📚 Formula Theorem Quick Reference for A-Level Edexcel Geography | A-Level Edexcel 地理：公式定理速查手册
This quick reference compiles all the essential formulas, equations, and statistical techniques required for A-Level Edexcel Geography. Whether you are analysing river discharge, calculating demographic rates, or testing relationships with Spearman’s rank, having these mathematical tools at your fingertips will strengthen your geographical skills and boost your confidence in the exam. Each formula is explained in plain language and shown in a clear, memorisable format.
本速查手册汇编了 A-Level Edexcel 地理考试中所有必备的公式、方程和统计技术。无论你是在分析河流流量、计算人口统计比率，还是用斯皮尔曼等级相关系数检验关系，掌握这些数学工具都能强化你的地理技能，并提升考试信心。每条公式均以简明语言解释，并以清晰易记的格式呈现。
1. River Discharge and Flow | 河流流量与流速
River discharge is the volume of water flowing past a cross‑section per unit time. Understanding this is fundamental to flood risk and water resource management.
河流流量是单位时间内流过过水断面的水体体积。理解这一概念对洪水风险和水资源管理至关重要。
Q = A × v
where Q = discharge (m³ s⁻¹), A = cross‑sectional area (m²), and v = mean velocity (m s⁻¹). Mean velocity is usually measured with a flowmeter at 0.6 of the depth from the surface when the river is shallow, or averaged from 0.2 and 0.8 depth readings in deeper channels.
其中 Q = 流量（m³ s⁻¹），A = 过水断面面积（m²），v = 平均流速（m s⁻¹）。浅水河流通常在距水面 0.6 水深处用流速仪测量平均流速；较深河道则取 0.2 和 0.8 水深读数求平均。
Manning’s equation is a widely used empirical formula for estimating velocity in open channels:
曼宁公式是一种广泛用于明渠流速估算的经验公式：
v = (R2/3 × S1/2) / n
Here R is the hydraulic radius (cross‑sectional area / wetted perimeter), S is the channel slope (gradient), and n is Manning’s roughness coefficient. A smaller n means a smoother channel and faster flow.
式中 R 为水力半径（过水断面面积 / 湿周），S 为河道坡度（比降），n 为曼宁粗糙系数。n 越小表示河道越光滑，流速越快。
2. Drainage Basin Morphometry | 流域形态测量
Morphometric indices help compare the shape and efficiency of drainage basins. The two most common are drainage density and bifurcation ratio.
形态测量指数有助于比较流域的形状和排水效率。最常用的两个指标是河网密度和分叉比。
Drainage Density (Dd) = ΣL / A
Total length of all stream channels (ΣL, in km) divided by the basin area (A, in km²). High drainage density indicates impermeable surfaces, steep slopes, or sparse vegetation, and is associated with rapid runoff.
所有河道总长度（ΣL，km）除以流域面积（A，km²）。高河网密度表明地表不透水、坡度陡峭或植被稀疏，并与快速产流相关。
Bifurcation Ratio (Rb) = Nu / Nu₊₁
where Nu is the number of streams of a given order and Nu₊₁ is the number of streams of the next higher order. In a typical natural basin, Rb ranges from 3 to 5. A consistent Rb suggests uniform geology.
其中 Nu 为某一级支流的数量，Nu₊₁ 为下一更高级支流的数量。在典型天然流域中，Rb 在 3 至 5 之间。分叉比稳定表明地质条件均匀。
3. Demographic Rates | 人口统计学比率
Population change is central to human geography. The fundamental equation sums natural change and net migration.
人口变化是人文地理学的核心内容。基本方程概括了自然变化和净迁移。
Population Change = (Births − Deaths) + (Immigration − Emigration)
To compare across countries, rates per 1000 population are used:
为了跨国比较，使用每千人的比率：
Crude Birth Rate (CBR) = (Total births / Total population) × 1000
Crude Death Rate (CDR) = (Total deaths / Total population) × 1000
Rate of Natural Increase (%) = ((CBR − CDR) / 1000) × 100
Net Migration Rate = ((Immigration − Emigration) / Total population) × 1000
Doubling time can be estimated by the ‘Rule of 70’:
翻倍时间可用“70 法则”估算：
Doubling Time (years) ≈ 70 / Population Growth Rate (%)
4. Urban and Economic Measures | 城市与经济度量
Urban geography frequently uses indices to describe economic specialisation and connectivity. The location quotient (LQ) is a simple ratio that identifies regional industrial concentration.
城市地理学常用指数来描述经济专业化和连接度。区位商（LQ）是一个可识别区域产业集中度的简单比率。
LQ = (ei / e) / (Ei / E)
where ei = local employment in industry i, e = total local employment, Ei = national employment in industry i, and E = total national employment. LQ > 1 suggests that region specialises in that industry.
