Lecture

2023-09-25

Today

Downscaling

Downscaling: theory

Project 1

- Stormwater management (long-term desigm)
- Water resources management (subseasonal to multi-year planning)
- Fire propagation (hourly to weekly)

- enhanced spatial detail
- mitigation of systematic ESM
^{1}biases - generation of variables not explicitly rendered by GCMs

- ESMs are tuned to get energy balance and large-scale circulation right, not local extremes
- ESMs average over space and time
- Local-scale precipitation can be tricky to model well

Today

Downscaling

Downscaling: theory

Project 1

- Input: pairs \((X_i, y_i)\)
- \(X_i\): predictors (e.g., gridded rainfall)
- \(y_i\): predictand (e.g., gauge rainfall)

- Goal: learn a function \(f\) such that \(f(X_i) \approx y_i\)
- Measure quality of approximation through a
*loss function*(more later)

- Measure quality of approximation through a
- Key point: the \(X_i\) and \(y_i\) are observed at the same time
- Example: map satellite to radar data

- ESMs simulate from the distribution of weather, given climate boundary conditions. For example:
- Run 100 ESM
*ensemble members*over 20th century conditions - Study December 1, 1980 in all draws
- Some will be rainy, some will be dry; some cool, some warm
- Statistically meaningful, but not a forecast!

- Run 100 ESM
- No pairs \((X_i, y_i)\). Instead we have samples \(\left\{X_1, \ldots, X_N \right\}\) and \(\left\{y_1, \ldots, y_K \right\}\)

- Gauge data
- Gridded observational products
- For example: radar measurments are processed to produce gridded rainfall estimates

- Reanalysis products
- Use
*assimilation*to “digest” observations using a model - Gridded reconstructions of past weather
- State of the art is ERA5

- Use
- ESM outputs
- Historical runs
- CMIP: compare multiple models on standardized scenarios (e.g., RCP 2.6, 4.5, 8.5)
- Simulate from weather,
**conditional on boundary conditions**

Simplest form of downscaling. Usually \(X\) are samples from a climate model and \(y\) are observations. \[ \begin{aligned} \text{bias} &= \mathbb{E}[X] - \mathbb{E}[y] \\ \hat{y} &= X - \text{bias} \end{aligned} \]

**Note**

Is this a distributional or supervised method?

Stationarity means the relationship between \(X\) and \(y\) does not change over time

- Supervised: \(p(y | X)\) or \(y = f(X)\) does not change over time
- Distributional: Corrections to the distribution do not change over time

This is never a perfect assumption

Today

Downscaling

Downscaling: theory

Project 1

- Given:
- Hourly gridded rainfall data (small area)
- Hourly large-scale pressure and temperature fields
- Hourly gauge rainfall data at a single station

- Develop a model to predict hourly gauge rainfall from the available datasets

- Try and compare at least two different approaaches
- Quantitative and qualitative evaluation of the models
- You can use any methods you like:
- Those we cover in class
- Those you already know## References

Lanzante, J. R., Dixon, K. W., Nath, M. J., Whitlock, C. E., & Adams-Smith, D. (2018). Some Pitfalls in Statistical Downscaling of Future Climate. *Bulletin of the American Meteorological Society*, *99*(4), 791–803. https://doi.org/10.1175/bams-d-17-0046.1

Price, I., & Rasp, S. (2022, March 23). Increasing the accuracy and resolution of precipitation forecasts using deep generative models. https://doi.org/10.48550/arXiv.2203.12297