# Using Logistic Regression to Identify the Key Hydrologic Controls of Ice-Jam Flooding near the Peace–Athabasca Delta: Assessment of Uncertainty and Linkage with Physical Process Understanding

## Abstract

**:**

## 1. Introduction

## 2. Background Information

_{1}, x

_{2}, x

_{3}, …)} = 1/(1 + exp(−X))

_{1}, x

_{2}, and x

_{3}are the explanatory variables and

_{0}+ b

_{1}x

_{1}+ b

_{2}x

_{2}+ b

_{3}x

_{3}+ …

_{0}, b

_{1}, b

_{2}, and b

_{3}being coefficients, of which the numerical values are furnished by the regression. The form of Equation (1) ensures that P always varies between 0 and 1 (as it should), regardless of the value of X, while Equation (2) assumes that the effects of the explanatory variables are linear and additive. Together, Equations (1) and (2) imply that X = ln{P/(1 − P)} = natural logarithm of the “odds ratio” and is called the “logit” in the statistical literature. To those who are accustomed to physics-based research, considerable serendipity appears to be needed to come across linear and additive logits (see also Section 4). This is a key point because the numerical values of b

_{0}, b

_{1}, b

_{2}, … and their associated p-values depend on the structure of Equation (2).

## 3. Epistemic Uncertainty

_{io}) alone indicate statistical significance (at the 0.05 p-value threshold) for Q7, near-significance for HF, and a lack of significance for h

_{io}(Table 1). Simultaneous regressions of (Q7, HF) and (Q7, HF, h

_{io}) indicate significance for Q7 and HF but a lack of significance for h

_{io}. Regardless of the assumed model, the regressions indicate that Q7 and HF have positive and negative effects on the chances of an IJF, respectively. However, the model of h

_{io}alone points to a negative effect (b

_{1}< 0), while the model that combines h

_{io}with Q7 and HF indicates a positive effect (b

_{3}> 0).

_{io}and S5, the remainder of this paper focuses on the primary variables Q7 and HF, which are practically uncorrelated (Pearson’s r ≈ 0.07).

## 4. Structural Uncertainty

_{0}+ b

_{1}Q7 + b

_{2}DF

_{0}+ b

_{1}√Q7 + b

_{2}(DF)

^{3}

_{0}+ b

_{1}ln(Q7) + b

_{2}ln(DF)

^{b1}(DF)

^{b2}. Table 2 summarizes the results of logistic regressions with these models and shows that all p-values are lower or slightly higher than the conventional significance limit of 0.05. The same applies to various other models that were also tried, each involving different functions of Q7 and DF. It is concluded that these two variables are indeed statistically significant controls of IJF occurrence.

## 5. Other Sources of Uncertainty

^{3}/s [33]. Consequently, P(IJF) is nil if Q7 is less than, or equal to, 4000 m

^{3}/s. However, all Q7-related regressions that have been performed so far predict nonzero probabilities for all values of Q7. It is not known how to account for this discrepancy in the logistic regression. A possible option could be to discard the years in which Q7 was less than 4000 m

^{3}/s. However, this approach would severely reduce the sample size, exacerbating the parametric uncertainty.

## 6. Physics of the Spring Breakup Process in the Lower Peace River

_{io}. For the vicinity of the Peace Point gauge, the effects of these two variables are combined in a dimensionless quantity, R

_{tf}, which can be closely approximated using the following expression [26]:

_{tf}≈ h

_{io}

^{1/2}f(DF)

_{io}at Peace Point is limited (~0.7 to 1.2 m), the variability of Rtf is dominated by HF. This is illustrated in Figure 8, where Rtf is seen to vary almost uniquely with HF; the relatively small scatter is caused by the variability of h

_{io}. The influence of resistance is evident in Table 3, which summarizes the breakup travel times for the four notable events mentioned in the previous paragraph. Despite the extreme 2020 breakup flow, the advance of the front was relatively slow and an IJF did not materialize.

- −
- Weather conditions before and during the advance of the front influence the competence of the downstream ice cover and its capacity to cause ice jams or add rubble to them. Internal ice structure, air temperature, solar radiation, and melt rate of the snow cover are relevant weather-related variables [43];
- −
- The temporal gradient of the flow hydrograph as the breakup front advances along the river: increasing flow facilitates the release of ice jams that may form along the way, reducing residence times and enhancing celerity. The opposite will apply if the flow peak “overtakes” the front, which will then advance under a negative flow gradient.

## 7. Discussion

## 8. Summary and Conclusions

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plan view of Peace River and Peace–Athabasca Delta (showing only the northern portion of Athabasca River). From [13], with changes.

**Figure 2.**IJF occurrence (binary variable = 1) or non-occurrence (binary variable = 0) versus predicted probability using the regression model of [22]. Note the large overlap range that includes 5 IJFs and 6 non-floods.

**Figure 3.**Peak 7-day running average flow (Q7) at Peace Point during the breakup period versus Grande Prairie winter precipitation (WP), as determined in [22]. The red circles mark IJF years. Data range: 1962–2020, excluding the reservoir-filling years 1968–1971. Flow data source: WSC archives for the Peace Point gauge. https://wateroffice.ec.gc.ca/. Accessed on 28 October 2023.

