State-space Modeling of Climate Change

This page discusses about some of my recent work (including from my postdoc and my current work at Tetra Tech) related to the State-space models (SSM) developed for global average temperature (or global warming level, GWL) in Climatic Change and Journal of Hydrology: Regional Studies.

See other pages about my earlier work related to statistical forecasting of local climate change and using climate models to inform future conditions.

In my view, this is a more sophisticated approach to my previous work of using ARIMA to forecast future conditions (as described on the statistical forecasting page) because the SSM provides a more natural way to inform underlying climate-change parameters using process-based modeling results and to constrain future conditions with historical data.

An explanatory video for this work on the Colorado River Basin

Please note that this video was generated by AI (NotebookLM) and is intended to provide some general background information. See discussions below for more technical details.



In this work, we have developed a comprehensive probabilistic modeling framework designed to bridge the gap between global climate science and practical, regional decision-making. Our work centers on the development and application of a physical-parameter-based SSM combined with Bayesian inference. This approach allows us to project long-term trajectories of climate change—specifically GWL—and extend those projections to assess regional water supply risks in the Colorado River Basin (CRB).

1. Objectives

Our primary goal is to reduce projection uncertainty and facilitate adaptation strategies by integrating physical climate parameters with statistical learning. We recognize that while Global Climate Models (GCMs) provide essential long-term scenarios, they often exhibit large uncertainties and may not perfectly align with historical records. Therefore, we aim to:

  • Quantify the value of information: We seek to demonstrate how processing observational data (both historical and future "pseudo-observations") sequentially reduces uncertainty in key parameters like Equilibrium Climate Sensitivity (ECS) and future temperature projections.
  • Constrain regional projections: By extending our global framework to the CRB, we aim to use historical streamflow and climate data to constrain hydrologic projections, providing more accurate estimates of future water availability.
  • Support decision-making: In the case of CRB, we integrate these projections with river system models to help policymakers anticipate long-term water shortages and evaluate the sufficiency of conservation efforts.

2. Methodology

Our approach relies on a Bayesian inference that treats GCM simulations as informative priors and historical observations for further calibration.

Physical-Parameter-Based SSM

At the core of our methodology is a two-layer energy-balance model that describes the Earth's temperature response (as in GWL) to radiative forcing, and additionally how GWL leads to regional changes. We translate such equations into a state-space form where the state variables include:

  • Global Variables: Surface and deep ocean temperature anomalies, driven by radiative forcing (Greenhouse Gases and aerosols) and governed by parameters such as ECS and ocean heat capacity.
  • Regional Variables: For the CRB application, we expand the state vector to include regional temperature and precipitation changes, modeled as responses to GWL (linear scaling) plus autoregressive noise to capture natural variability.
  • Hydrologic Variables: We further link these climate variables to subbasin streamflow changes, accounting for temperature and precipitation sensitivities.

Sequential Bayesian Learning (or Transfer Learning)

We employ a "learning" process to estimate the model parameters:

  • Step 1 (GCM-Informed Priors): We first process long-term simulations from ensembles of GCMs (CMIP5 or CMIP6) as well as GCM-coupled modeling results. This allows us to capture the physics and covariance structures inherent in these complex models, effectively "transferring" their knowledge into our statistical framework.
  • Step 2 (Observational Constraints): We then update these priors using historical observations (e.g., global temperature records, CRB streamflow data). This step constrains the parameters, pulling the projections closer to observation data and reducing the spread of uncertainty often found in raw GCM ensembles.

Integration with System Models

For the regional analysis, we feed our probabilistic streamflow projections into the Colorado River Simulation System (CRSS) model. This allows us to translate hydrologic changes into system-specific metrics, such as the water elevation of Lake Mead and the reliability of water deliveries.

3. Results

Our combined analyses reveal that observational constraints can largely narrow the range of future climate possibilities and also point toward a drier future for the CRB.

Global Uncertainty Reduction
We demonstrate that the uncertainty in global temperature projections decreases as we incorporate more data over time. For instance, when projecting GWL under the SSP2-4.5 scenario, the range of the 95% prediction interval drops from 1.9°C (using data up to 2020) to 0.6°C (using data up to 2080). This confirms that our SSM framework successfully captures the "learning" of climate change parameters, such as ECS, as the climate signal emerges from the noise of natural variability.

Regional Hydrologic Insights (Colorado River Basin)
When applying this framework to the CRB, we find that historical observations paint a more concerning picture than many GCMs:

  • Higher Temperature Sensitivity: Our inference suggests a stronger negative response of streamflow to warming than many process-based models estimate. While GCMs vary, our observation-constrained results indicate a high sensitivity, aligning with recent studies suggesting larger flow reductions per degree of warming.
  • Projected Decline: We project a long-term decline in annual streamflow, both in the average and the confidence intervals. Specifically, we estimate that the "millennium drought" conditions observed during 2000–2023 will likely become the average condition by approximately 2040, with further declines thereafter.
  • Precipitation Trends: Unlike some GCMs that project precipitation increases, our SSM suggests a modest decrease in annual precipitation in the Upper Basin due to climate change (e.g., 2.3% lower than the long-term average by 2050).

Implications for Water Management
Our integration with the CRSS model highlights critical vulnerabilities. We find that despite extraordinary conservation efforts in recent years, current baseline policies may be insufficient. The probability of significant water shortages increases substantially after 2040. Even an additional conservation of 1 million acre-feet per year might only stabilize the system temporarily before long-term declines resume due to accelerated warming.