Seismic Hazards: Mathematical Modelling for A-Level Maths | 地震灾害：A-Level 数学建模

📚 Seismic Hazards: Mathematical Modelling for A-Level Maths | 地震灾害：A-Level 数学建模

Seismic hazards are usually studied in geography, but they also provide excellent applied contexts for A-Level Mathematics. In the Edexcel specification, real-world modelling questions often use earthquakes to test logarithms, exponential growth and decay, probability distributions, regression and calculus. This article builds a complete revision pathway from magnitude scales to hazard curves, so you can handle both pure-mathematical techniques and contextual exam problems.

地震灾害通常在地理学科中研究，但它也为 A-Level 数学提供了极好的应用背景。在 Edexcel 考试中，真实情境建模题经常以地震为素材考查对数、指数增长与衰减、概率分布、回归和微积分。本文构建从震级标度到灾害曲线的完整复习路径，帮助你同时掌握纯数学技巧和情境化考试题。

1. Magnitude Scales and Logarithms | 震级标度与对数

Seismic magnitude scales are logarithmic: a one-unit increase in magnitude corresponds to a tenfold increase in recorded wave amplitude. This is why a magnitude 7 earthquake shakes the ground much more strongly than a magnitude 6 event.

地震震级标度是对数标度：震级每增加 1 级，记录到的地震波振幅增大为原来的 10 倍。这就是为什么 7 级地震的地面震动远强于 6 级地震。

The original Richter scale is defined by the logarithm of the maximum trace amplitude relative to a reference amplitude.

最初的里氏震级通过最大波形振幅与参考振幅之比的对数来定义。

M = log₁₀(A ÷ A₀)

Here M is the magnitude, A is the maximum amplitude recorded by a seismograph, and A₀ is a standard reference amplitude. Because the scale is logarithmic, comparing two earthquakes uses the difference:

其中 M 是震级，A 是地震仪记录到的最大振幅，A₀ 是标准参考振幅。因为该标度是对数的，比较两次地震时使用差值：

ΔM = M₂ − M₁ = log₁₀(A₂ ÷ A₁)

If the amplitude ratio is 100, the magnitude difference is log₁₀(100) = 2, so the larger event is 2 magnitude units stronger.

如果振幅之比为 100，则震级差为 log₁₀(100) = 2，因此较大地震强 2 个震级单位。

2. Gutenberg-Richter Frequency-Magnitude Law | 古登堡-里希特频度-震级定律

A key empirical law in seismology is the Gutenberg-Richter relation. It links the number of earthquakes above a given magnitude with magnitude itself, and it is a very common context for linear regression in A-Level Maths.

地震学中一个关键的经验定律是古登堡-里希特关系。它把超过某一震级的地震次数与震级本身联系起来，也是 A-Level 数学中线性回归非常常见的背景。

log₁₀ N = a − bM

Here N is the number of earthquakes with magnitude at least M in a fixed time window. The parameter a depends on the region and time interval, while b is the slope. Globally, b is often close to 1.0.

其中 N 是在固定时间窗口内震级不小于 M 的地震次数。参数 a 取决于区域和时间区间，b 是斜率。全球范围内 b 通常接近 1.0。

For example, if a = 5.0 and b = 0.9, then for M = 4.0 the expected number is:

例如，如果 a = 5.0 且 b = 0.9，那么 M = 4.0 时的预期次数为：

N = 10^(5.0 − 0.9 × 4.0) = 10^(1.4) ≈ 25.1

You should be able to rearrange this equation to find M given N, and to interpret the gradient b as the rate at which the logarithm of frequency decreases per magnitude unit.

你应当能够对该方程进行变形，在已知 N 时求 M，并解释斜率 b 是频率的对数每增加一个震级单位而下降的速率。

3. Energy-Magnitude Relationship | 能量-震级关系

Earthquake energy release is not linearly related to magnitude. A widely used empirical relation is:

地震释放的能量与震级不是线性关系。一个广泛使用的经验关系是：

log₁₀ E = 4.8 + 1.5M

Here E is energy in joules. This equation shows that adding 1 to the magnitude multiplies the energy by 10^1.5, which is about 31.6.

其中 E 是以焦耳为单位的能量。该方程表明，震级每增加 1，能量乘以 10^1.5，约为 31.6

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