# Optimizing Analog Ensembles for Sub-Daily Precipitation Forecasts

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

- each station has at least 90% of data available, and missing data is not systematically distributed (e.g., in the same season, at the same time of day, or over a large consecutive period); and
- outliers are reasonable considering the synoptic situation (e.g., convection), nearby stations, and the available station climatology.

#### 2.2. Analog Ensemble Methodology

#### 2.2.1. Predictor Selection Procedures

- All-EFS: Using the EFS to test all 40 variables in addition to PCP as predictors.
- DC-EFS: Using the EFS to test the same subset of 10 predictor candidates as in DC-FS.
- DCV-EFS: Using the EFS to test a subset of 10 variables as predictors, except here, the predictor candidates are based on the best DCorr, as well as the variance inflation factor (VIF, a measure of multicollinearity among variables). Specifically, we grow a set of 10 predictor candidates by sequentially adding one variable at a time, starting from the best ranking DCorr, provided the VIF among the growing set of predictor candidates stays below a threshold value of 10. If this threshold is exceeded it means that the variable exhibits strong correlation with other variables that were already selected and we assume that this variable contributes no additional value as a predictor for the AnEn. Since some of our 41 variables are related (e.g., the same variables at different vertical levels), the VIF check limits the use of correlated and presumably redundant variables in the FS.

#### 2.2.2. The Supplemental-Lead-Time (SLT) Approach

## 3. Results and Discussion

#### 3.1. Predictor Selection Optimization

#### 3.2. Temporal Trend Similarity

#### 3.3. Supplemental Lead Times (SLTs)

#### 3.4. Verification

- the control AnEn using reference predictors, no TTS, and no SLTs (Control),
- the AnEn with optimized predictors, but no TTS, and no SLTs (Step 1),
- the AnEn with optimized predictors and optimized TTS consideration, but no SLT (Step 2), and
- the AnEn with optimized predictors and optimized TTS consideration, and using SLTs in a window of $\pm 6$ (Step 3).

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AnEn(s) | Analog Ensemble(s) |

AnFcst(s) | Analog Forecast(s) |

AnObs | Analog Observation(s) |

TaFcst(s) | Target Forecast(s) |

VerifObs | Verifying Observation(s) |

TTS | Temporal Trend Similarity |

SLT(s) | Supplemental Lead Time(s) |

FS | Forward Selection |

EFS | Efficient Forward Selection |

## Appendix A. Percentiles

**Figure A1.**Top row: Histogram of binned daily observed 75th percentiles (75p) at all stations (top left), and corresponding geographic distribution of the 75p relative to topography (top right); Bottom row: Histograms of 75p and 90p for other accumulations. Although not plotted here, these frequency distributions show similar geographic variations to the top right panel.

## Appendix B. Evaluation

#### Appendix B.1. Threshold-Weighted Continuous Ranked Probability Score

#### Appendix B.2. Statistical Tests

## Appendix C. Correlation Analysis

**Figure A2.**Distance Correlation coefficients (DCorr) of all model variables (see Table 1) with observed precipitation. Variability across 46 stations aggregated over time (left), and variability across months aggregated over stations (right). The time period covers four complete years from the optimization period (rather than the full 4.75-year period) to ensure similar sample size across months.

## Appendix D. Predictor Weights

**Figure A3.**Station average of the final predictor weights resulting from the All-EFS method (see Section 2.2.1) for each season. Variables that are in Table 1 but not in the x-axis were never selected by any station at any season.

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**Figure 1.**Domain of interest in southwest British Columbia with locations of 46 stations that provide hourly precipitation observations from two networks.

**Figure 3.**Graphic of the supplemental-lead-time approach (SLT; bottom), compared to the original approach (top). The circles along the arrows represent lead times of an initialization. This example illustrates the analog search at lead time t. For the first past-forecast (PaFcst) initialization, the SLT approach using ±1 SLTs selects the analog forecast (AnFcst) at lead time t as in the original approach. For the second PaFcst initialization, the SLT approach finds a better AnFcst at lead time $t+1$.

**Figure 4.**Box-and-whisker plots of 75p twCRPSS distributions across stations (46 stations in each boxplot, except in summer) after predictor optimization with four methods. The dotted zero line separates values that indicate improvement (positive values) vs. deterioration (negative values) compared to the reference twCRPS using control predictors. Performance differences between training (lighter colors) and testing (darker colors) informs about the degree of overfitting.

