📚 Water Security: Applied Mathematics for Resource Management | 水安全：资源管理中的应用数学
Water security is a global challenge that requires precise quantitative analysis. In A-Level Edexcel Mathematics, we can use functions, statistics, probability, calculus and optimisation to model water supply, demand and storage. This article links core mathematical techniques to realistic water security problems, helping you apply pure and applied mathematics in context.
水安全是一项需要精确定量分析的全球性挑战。在 A-Level Edexcel 数学中，我们可以使用函数、统计、概率、微积分和优化来模拟水的供应、需求和储存。本文将核心数学技术与现实水安全问题联系起来，帮助你在实际情境中应用纯数学和应用数学。
1. Functions and Variables in Water Supply Models | 供水模型中的函数与变量
In water security modelling, we define supply S(t), demand D(t) and storage V(t) as functions of time t. The net change in storage equals the difference between inflow and outflow.
在水安全建模中，我们将供水量 S(t)、需水量 D(t) 和储水量 V(t) 定义为时间 t 的函数。储水量的净变化等于流入量与流出量之差。
V(t) = V(0) + ∫₀ᵗ [S(u) − D(u)] du
This fundamental balance equation is the starting point for many A-Level modelling questions. You may be asked to evaluate V(t) for given linear or quadratic functions of S(u) and D(u).
这个基本平衡方程是许多 A-Level 建模题的起点。你可能会被要求对给定的线性或二次函数 S(u) 和 D(u) 求 V(t)。
- If S(t) = 3t + 2 and D(t) = t², integrate term by term.
- 如果 S(t) = 3t + 2，D(t) = t²，则逐项积分。
2. Statistical Measures of Rainfall Data | 降雨量数据的统计量
Annual rainfall data are often summarised using the mean μ and standard deviation σ. For grouped frequency tables, use class midpoints to estimate these measures.
年降雨量数据通常用均值 μ 和标准差 σ 来概括。对于分组频数表，使用组中值来估计这些度量。
μ = Σxᵢ / n, σ = √[Σ(xᵢ − μ)² / n]
These statistics describe the central tendency and spread of rainfall. A low mean and high standard deviation indicate unreliable water supply, which is a key water security risk.
这些统计量描述了降雨量的集中趋势和离散程度。均值低且标准差高表明供水不可靠，这是水安全的关键风险。
- Use Σxᵢfᵢ for grouped data, then divide by Σfᵢ.
- 对于分组数据，使用 Σxᵢfᵢ，然后除以 Σfᵢ。
- Variance = Σxᵢ²fᵢ / Σfᵢ − μ².
- 方差 = Σxᵢ²fᵢ / Σfᵢ − μ²。
3. Probability Distributions for Drought Events | 干旱事件的概率分布
The number of drought years in a fixed period can be modelled by a binomial distribution B(n, p) if years are independent. For rare events over a long interval, a Poisson distribution Po(λ) is appropriate.
如果年份相互独立，则固定时期内干旱年数可以用二项分布 B(n, p) 建模。对于长时间间隔内的稀有事件，泊松分布 Po(λ) 更合适。
P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ
For example, if the probability of a drought year is 0.2 and we observe 10 years, find P(X = 2). This gives the likelihood of exactly two drought years.
例如，如果干旱年的概率为 0.2，我们观察 10 年，求 P(X = 2)。这给出了恰好两年干旱的可能性。
- For Poisson: P(X = k) = e⁻λ λᵏ / k!
- 对于泊松分布：P(X = k) = e⁻λ λᵏ / k!
4. Exponential Growth of Water Demand | 水需求的指数增长
Water demand often grows exponentially with population. The model D(t) = D₀ eᵏᵗ is common, where D₀ is initial demand and k is the growth rate.
水需求通常随人口呈指数增长。模型 D(t) = D₀ eᵏᵗ 很常见，其中 D₀ 是初始需求，k 是增长率。
D(t) = D₀ eᵏᵗ
You may need to find k from two data points using logarithms: ln[D(t) / D₀] = kt. This tests your skills with exponential and logarithmic functions.
你可能需要用对数从两个数据点求 k：ln[D(t) / D₀] = kt。这考察你对指数函数和对数函数的技能。
- If D doubles in 15 years, solve eᵏ×15 = 2 to get k = ln 2 / 15.
- 如果 D 在 15 年内翻倍，解 eᵏ×15 = 2 得 k = ln 2 / 15。
5. Differential Equations for Reservoir Levels | 水库水位的微分方程
The rate of change of reservoir storage is the difference between inflow and outflow. This leads to the differential equation dV/dt = S(t) − D(t).
