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Lindsay Fitzpatrick edited this page Jun 23, 2026 · 4 revisions

This section provides detailed explanations of the methods, equations, and terminology used in the CNBS Predictor workflow. It includes definitions of forecast variables, derived metrics, preprocessing steps, and probabilistic products.

Examples include:

  • How evaporation, runoff, and net basin supply (NBS) are calculated
  • How climatology exceedance probability (CEP) is derived and interpreted
  • Definitions of forecast metrics such as RMSE, NSE, KGE, and CRPS
  • Assumptions used in data processing and model evaluation

Climatology Exceedance Probability (CEP)

Definition

Climatology Exceedance Probability (CEP) is a probabilistic metric used to compare a forecasted value against a historical climatological distribution. It represents the fraction of climatological values that exceed the forecast.

CEP provides a standardized way to assess whether a forecast is wetter, drier, higher, or lower relative to historical conditions.

For the CNBS Predictor, CEP is used to evaluate forecasted Net Basin Supply (NBS) relative to climatology for each Great Lake and forecast month.

Source Data

CEP values are derived using probability-of-exceedance lookup tables compiled by the United States Army Corps of Engineers (USACE). These files are periodically updated by USACE as new observational data become available.

Each probability file contains monthly climatological NBS thresholds for a specific lake and exceedance probability.

Example file structure:

Probabilities of exceedance values for ERI
# Computed from RNBS values from 1900 to 2024
# Units: cms

"Probability Of Exceedance","Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"
"0.99",-924,-821,395,180,192,-133,-618,-1086,-1400,-1738,-1218,-1040
"0.98",-646,-512,638,340,268,-72,-586,-1035,-1365,-1535,-1175,-762
"0.97",-392,-349,709,412,317,-16,-546,-965,-1331,-1458,-1148,-657

These tables are generated using historical Residual Net Basin Supply (RNBS) records spanning 1900–present.

Each row represents:

  • A probability of exceedance (0.01–0.99)
  • The corresponding NBS threshold for each month

Calculation

CEP is defined as:

CEP = P(X_clim > X_forecast)

where:

  • CEP = climatology exceedance probability
  • X_clim = climatological distribution of historical values
  • X_forecast = forecasted value

Rather than explicitly calculating the probability distribution during runtime, the NBS-Predictor uses the precomputed USACE lookup tables.

For each forecasted value:

  1. Determine the forecast month
  2. Determine the lake
  3. Select the corresponding USACE probability table
  4. Compare the forecasted NBS value to the monthly climatological thresholds
  5. Assign the closest exceedance probability

This can be expressed as:

CEP = argmin | X_lookup(p) - X_forecast |

where:

  • X_lookup(p) is the climatological threshold associated with exceedance probability p
  • The selected CEP is the probability whose threshold is closest to the forecast value

Implementation

CEP is calculated using the calculate_cep() method.

The function:

  • Validates required forecast columns
  • Converts forecast dates to calendar months
  • Standardizes lake names
  • Uses a lookup function to determine CEP for each forecast

Required forecast dataframe columns:

  • forecast_month
  • model
  • lake
  • nbs

The output dataframe contains:

  • date
  • forecast_month
  • model
  • lake
  • nbs
  • cep

Example output:

date forecast_month model lake nbs cep
2026-06-01 2026-06 RF superior 4210 0.72

Interpretation

CEP values range from 0 to 1.

  • CEP = 0.5 → Forecast is near climatology (normal conditions)
  • CEP > 0.5 → Forecast indicates drier-than-normal conditions
  • CEP < 0.5 → Forecast indicates wetter-than-normal conditions

Assumptions

Several assumptions are inherent in CEP calculation:

  1. USACE probability tables adequately represent historical climatological variability.
  2. Historical RNBS distributions are assumed sufficiently stable for probabilistic comparison.
  3. Forecasts are compared only against climatology from the same calendar month to preserve seasonality.
  4. Selecting the nearest lookup value provides a sufficiently accurate estimate of exceedance probability.
  5. CEP describes relative climatological position and does not directly quantify forecast skill or uncertainty.

Notes

CEP provides a normalized metric that allows direct comparison across lakes and seasons, even when absolute NBS values differ substantially.

Unlike raw forecast values, CEP answers the question:

"How unusual is this forecast relative to historical conditions?"

This makes CEP particularly useful for:

  • Drought monitoring
  • Water level outlooks
  • Seasonal anomaly assessment
  • Communicating forecast severity

Note: CEP is referred to as Probability of Exceedance (POE) by the USACE.

Evaporation

Evaporation Calculation from Latent Heat Flux

Evaporation was calculated from model-derived latent heat flux using an energy balance approach. The Climate Forecast System (CFS) provides surface latent heat flux as an energy flux in units of W m⁻², equivalent to J s⁻¹ m⁻². This latent heat flux represents the rate of energy transfer associated with phase change during evaporation.

Evaporation rate was calculated as:

$$ E = \frac{LE}{\lambda} $$

where:

  • E = evaporation rate (kg m⁻² s⁻¹)
  • LE = latent heat flux (W m⁻²)
  • λ = latent heat of vaporization (J kg⁻¹)

Temperature-Dependent Latent Heat of Vaporization

The latent heat of vaporization was allowed to vary with air temperature following Allen et al. (1998):

$$ \lambda = 2.501 - (2.361 \times 10^{-3})T $$

where:

  • T = air temperature (°C)
  • λ = latent heat of vaporization (MJ kg⁻¹)

Air temperature provided in Kelvin was converted to degrees Celsius prior to calculation.

Latent heat flux was converted from W m⁻² to MJ s⁻¹ m⁻² by multiplying by:

$$ 10^{-6} $$

Unit Conversion

The resulting evaporation rate is expressed as:

$$ kg\ m^{-2}\ s^{-1} $$

which is physically equivalent to:

$$ mm\ s^{-1} $$

under the assumption that the density of liquid water is approximately:

$$ 1000\ kg\ m^{-3} $$

Thus:

$$ 1\ kg\ m^{-2} = 1\ mm $$

Monthly Aggregation

Monthly evaporation totals were calculated by multiplying the evaporation rate by:

  • Total number of seconds in the month
  • Grid-cell area

This provides an estimate of total volumetric water loss over each grid cell.

Assumptions

Several assumptions are inherent in this approach:

  1. The latent heat flux provided by the CFS accurately represents surface evaporation energetics and includes all relevant surface–atmosphere exchange processes resolved by model physics.

  2. Water density was assumed constant such that:

$$ 1\ kg\ m^{-2} = 1\ mm $$

  1. Latent heat of vaporization was assumed to vary only as a function of air temperature following Allen et al. (1998).

  2. Temporal averaging assumptions inherent in monthly mean latent heat flux fields were preserved during aggregation to monthly evaporation totals.

  3. Sublimation, blowing snow transport, and phase-dependent latent heat variations beyond liquid-water evaporation were not explicitly separated.

Notes

This approach directly leverages the physically based surface energy balance simulated by the atmospheric model and avoids the need to independently estimate evaporation using empirical bulk aerodynamic or Penman–Monteith formulations.

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