其中 ei = 某地 i 产业就业人数，e = 该地总就业人数，Ei = 全国 i 产业就业人数，E = 全国总就业人数。LQ > 1 表明该地区在此产业上具有专业化优势。
Gini coefficient is a measure of inequality often applied to income or wealth, and can be computed from a Lorenz curve:
基尼系数是衡量收入或财富不平等的常用指标，可由洛伦兹曲线计算：
Gini Coefficient = A / (A + B)
where A is the area between the line of perfect equality and the Lorenz curve, and B is the area under the Lorenz curve. Values range from 0 (perfect equality) to 1 (perfect inequality).
其中 A 为完全平等线和洛伦兹曲线之间的面积，B 为洛伦兹曲线下的面积。取值从 0（完全平等）到 1（完全不平等）。
5. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank correlation (rₛ) tests the strength and direction of a relationship between two variables when data is ordinal or non‑normally distributed. It is widely used in geographical fieldwork.
斯皮尔曼等级相关系数（rₛ）用于在数据为顺序级或非正态分布时，检验两个变量之间关系的强度和方向。其在地理实地调查中应用广泛。
rₛ = 1 − (6 × Σd²) / (n × (n² − 1))
Step 1: Rank the data for Variable 1 and Variable 2 separately. Step 2: Calculate the difference d between the two ranks for each paired observation. Step 3: Square these differences (d²) and sum them. Step 4: Substitute into the formula, where n is the number of paired observations. The critical value table with degrees of freedom n must be consulted to decide significance.
步骤 1：分别对变量 1 和变量 2 的数据进行排序赋秩。步骤 2：计算每对观测值的等级差 d。步骤 3：将差值平方（d²）并求和。步骤 4：代入公式，其中 n 为成对观测的数量。须查阅自由度为 n 的临界值表以判定显著性。
6. Measures of Central Tendency and Dispersion | 集中趋势和离散程度的测量
Accurate description of geographical datasets requires mean, median, mode, and measures of spread such as range and interquartile range (IQR).
准确描述地理数据集需要均值、中位数、众数以及极差、四分位距（IQR）等离散程度的度量。
Mean (x̄) = Σx / n
Range = Maximum value − Minimum value
IQR = Q₃ − Q₁
where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile). Standard deviation (s) shows the average distance of data points from the mean:
其中 Q₁ 为下四分位数（第 25 百分位），Q₃ 为上四分位数（第 75 百分位）。标准差（s）表示数据点与均值的平均距离：
s = √(Σ(xi − x̄)² / (n − 1))
Always use n − 1 for a sample; for a population the denominator is n. In geographical investigation, the coefficient of variation (CV = s / x̄ × 100%) is helpful for comparing variability between datasets with different units.
样本计算时始终使用 n − 1；总体则分母为 n。在地理调查中，变异系数（CV = s / x̄ × 100%）有助于比较不同单位数据集之间的波动程度。
7. Carbon Cycle and Ecosystem Productivity | 碳循环与生态系统生产力
Net primary productivity (NPP) is a key concept in the carbon cycle and energy security topics. It measures the rate at which plants store carbon after respiration losses.
净初级生产力（NPP）是碳循环与能源安全主题中的关键概念，用于衡量植物在呼吸损耗后储存碳的速率。
NPP = GPP − R
Gross primary productivity (GPP) is the total rate of carbon fixation by photosynthesis, and R is the autotrophic respiration rate. Both are usually expressed in gC m⁻² yr⁻¹. High NPP is found in tropical rainforests, while deserts have low NPP.
总初级生产力（GPP）为光合作用固定碳的总速率，R 为自养呼吸速率。两者通常均以 gC m⁻² yr⁻¹ 表示。热带雨林 NPP 较高，而沙漠 NPP 较低。
Soil carbon storage can be estimated by multiplying bulk density, organic carbon fraction, and soil depth:
土壤碳储量可通过容重、有机碳含量和土壤厚度相乘来估算：
Soil Carbon (kg m⁻²) = Bulk Density (kg m⁻³) × Organic Carbon Proportion × Depth (m)
8. Storm Hydrograph Analysis | 暴雨径流分析
Storm hydrographs depict river discharge over time in response to a rainfall event. Key parameters include lag time, peak discharge, and rising limb gradient.
暴雨径流曲线图描述了河流流量随降水事件的时序响应。关键参数包括滞时、洪峰流量和涨水段坡度。
Runoff Depth (mm) = (Total Storm Discharge Volume / Catchment Area) × 1000
Convert total discharge (m³) to depth over the catchment (m²) to compare with rainfall depth. The runoff coefficient is:
将总径流量（m³）换算为流域面积（m²）上的水深，以便与降雨量比较。径流系数为：
Runoff Coefficient = Total Runoff Depth / Total Rainfall Depth
A coefficient close to 1 indicates high runoff due to impermeable surfaces or saturated soil; lower values suggest high infiltration. Urbanisation increases the peak discharge and shortens lag time.
