tree rings

Bayesian Hierarchical Nonhomogeneous Hidden Markov Model - Study Region / Data

Bracken et al., 2016 (WRR)

A doubly embedded stochastic process that can be used to model time series data

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1. Each time-series observation is a stochastic sample from a corresponding, underlying distribution

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2. The sequence of underlying distributions, or hidden states, follow a Markov chain

Observation 1

t=1

Observation 2

t=2

Observation 3

t=3

Observation 4

t=4

State 1

State 1

State 1

State 2

TPM

Coles (2001), Zucchini (2016)

Hidden Markov Model

• Each State is a PDF with parameters
• State transition probability matrix (TPM)
• TPM can be nonstationary (e.g., Bracken et al., 2015, 2016)
• EM, Baum-Welch algorithm

Y_st are streamflow at location s, in year t

Data Layer:

Pocess Layer:

State Transitions

where                is the joint cdf of an S-dimensional multivariate normal distribution with covariance matrix Σ

Spatial Model

• Suitable Priors
• Likelihood and MCMC

to obtain posteriors of

the parameters the flows

BHNHMM

• In the historical period

(1906 – 1997) BHNMM Captures

• Basic Statistics
• Reonstructions are closer

to observed, relative to a

linear regression-based

model

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Network Model - Study Region / Data

Spatial correlations among the sites are

Very well captured due to Copulas

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Past streamflow reconstruction

Bracken et al., 2016

Multivariate Extremes

Application - Taylor Park Dam

Bracken et al. 2018 (WRR)

106°20'0"W

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Taylor Park model structure

(y(t), z(t), h(t)) ⇠ C_g (S; {µ_y (t), s_y, x_y, µ_z (t), s_z, x_z, µ_h (t), s_h, x_h }) y(t) ⇠ GEV(µ_y (t), s_y, x_y )

z(t) ⇠ GEV(µ_z (t), s_z, x_z )

(11)

(12)

(13)

(14)

(15)

(16)

(17)

h(t) ⇠ GEV(µ_h (t), s_h, x_h )

µ_y (t)= µ_y0 + x(t)^T b_y

µ_z (t)= µ_z0 + x(t)^T b_z

µ_h (t)= a - exp(-bµ_z (t))

where x(t)^T is a vector of climate covariates.

Results - Joint distributions

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Results - Nonstationary return levels

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Network model for daily streamflow

Ossandon et al. (2021a,b, 2022a,b, 2023)

Outline

• Bayesian Hierarchical Model for Daily Streamflow
• BHNM
• Post-processing of Physical Models & Combination
• Real-time Forecasting
• Seasonal Maximum

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Streamflow

• Daily summer (July-August) streamflow at four gauge stations
• India Water Resource Information System (IWRIS)
• Period 1978 – 2014

Hydro-Meteorological Variable

• Gridded daily summer (July-August) precipitation from the India Meteorology Department (IMD)
• 0.25° spatial resolution

Network Model - Study Region / Data

In the Bayesian Hierarchical Network Model (BHNM) streamflow at each gauge follows a Gamma distribution. Data Layer:

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Model Structure

Process Layer: Parameters vary in space and time

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• The priors of β and φ are Multivariate Normal distribution (MVN)

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• Posterior distributions of parameters and streamflow via Markov Chain Monte Carlo (MCMC)�

Priors: Parameters vary in space and time

As potential covariates, we considered:

• Daily streamflow from upstream gauges depending on effective travel times

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• 1-day, 2-day, or 3-day spatial average precipitation from contributing areas between two stations
• reflects the antecedent land conditions.

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• These variables are considered at 1-day lead time for forecast.

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• The best set of covariates for each station gauges were obtained based on the highest linear correlation coefficient (R)

Model Covariates

• Candidate models (i.e., combinations of covariates,

Data layer distribution – Gamma, Lognormal etc.)

are fitted and the best model selected using

DIC (Deviance Information Criteria)

Implementation and Model Fitting

Calibration

• Predictive posterior

distribution of simulated

July-August daily streamflow for Mandleshwar gauge

(a) For 1978-2014

(b) scatterplot

(c) For 2013

• Scatterplots of the posterior median and observed flows along with the 1:1 line and R values in inset panels
• The black box in panel a shows the temporal windows for time series in panel b.

