Statistical downscaling: the delta method, explained

The statistical method we use is often called the delta method, as it adds the difference (delta) between a historical period and a modeled value to a known historical climatology. Thus, only the change in climate (as expressed in the model) is incorporated into a known historical baseline. 

Other features:

  • This method uses only subtraction and division, which helps in interpreting and explaining downscaling results
  • Its low computational demand makes it very efficient to downscale many GCMs and emission scenarios over hundreds of years
  • Monthly temporal resolution
  • Spatial resolution is 2-km x 2-km pixels across Alaska and Western Canada
  • Variables are limited those that have long observational records, including surface temperature and precipitation.

General steps

  1. Determine which time series you want to downscale and which climatology dataset you will downscale to. We downscale GCM and CRU time series, and use PRISM and CRU climatologies as downscaling climatologies.
  2. Calculate changes in the monthly time series (temperature or precipitation) in relation to the time series average climate during the time period for which the climatology is available.
  3. Interpolate those changes—also referred to as deltas or anomalies—to match the climatology’s spatial resolution. Then, add them to (for temperature) or multiply them by (for precipitation) the climatology values for the same month.

SNAP delta downscaling code can be obtained from our code repository.

Example: downscaling GCM data to PRISM climatology

  1. Standardize GCM and PRISM data

    1. Rotate grid and set latitude and longitude values to standard WGS84 geographic coordinate system. This sets North as the top of the grid and converts original lat/long values from 0°-360° to -180° -180°.
    2. Convert GCM units to more common units:

      1. Convert temperature from Kelvin to Celsius
      2. Convert precipitation values from kg/m²/sec to mm/month
  2. Calculate GCM climatologies to determine projected changes in climate and the amount of model bias inherent in that change. Climatologies are GCM mean monthly values across a reference period (usually 30 years) from the 20c3m scenario outputs. The values represent modeled data and contain an expected model bias that we later adjust.

    1. Determine a reference state of the climate according to the GCMs by using 20th-century (20c3m) scenario GCM data values to calculate climatologies for the same time span used in the high resolution data we are downscaling to.
    2. We do this calculation for a worldwide extent at the coarse GCM spatial resolution (range: 1.875°–3.75°).
  3. Calculate anomalies: monthly absolute (for temperature) or proportional (for precipitation) anomalies for a worldwide extent at the coarse GCM spatial resolution.

    1. Temperature: future monthly value (e.g., May 2050 A1B scenario) - 20c3m climatology
    2. Precipitation: future monthly value (e.g., May 2050 A1B scenario) / 20c3m climatology
  4. Downscale, and remove bias

    1. Interpolate temperature and precipitation anomalies with a first-order bilinear spline technique across an extent larger than our high-resolution climatology dataset. We use a larger extent to account for the climatic variability outside of the bounds of our final downscaled extent.
    2. Our GCM anomaly datasets are now at the same spatial resolution as our high resolution climatology dataset.
  5. For temperature, add the interpolated temperature anomalies to the high-resolution climatology data (e.g., PRISM). For precipitation, multiply the interpolated anomalies by the climatology data. This step effectively downscales the data and removes model biases by using observed data values as baseline climate.

    1. The final products are monthly downscaled high resolution (2-km or 771-m for PRISM) data.