Writing the 24-hour delta — Johnathan Radojevich

The research note treats rapid intensification as a $\ge 30$ kt gain in $24$ hours. That sentence is short on purpose. The failure mode is not the threshold. It is having the threshold in a paper and a different one in a notebook.

I write the inequality first, then I write the function that is allowed to implement it. If they disagree later, the function is wrong.

The inequality

Let $v_{\max}(t)$ be the best-track maximum sustained wind at a 6-hour fix. The 24-hour forward delta is four steps on that cadence:

\Delta v_{24}(t) = v_{\max}(t + 24\,\mathrm{h}) - v_{\max}(t)

A raw RI flag is then an indicator that also refuses post-landfall points. Those are a different physical problem, and they inflate a seasonal count if you leave them in.

\mathrm{RI}(t) = \begin{cases} 1 & \text{if }\Delta v_{24}(t)\ge 30\,\mathrm{kt}\text{ and }t\text{ is pre-landfall}\\ 0 & \text{otherwise} \end{cases}

A storm that stays above the threshold for $18$ hours is still one event. Consecutive trues collapse to the first fix:

\mathrm{onset}(t) = \mathrm{RI}(t)\,(1 - \mathrm{RI}(t-6\,\mathrm{h}))

The seasonal count $N_y$ is the number of storms in season $S_y$ with at least one onset, not the number of 6-hour flags.

N_y = \sum_{s\in S_y}\mathbf{1}\{\exists\,t\in s:\mathrm{onset}(t)=1\}

That last line is the one people skip when they groupby a boolean and call it a frequency.

A series you can see

Six-hour $v_{\max}$ for a toy storm that jumps through $30$ kt between $t$ and $t+24\mathrm{h}$ , then holds:

$t$	$v_{\max}$ (kt)	$\Delta v_{24}$	$\mathrm{RI}$	$\mathrm{onset}$
$00\mathrm{Z}$	$55$	$+20$	$0$	$0$
$06\mathrm{Z}$	$60$	$+35$	$1$	$1$
$12\mathrm{Z}$	$70$	$+30$	$1$	$0$
$18\mathrm{Z}$	$85$	$+15$	$0$	$0$
$00\mathrm{Z}^{+1}$	$95$	—	—	—
$06\mathrm{Z}^{+1}$	$100$	—	—	—

The useful timestamp is $06\mathrm{Z}$ , not $12\mathrm{Z}$ . If you plot every true $\mathrm{RI}$ as a pin on a basin map, you draw a smear. Onset is the pin.

The function

The pandas that matches the three equations, and nothing else:

import pandas as pd

RI_KT = 30
STEPS = 4  # 24 h at a 6-hour cadence

def ri_onset(vmax: pd.Series, land: pd.Series) -> pd.Series:
    """True on the first 6-hour fix of a ≥30 kt / 24 h jump."""
    delta = vmax.shift(-STEPS) - vmax
    raw = (delta >= RI_KT) & ~land.astype(bool)
    return raw & ~raw.shift(1, fill_value=False)


def seasonal_count(onset: pd.Series, storm: pd.Series, year: pd.Series) -> pd.Series:
    flagged = onset.groupby([year, storm]).any()
    return flagged.groupby(level=0).sum()

shift(-4) is the $t+24\mathrm{h}$ in $\Delta v_{24}$ . The ~raw.shift(1) is the $(1-\mathrm{RI}(t-6\mathrm{h}))$ . If you find yourself adding a rolling max or a “sustained for two fixes” clause, you are writing a second definition. Put it in the math first.

The same onset, if the record arrives as a typed fix instead of a frame:

export type IntensityFix = {
  t: string
  vmaxKt: number
  inland: boolean
}

export function onsetIndex(fixes: IntensityFix[]): number[] {
  const hits: number[] = []

  for (let i = 0; i + 4 < fixes.length; i++) {
    if (fixes[i].inland) continue
    const jump = fixes[i + 4].vmaxKt - fixes[i].vmaxKt
    const already = i > 0 && !fixes[i - 1].inland
      && fixes[i + 3].vmaxKt - fixes[i - 1].vmaxKt >= 30
    if (jump >= 30 && !already) hits.push(i)
  }

  return hits
}

I keep both. The frame is what I run on HURDAT2. The typed loop is what I want next to a service that has to emit “this storm just jumped” as an event, not as a column that someone will re-derive at 2 a.m.

What I will not do in the function

I will not smooth $v_{\max}$ before the subtraction. A 6-hour best track is already an analysis. If a spike is wrong, that is a data problem, and it should fail a holdout, not disappear inside a rolling(3).mean().

I will not change $30$ to $25$ because a plot looks sparse. The Kaplan–DeMaria threshold is a convention. Conventions are allowed to be blunt. They are not allowed to drift per figure.

I will not count $\sum_t \mathrm{RI}(t)$ and call it $N_y$ . That quantity is hours spent intensifying, useful, and a different paper.

The earlier note is why the number matters after landfall. This one is only the contract: one inequality, one onset, one function, and a seasonal count that still means storms.