# A Re-Evaluation of the Swiss Hail Suppression Experiment Using Permutation Techniques Shows Enhancement of Hail Energies When Seeding

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## Abstract

**:**

## 1. Introduction

^{2}in the centre of Switzerland, ${47}^{\circ}$ North, ${8}^{\circ}$ East. The region was surveyed by two radars and a network of hailpads, the seeding performed by Soviet rockets and launchers. A total of 37 experimental days were drawn for seeding, 46 for non-seeded controls, containing 113 and 140 cells, respectively. The total hail energy on ground (${E}_{GR}$) was determined for each cell by radar. Everything was done to reproduce exactly the procedure used at that time in the USSR. However, the performance of the seeding was unsatisfactory as only one half of the prescribed number of rockets were launched successfully. From the logbooks for each cell the “seeding coverage” ($sc$), the ratio of successful launches to the prescribed number of rockets or, in other terms, the fraction of the duration of correct seeding was determined. One rocket had to be fired every 5 min as long as the seeding criterion was fulfilled. The shortcoming of the seeding makes it questionable whether Grossversuch IV is a representative test for the concept of Sulakvelidze, but thanks to $sc$ it is still a useful and important experiment as will be shown here.

- Unsatisfactory seeding was not taken into account in [7]. The magnitude of the treatment variable $sc$, varying from 0 to 1, contains information on how well seeding was done. Instead of using the objective values of $sc$, in [7] it was replaced by $sc=1$ whenever seeding was planned, while some 20% of the cells planned for seeding were not at all seeded.
- In [7], the response variable ${E}_{GR}$ was converted to its logarithm $ln({E}_{GR}+1)$. This non-linear transformation reduces ${E}_{GR}$ of severe hailstorms nearly to the level of the many light storms. It aborts the physical meaning of ${E}_{GR}$ and its tight correlation to crop damage and changes the probability to reject ${H}_{0}$. It will be shown that conflicting results can be obtained for the original and the transformed variable (see Section 4.1).
- Some evaluations used a predictor based on meteorological data. This introduced complexity and errors in the statistical analysis.
- The data of the hailpads is not representative enough to calculate hail kinetic energies, as will be shown by statistical evaluations.

## 2. The Hail Suppression Experiment “Grossversuch IV”

^{2}was surveyed by radar and by hailpads. On 83 experimental days 253 convective cells were found to comply with the conditions for seeding, 154 were thermal and 99 frontal thunderstorms. For every cell ${E}_{GR}$ was estimated by radar. A visualization of the data is shown in Figure 1. ${E}_{GR}$ is stratified by the lifetime of the cells, i. e. the time between the criterion of seeding first and last met. The lifetime of the cells is typically 10 to 100 min. Some of the shorter lifetimes may be due to cells moving into or out of the experimental zone.

^{−3}ice crystals into the region important for hail embryo growth.

^{2}per hailpad representing 3.8 to 4 km

^{2}and maybe other errors led to stochastic variations which made it improbable to reach statistical significance for the demanding variable ${E}_{GR}$.

^{−2}s

^{−1}.

^{2}large and with a mesh area of 3.8 to 4.0 km

^{2}are found in the appendix of [7]. The results correlate with those from the radar observation but the stochastic variations are too large to reach statistical significance for hail energies. Evidence for this statement is found towards the end of Section 3.4. More reliable are the results for a less demanding variable, such as the area touched by hail, see ${S}_{G}$ in Table 13 in [7]. However, a decrease of the number of hailstones or an increase of the number of pads hit when seeding does not allow to draw conclusions about the total hail energy.

## 3. Methods and Results

#### 3.1. The Variables and Parameters

#### 3.2. The Calculation of Probabilities

#### 3.3. Confidence Intervals and Standard Error

#### 3.4. Re-Evaluated Results of Grossversuch IV

- Each cell contributes the same amount to the daily ${E}_{GR}$ of non-seeded cells, corresponding to total intraday autocorrelation. The result of the permutation test for the 253 cells is $\mathit{rr}$ = 3.0, $P\left({H}_{0}\right)$ = 0.27%.
- The daily total comes from only one cell, the other cells of the same day are without hail. In this case $\mathit{rr}$ = 3.0, $P\left({H}_{0}\right)$ = 0.74% is obtained.

## 4. Discussion

#### 4.1. What Transformations Do

- 100 seeded and 20 non-seeded cells with both ${E}_{GR}$ and $ln(1+{E}_{GR})$ = 0
- 100 non-seeded cells with ${E}_{GR}$ =150, $ln(1+{E}_{GR})$ = 5
- 20 seeded cells with ${E}_{GR}$ = 3000, $ln(1+{E}_{GR})$ = 8

#### 4.2. Multiplicity Effects

#### 4.3. Possible Mechanisms

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Modeling an Experiment by Permutation or Bootstrap

**Figure A1.**Eight cups of tea tasted by the lady in Fisher’s experiment (see [23] (p. 59)). The probability for accidental hits assuming ${H}_{0}$ true is calculated either by permutation (blue) or by bootstrap (red). The blue cross indicates the statistical significance when the partition is known, the red cross when not known and everything is possible.

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**Figure 1.**Visualization of the data from the Swiss hail suppression experiment “Grossversuch IV” [7] that builds the basis of the re-evaluation presented in this paper. For better readability some overlapping points have been slightly separated on the x-axis and a logarithmic scale is used, necessitating to add 1 to the data.