**Figure 5.**Heatmaps of station-averaged twCRPSS for hourly to 12-hourly discrete accumulation windows using $\tau $ between 1 and 5 to consider temporal trend similarity (TTS) for all seasons and forecast windows. Blue (red) colors indicate better (worse) average twCRPS compared to the reference using $\tau =0$ (no TTS). Crosses “X” mark significant differences in twCRPS station distributions compared to the reference. Empty circles mark the value $\tau $ that exhibits best improvement overall, and filled circles correct the position of best $\tau $ if the value in the empty circle is not significant.

**Figure 6.**As in Figure 5 but for 3-hourly to daily rolling windows. Again, empty circles mark the value $\tau $ that exhibits best improvement overall, and filled circles correct the position of best $\tau $ if the value in the empty circle is not significant. Forecast day 1 for daily accumulations is removed, since it contains only 1 instead of 24 lead-time samples as on the other forecast days.

**Figure 7.**Lead-time aggregated (

**top**) and station-aggregated (

**bottom**) twCRPSS from the supplemental-lead-time (SLT) experiments. The reference for twCRPSS is the original approach with optimized predictors but without SLT. Positive twCRPSS indicate improvement compared to the reference.

**Figure 8.**Station mean of the mean absolute error skill score (MAESS) by forecast day and for hourly to daily rolling windows, relative to the raw NWP forecast.

**Figure 9.**Station-aggregated 90p reliability diagrams on all forecast days for hourly discrete to daily rolling accumulation windows. The dashed black line is the reference for perfect reliability, the grey dotted lines show climatological probability. The inset in the lower right corner displays the corresponding sharpness diagram, which shows the relative frequency of forecasts that fall into each bin. Due to the skewed nature of precipitation distributions, the y-axis in the sharpness diagram is plotted on a logarithmic scale. The reliability diagram displays only bins that include at least 50 samples in total (i.e., only those points above the dashed grey line in the sharpness diagram).

**Figure 10.**Station-aggregated 90p receiver operating characteristic (ROC) diagrams for all forecast days and hourly discrete to daily rolling accumulation windows. The area under the curve (AUC) is given in each legend and has a perfect score of 1. The dashed black line represents the line of no skill with AUC = 0.5 corresponding to climatology.

Variable | Abbreviation | Levels |
---|---|---|

Total Precipitation | PCP | Surface |

Integrated Water Vapor * | IWV | Column |

Integrated Vapor Transport * | IVT | Column |

Water Vapor Mixing Ratio | r | 2 m, 70 kPa, 50 kPa |

Specific Humidity * | SH | 70 kPa, 50 kPa |

Relative Humidity * | RH | 70 kPa, 50 kPa |

Moisture Index 1 * ($R{H}_{70kPa}\times IWV$) | MI1 | |

Moisture Index 2 * ($R{H}_{70kPa}\times {W}_{70kPa}$) | MI2 | |

Temperature | T | 2 m, 70 kPa, 50 kPa |

Potential Temperature | Th | 2 m, 70 kPa, 50 kPa |

Dewpoint Temperature | Td | 70 kPa, 50 kPa |

Total Totals Index * | TT | |

K-Index * | KI | |

U-component Wind | U | 10 m, 70 kPa, 50 kPa |

V-component Wind | V | 10 m, 70 kPa, 50 kPa |

W-component Wind | W | 70 kPa, 50 kPa |

Wind Direction * | WD | 10 m, 70 kPa, 50 kPa |

Wind Speed * | WS | 10 m, 70 kPa, 50 kPa |

Sea Level Pressure | SLP | Sea Level |

Surface Pressure | SfcP | Surface |

Geopotential Height | GPH | 70 kPa, 50 kPa |

Boundary Layer Height | PBLH |

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**MDPI and ACS Style**

Jeworrek, J.; West, G.; Stull, R.
Optimizing Analog Ensembles for Sub-Daily Precipitation Forecasts. *Atmosphere* **2022**, *13*, 1662.
https://doi.org/10.3390/atmos13101662

**AMA Style**

Jeworrek J, West G, Stull R.
Optimizing Analog Ensembles for Sub-Daily Precipitation Forecasts. *Atmosphere*. 2022; 13(10):1662.
https://doi.org/10.3390/atmos13101662

**Chicago/Turabian Style**

Jeworrek, Julia, Gregory West, and Roland Stull.
2022. "Optimizing Analog Ensembles for Sub-Daily Precipitation Forecasts" *Atmosphere* 13, no. 10: 1662.
https://doi.org/10.3390/atmos13101662