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BHNW - Calibration

• BHNW shows very good performance in calibration

Calibration

Pairs plot (upper triangular matrix) and Pearson correlation coefficients (R, lower triangular matrix) of July-August daily

• observed streamflow and

(b) ensembles mean forecast streamflow from BHNM for the three gauges analyzed in this study.

Streamflow values are in (100cms).

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BHNW - Calibration

• BHNW captures spatial correlation very well

Cross Validation

• Drop two-year

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Predictive posterior distribution of July-August daily streamflow at

Mandaleshwar gauge

(a) For 1978-2014

(b) scatterplot

(c) For 2013

BHNW – Cross Validation

• BHNW shows excellent performance in

cross validation also

BHNW – Skill Scores

• BHNW shows high median skill score in CRPSS (1 is perfect

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• EES indicates good spatial skill over the entire River network

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compared to traditional Linear Model (GLM)

BHNW – Skill Scores

• Rank histogram shows from the
• ensembles shows less discrepancy
• for BHNW
• Indicating that the posterior distributions are well reliable and capture the uncertainty robustly

BHNM - Summary

• The model captures the median observed streamflow very well, along with uncertainties via the posterior distribution

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• Scatterplots of observed flows vs Posterior Median flows at all the locations show good skill in the flow volume and timing

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• The skill scores compared to traditional Linear Model are higher besides very good spatial skill computed via the EES metric.

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• Adapt the BHNW for post-processing of streamflow simulations from physical at multiple sites

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• Enhances the skill of the physical model outputs

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• Correct systematic and nonsystematic biases

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• Uncertainty quantification

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Physical hydrological model

Meteorological forcing

Streamflow

Bayesian Hierarchical model (BHM)

Other covariates

Q(t)

Q(t)

Streamflow ensembles

BNHM – Extensions

Physical Model – Post Processing

Ossandon et al. 2022, AMS J. of Hydromet

Model Structure

Relative to Single ensemble performance(mean)

• Increase of the of the correlation (R)

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• Significant reduction of the bias (at least 20%)

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Skill of model combination higher than VIC

Results – Cross-Validation

Two model combination shows higher ensemble performance than single model

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• Increase of the of the correlation (R)

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• Significant reduction of the bias (at least 20%)

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Results – Two Models – VIC and SAC

Thota et al., 2023 – in preparation

To develop a Bayesian Hierarchical model combination (BHMC) approach for skillful real-time forecast of daily streamflow at multiple lead times

For k-day lead times

BNHM – Extensions

Realtime Forecasting

Ossandon et al., HESS 2022

Ossandon et al., Earths Future 2023

• BHMC captures most of the observations inside of the 50% credible interval (orange bands)

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• high correlation and low BIAS, up to 3-day lead time

Handia

Realtime forecast of 2021 Season

BHM for seasonal maximum flow projections in the UCRB (Ossandon et al., 2022, 2023)

• Develop Bayesian Hierarchical model framework for projections of seasonal streamflow extremes on a river network

Climate covariates

Forecast precipitation

BHM

BHM

Peak monsoon season

BNHM – Extensions

Seasonal Maximum flow

Ossandon et al., 2022, HESS

Colorado River spring season maximum

Ossandon et al., WRR, 2023

Covariates selection for each lead time

• Year to year variability of the rainfall over India is driven largely by ENSO, IOD, and PWPR

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Covariate Selection

• IOD, PWPR, Nino1+2, Nino 3.4
• We consider our own index (NI) for each lead time based on the region of highest correlation between the first PC of maximum streamflow and SST
• Basin average monsoon (July-August) total precipitation (AMTP) forecast

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Covariates Selection

• BHM captures all the observations inside ensemble spread up to 1-month lead time (available on June 1)

Handia

Results – Cross Validation

Model performance

Distributional performance

• Skill decreases as the lead time increases
• Coherent forecast (no worse than climatology)
• For high flow years median CRPSS values above 0.1 up to 1-month lead
• Reliable streamflow forecast ensembles for all the lead times (alpha values above 0.9)

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Results – Cross Validation

Space-Time Bayesian Hierarchical (Hot off the Press)

• SST field modeled as Multivariate Normal Distribution – Data Layer

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Process Layers

• The mean and standard deviation varies in space-time

as a linear functional model of ‘ covariates’.