水库储水量的变化率是流入量与流出量之差。这导出微分方程 dV/dt = S(t) − D(t)。
dV/dt = S(t) − D(t)
Solving this ODE by separation of variables or direct integration gives V(t). Boundary conditions such as V(0) = V₀ define the particular solution.
通过分离变量法或直接积分求解该常微分方程可得到 V(t)。边界条件如 V(0) = V₀ 确定特解。
- If S(t) = a and D(t) = bV, then dV/dt = a − bV is a standard linear ODE.
- 如果 S(t) = a 且 D(t) = bV，则 dV/dt = a − bV 是一个标准线性常微分方程。
6. Linear Programming for Water Allocation | 水资源分配的线性规划
Linear programming helps allocate limited water between agriculture, industry and domestic use. The objective function Z = c₁x₁ + c₂x₂ is maximised subject to linear constraints.
线性规划有助于在农业、工业和家庭用水之间分配有限的水资源。目标函数 Z = c₁x₁ + c₂x₂ 在满足线性约束条件下最大化。
Maximise Z = c₁x₁ + c₂x₂ subject to a₁x₁ + a₂x₂ ≤ b, x₁, x₂ ≥ 0
Draw the feasible region and test vertices. This is a key Decision Mathematics skill and appears frequently in Edexcel D1 questions.
画出可行域并测试顶点。这是决策数学的关键技能，经常出现在 Edexcel D1 考题中。
- Use integer constraints when water units are discrete.
- 当水量单位是离散的时，使用整数约束。
7. Numerical Methods for Supply Models | 供水模型的数值方法
Some water supply equations cannot be solved algebraically. The Newton-Raphson method gives iterative approximations to roots of f(x) = 0.
有些供水方程无法用代数方法求解。Newton-Raphson 方法对 f(x) = 0 的根给出迭代近似值。
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
For example, solving eˣ − 3x = 0 might model equilibrium between supply and demand. Use a starting value and repeat until convergence.
例如，求解 eˣ − 3x = 0 可以模拟供需之间的平衡。使用初始值并重复迭代直到收敛。
- Check f'(xₙ) is not zero to avoid division by zero.
- 检查 f'(xₙ) 不为零，以避免除零错误。
8. Hypothesis Testing of Water Treatment | 水处理效果的假设检验
A water treatment plant claims the mean contaminant level is below a legal limit μ₀. We test H₀: μ = μ₀ against H₁: μ < μ₀ using a sample mean.
一家水处理厂声称平均污染物水平低于法定限值 μ₀。我们使用样本均值检验 H₀: μ = μ₀ 与 H₁: μ < μ₀。
z = (x̄ − μ₀) / (σ / √n)
Compare the test statistic with the critical value from the normal distribution. This is standard S2 hypothesis testing applied to water quality.
将检验统计量与正态分布的临界值进行比较。这是应用于水质的标准 S2 假设检验。
- Use a one-tailed test when the direction is specified as lower is better.
- 当方向明确为越低越好时，使用单尾检验。
9. Correlation and Regression for Storage | 储水量的相关与回归
We can investigate the linear relationship between rainfall and reservoir storage using the product moment correlation coefficient r and least squares regression.
我们可以使用积矩相关系数 r 和最小二乘回归来研究降雨量与水库储量之间的线性关系。
y = a + bx, where b = Sxy / Sxx
Here Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ) and Sxx = Σ(xᵢ − x̄)². Regression predicts storage for a given rainfall, but beware extrapolation beyond the data range.
此处 Sxy = Σ(xᵢ − x̄)(yᵢ − ȳ)，Sxx = Σ(xᵢ − x̄)²。回归可以预测给定降雨量下的储水量，但要注意超出数据范围的外推。
- r close to 1 indicates strong positive linear correlation.
- r 接近 1 表明强正线性相关。
10. Composite Water Security Index | 综合水安全指数
A composite index combines multiple indicators such as availability, access, quality and stability. Standardise each indicator using z-scores, then compute a weighted average.
综合指数结合了多个指标，如可用性、可及性、质量和稳定性。使用 z 分数对每个指标进行标准化，然后计算加权平均值。
Index = w₁z₁ + w₂z₂ + w₃z₃ + w₄z₄, Σwᵢ = 1
This technique reviews mean, standard deviation and linear combinations of random variables, which are essential A-Level statistics topics.
该技术复习了均值、标准差和随机变量的线性组合，这是 A-Level 统计的重要主题。
- Positive z-scores mean above-average performance.
- 正 z 分数表示高于平均水平的表现。
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