**Figure 2.**Visualization of the seeding coverage versus the duration of cell lifetime, which is defined as the time with radar reflectivity exceeding 45 dBZ. The dots correspond to the 113 cells on the days that have been selected for seeding in the randomization process.

**Figure 3.**Cumulated probability P to obtain a correlation coefficient R more extreme than the value indicated on the x axis, provided that the null hypothesis ${H}_{0}$ is true. Three curves show the cumulative distribution function of min$(P,1-P)$ to read off $P\left(R\right|{H}_{0})$ for the methods Fisher’s z (green, cross), permutation (blue, cross) and bootstrapping the scores of ${E}_{GR}$ (red, circle). R and the curves are calculated for ${E}_{GR}$ and $sc$ of the 83 experimental days.

**Figure 4.**Cumulated probability P to obtain a correlation coefficient R more extreme than the value indicated on the x axis, provided that the alternative hypothesis ${H}_{1}$ is true. Three curves cdf of min$(P,1-P)$ to read off $CI$ for the methods based on Fisher’s z (green), permutation CIP (blue) and bootstrap (red). The two black circles indicate the $CI$ obtained by BCa bootstrapping. The crosses remind $P\left({H}_{0}\right)$ of Figure 3. The blue square indicates R at a probability of 15.9% (see end of Section 3.3). R and the curves are calculated for ${E}_{GR}$ and $sc$ of the 83 experimental days.

**Table 1.**Parameters $\mathit{dif}$ in MJ per cell and risk ratio $\mathit{rr}$ calculated by two models: regression or weighted average based on $\mathit{avs}$, $\mathit{avn}$. Conversion of $\mathit{dif}$ for days to MJ/cell by the factor 83/253. The probabilities calculated later in Section 3.2 are added.

Model | n | $\mathit{dif}(\mathbf{MJ}/\mathbf{Cell})$ | $\mathit{P}\left(\mathit{dif}\right|{\mathit{H}}_{0})$ | $\mathit{rr}$ | $\mathit{P}\left(\mathit{rr}\right|{\mathit{H}}_{0})$ |
---|---|---|---|---|---|

regression (days) | 83 | 1612 | 0.38% | 3.27 | 0.38% |

regression (cells) | 253 | 1583 | 0.38% | 3.01 | 0.38% |

$\mathit{avs}$, $\mathit{avn}$ (days) | 83 | 1721 | 0.53% | 3.01 | 0.87% |

$\mathit{avs}$, $\mathit{avn}$ (cells) | 253 | 1942 | 0.29% | 3.50 | 0.31% |

Experiment | Unit | Seeded | Non-s. | $\mathit{dif}/\mathbf{cell}$ | $\mathit{dif}-\mathit{\sigma}$ | $\mathit{P}\left({\mathit{H}}_{0}\right)$ | $\mathit{rr}$ | $\mathit{rr}-\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|---|

${E}_{GR}$ versus $sc$ | days | 34 | 49 | 1612 | 966 | 0.4% | 3.3 | 2.1 |

${E}_{GR}$ versus $sc$ | cells | 93 | 160 | 1583 | 880 | 0.4% | 3.0 | 1.8 |

Means of two groups | ||||||||

${E}_{GR}$ versus seeded, non-seeded | days | 34 | 49 | 1316 | 665 | 2.0% | 2.6 | 1.6 |

${E}_{GR}$ versus seeded, non-seeded | cells | 93 | 160 | 1615 | 964 | 0.5% | 3.1 | 2.0 |

Two groups (for comparison to [7]: cells planned for seeding but not seeded are attributed to seeded group) | ||||||||

${E}_{GR}$ versus planned, non-planned | cells | 113 | 140 | 3.7% | 2.2 | 1.4 | ||

Federer [7] Table 21, $C\left(\alpha \right)$ test | cells | 113 | 140 | 1.9% | 2.2 | 1.5 | ||

$2\times 2$ contingency table | ||||||||

hail, no-hail versus seeded, non-seeded | cells | 78 + 15 | 111 + 49 | 0.5% | 1.2 | 1.1 | ||

idem, for hailpads (213 cases) | cells | 45 + 29 | 64 + 75 | 2.1% | 1.3 | 1.2 |

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**MDPI and ACS Style**

Auf der Maur, A.; Germann, U. A Re-Evaluation of the Swiss Hail Suppression Experiment Using Permutation Techniques Shows Enhancement of Hail Energies When Seeding. *Atmosphere* **2021**, *12*, 1623.
https://doi.org/10.3390/atmos12121623

**AMA Style**

Auf der Maur A, Germann U. A Re-Evaluation of the Swiss Hail Suppression Experiment Using Permutation Techniques Shows Enhancement of Hail Energies When Seeding. *Atmosphere*. 2021; 12(12):1623.
https://doi.org/10.3390/atmos12121623

**Chicago/Turabian Style**

Auf der Maur, Armin, and Urs Germann. 2021. "A Re-Evaluation of the Swiss Hail Suppression Experiment Using Permutation Techniques Shows Enhancement of Hail Energies When Seeding" *Atmosphere* 12, no. 12: 1623.
https://doi.org/10.3390/atmos12121623