• Principal Components of SSTs of the ‘limited field’
• The Coefficients of the covariates modeled in space using

Gaussian processes

• Appropriate priors are placed on all the model parameters

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• MCMC for posteriors of the parameters and underlying SST field

Ossandon et al. (2024) Paleoceanography and Paleoclimate (in review)

Space-Time Bayesian Hierarchical (Hot off the Press)

Space-Time Bayesian Hierarchical

• Significant cooling in the

Central Pacific > 1 ^oC

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• Along with distinct warming

In the Western Pacific

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• Peak in cooling 6~4Ka

Before decreasing

Space-Time Bayesian Hierarchical

• Significant cooling in the

Central Pacific > 1 ^oC Peak in cooling 6~4Ka

Before decreasing

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• Along with distinct warming

In the Western Pacific

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Present day monsoon-ENSO teleconnection

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Percentage of Seasonal Rainfall

0 20 40 60 80 100 120 140

full

early

middle

late

70 – 100% increase in early season precipitation in north-central India and the Indo-Gangetic Plains;

30 – 80% increase in peak season precipitation in south-central India;

60 – 100% increase in late season precipitation in northern, northwestern, and central India.

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Percentage rainfall increase for each season (full, early, middle, or late) due to a 1°C cooling of the NINO3.4 region in the tropical Pacific:

Gill et al., 2015, JGR

Random Forest Based Flow forecasting

Woodson et al. (2021, 2024)

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Goal: Develop skillful forecasts of spring flow (April 1 – July 31 total flow) at lead times of 0 ~ 18 months

• Naturalized, Lees Ferry flow

Streamflow Forecasting Framework

Streamflow Forecasting Framework

• Mean annual Max Temp
• Runoff Efficiency
• AMO and PDO

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• ESP – Weather Service Forecast

Example hindcast:

• Apr. – July spring flow at Lees Ferry (naturalized)
• 8-, 12-, and 18-month leads
• Leave P out cross validation
• Current year, prior-2 and next-2 years dropped
• 500-member RF ensemble
• 25-member ESP
• After dropping 5-years to make blind

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Streamflow Hindcast

At longer lead times (5 ~ 18 months)

• Climate variables (AMO, PDO and forecast of Temp)

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At shorter lead times (0 ~ 4 months – ESP, short term forecast of precip and Temp from climate models)

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Random Forest Variable Importance

Ranked Probability Skill Score (RPSS)

• Mean square error but for probabilistic (ensemble) forecasts
• Calculated on terciles, i.e., how well do we predict high-, low-, or average flow terciles?
• Relative to climatological ensemble (1921-2017 flow terciles with forecast year dropped)
• 35 RPSS values per boxplot (one per hindcast year)

Ensemble Mean Root Mean Square Error (RMSE)

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Good performance

Bad performance

Random Forest Model Performance

• Bayesian Hierarchical/Statistical Learning Modeling
• Generalized framework for space-time modeling and forecast of hydroclimate (streamflow, precipitation) and extremes
• Daily, Seasonal and Paleo timescales
• Enhance the skill of deterministic forecasts
• Robust estimation of uncertainties
• Incorporate qualitative and quantitative priors

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• Can be applied to any class or problems
• Paleoclimate reconstruction of other variables

(vegetation, drought indices etc.), hot/cold

spells, tropical cyclones

• Extremes under climate change
• Water Quality modeling, Construction Safety

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Contact:

balajir@Colorado.edu

Parting Thoughts