**Figure 4.**Cross-sectional average ice thickness near the WSC Peace Point gauge, plotted versus accumulated degree-days of frost at Fort Chipewyan starting on October 1 of the year preceding the breakup year (Fort Chipewyan is located on the north shore of Lake Athabasca, ~90 km SE of Peace Point). The red circles mark IJF years. Adapted from [28]. Data range: 1962–2020. Thickness data source: WSC archives for the Peace Point gauge.

**Figure 5.**Variation in the probability of an ice-jam flood (PIJF) with Peace Point freezeup level and breakup flow, as calculated by logistic regression on Q7 and HF. Logistic regression was performed using the Firth bias-reduced algorithm supplied by Wessa [31].

**Figure 6.**Isolines for P(IJF) = 0.50, as generated by the three models in Table 2. The data points represent observed annual values of Q7 and HF at Peace Point for the period from 1962 to 2020, excluding the reservoir-filling years 1968–1971; the red circles mark IJFs. For 2018, the flow may have been higher than shown; for 2020, the freezeup level is only known to be within a range of elevations.

**Figure 7.**IJF occurrence (binary variable = 1) or non-occurrence (binary variable = 0) versus predicted probability using the third model of Table 2, which essentially involves the product of the powers of the explanatory variables.

**Figure 8.**Variation in the resistance component Rtf with the freezeup level for the vicinity of the Peace Point hydrometric gauge and for the years 1962 to 2020. Reservoir filling years are included because the flow hydrograph does not influence the relationship between Rtf and the variables HF and h

_{io}.

**Table 1.**Coefficients and p-values associated with logistic regression on different combinations of explanatory variables. Relevant hydrometric and climatic data are summarized in a supplement to this paper.

Model Variables | Model Coefficients (and p-Values) | |||
---|---|---|---|---|

b_{0} | b_{1} | b_{2} | b_{3} | |

Q7 (m^{3}/s) | −6.905 (0.00035) | 0.0011 (0.0025) | NA | NA |

HF (m) | 169.95 (0.067) | −0.804 (0.064) | NA | NA |

h_{io} (m) | −0.557 (0.844) | −1.398 (0.664) | NA | NA |

S5 (°C-days) | 1.680 (0.299) | −0.026 (0.045) | NA | NA |

Q7, HF | 321.83 (0.037) | 0.0015 (0.0065) | −1.550 (0.034) | NA |

Q7, HF, h_{io} | 360.50 (0.049) | 0.0019 (0.014) | −1.805 (0.044) | 14.752 (0.110) |

_{io}); reservoir filling years 1968–1971 are excluded; the year 2006 is excluded when the regression involves Q7 (insufficient data); Q7 and HF may have been underestimated in 2018 and 2020, respectively; their p-values decrease slightly if either one is increased.

**Table 2.**Exploring structural uncertainty for models that utilize breakup flow and freezeup level as the explanatory variables.

Model Variables | Model Coefficients (and p-values) | ||
---|---|---|---|

b_{0} | b_{1} | b_{2} | |

Q7, DF | −12.39 (0.0048) | 0.0014 (0.0058) | 1.53 (0.039) |

(Q7)^{0.5}, (DF)^{3} | −15.26 (0.0017) | 0.185 (0.0032) | 0.037 (0.045) |

ln(Q7), ln(DF) | −69.84 (0.0086) | 7.64 (0.0096) | 3.94 (0.066) |

**Table 3.**Breakup front travel times and relevant hydroclimatic controls for the years 1996, 1997, 2014, and 2020. T

_{bf}= travel time of breakup front between ~Sunny Valley and ~mouth of Peace River; T

_{jam}= duration of PAD jam.

Breakup Year | IJF? Yes/No | HF (m) | h_{io} (m) | Q7 (m ^{3}/s) | R_{tf} | Approx. T _{bf} (Days) | Approx. T _{jam}(Days) |
---|---|---|---|---|---|---|---|

1996 | Yes | 212.38 | 0.87 | 5514 | 1.12 | 3 | 8 |

1997 | Yes | 214.23 | 0.85 | 7596 | 2.50 | 6 | 7 |

2014 | Yes | 213.64 | 0.92 | 5591 | 2.05 | 5 | 8 |

2020 | No | 214.9 to 215.5 | 0.75 to 0.80 | 7857 | 3.05 to 3.73 | 13 | 1.5 |

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**MDPI and ACS Style**

Beltaos, S.
Using Logistic Regression to Identify the Key Hydrologic Controls of Ice-Jam Flooding near the Peace–Athabasca Delta: Assessment of Uncertainty and Linkage with Physical Process Understanding. *Water* **2023**, *15*, 3825.
https://doi.org/10.3390/w15213825

**AMA Style**

Beltaos S.
Using Logistic Regression to Identify the Key Hydrologic Controls of Ice-Jam Flooding near the Peace–Athabasca Delta: Assessment of Uncertainty and Linkage with Physical Process Understanding. *Water*. 2023; 15(21):3825.
https://doi.org/10.3390/w15213825

**Chicago/Turabian Style**

Beltaos, Spyros.
2023. "Using Logistic Regression to Identify the Key Hydrologic Controls of Ice-Jam Flooding near the Peace–Athabasca Delta: Assessment of Uncertainty and Linkage with Physical Process Understanding" *Water* 15, no. 21: 3825.
https://doi.org/10.3390/w15213825