系数接近 1 表明因不透水表面或土壤饱和而产生高比例径流；较低值则意味着高下渗量。城市化会增大洪峰流量并缩短滞时。
9. Coastal Sediment Budget | 海岸沉积物收支
Coastal systems rely on sediment inputs, outputs, and stores. The sediment budget equation evaluates whether a beach is eroding, accreting, or stable.
海岸系统依赖于沉积物的输入、输出和储存。沉积物收支方程可评估海滩是侵蚀、堆积还是稳定。
Net Sediment Change = (Cliff Erosion + River Input + Longshore Drift In) − (Longshore Drift Out + Offshore Loss + Dune Blowout)
Quantities are typically in m³ yr⁻¹. A positive net change suggests accretion; a negative net change indicates erosion and possible beach narrowing.
各量通常以 m³ yr⁻¹ 表示。正的净变化意味着堆积；负的净变化则表明侵蚀和可能发生的海滩变窄。
Longshore drift transport rate can be roughly estimated by the ‘white water’ technique or using tracers, but a formulaic approximation is:
沿岸漂沙输运率可通过“白水”法或示踪沙粗略估算，但通常可使用下式近似：
I = k × Hb² × Cb × sin(2αb)
where I is the immersed-weight transport rate, Hb is breaker wave height, Cb is wave speed at breaking, αb is breaker angle, and k is an empirical constant. This is part of the CERC formula, often simplified in A-Level contexts.
其中 I 为水下重量输运率，Hb 为破波波高，Cb 为破波波速，αb 为破波角度，k 为经验常数。这是 CERC 公式的一部分，在 A-Level 中常被简化使用。
10. Weather and Climate Indices | 天气与气候指数
Simple climatic metrics allow comparison of aridity, seasonality, and continentality. The de Martonne aridity index is quick to compute.
简单气候指标可用于比较干旱度、季节性和大陆度。德马尔干旱指数计算快捷。
Aridity Index = P / (T + 10)
where P is mean annual precipitation (mm) and T is mean annual temperature (°C). A value below 20 indicates a semi‑arid climate; below 10 indicates arid conditions. Another useful measure is:
其中 P 为年均降水量（mm），T 为年均温度（°C）。低于 20 表示半干旱气候；低于 10 表示干旱条件。另一个有用的指标是：
Continentality Index = Annual Temperature Range / Latitude (in degrees)
Larger values reflect greater distance from the moderating influence of oceans. For weather hazard topics, you may also calculate:
数值越大，反映距海洋调节作用越远。针对天气灾害专题，你也可以计算：
Storm Surge Height ≈ Atmospheric Pressure Anomaly (mb) / 10
(a rough rule of thumb: a 1 hPa drop raises sea level by about 1 cm).
（经验法则：气压每下降 1 hPa，海面约升高 1 cm）。
11. Environmental Quality and Sustainability Indices | 环境质量与可持续性指数
Qualitative environmental quality surveys are often converted to quantitative scores using weighted indices. For managing fieldwork data, these indices are essential.
定性环境质量调查常通过加权指数转化为定量评分。这类指数在管理实地考察数据时必不可少。
Environmental Quality Index (EQI) = Σ(wᵢ × sᵢ) / Σwᵢ
where wᵢ is the weight assigned to criterion i and sᵢ is the score for that criterion (e.g., 1–5 scale). This method can be applied to residential satisfaction surveys, urban green space assessments, and coastal zone evaluation.
其中 wᵢ 为指标 i 的权重，sᵢ 为该指标的得分（例如 1–5 分制）。此法可应用于居住满意度调查、城市绿地评估和海岸带评价。
The ecological footprint is a composite sustainability measure:
生态足迹是一种复合可持续性指标：
Ecological Footprint (gha) = Total Biologically Productive Area / (Population × Yield Factor × Equivalence Factor)
Simplified, it compares human demand to the planet’s biocapacity. A deficit indicates overshoot.
简而言之，它将人类需求与地球的生物承载力进行比较。赤字即意味着超载。
12. Quantifying Risk and Hazard Magnitude | 风险与灾害等级量化
In the tectonics and hazards module, risk is often expressed as the product of hazard and vulnerability divided by capacity to cope.
在构造与灾害模块中，风险通常表示为致灾因子强度与脆弱性的乘积，再除以应对能力。
Risk = (Hazard × Vulnerability) / Capacity
For earthquakes, the moment magnitude (Mw) is calculated from seismic moment, but a simplified formula linking energy release is:
对于地震，矩震级（Mw）根据地震矩计算，但一个联系能量释放的简化公式为：
Log₁₀E = 4.8 + 1.5 × M
where E is energy in joules, and M is magnitude (approx). This shows that each unit increase in magnitude multiplies energy release by about 31.6 times.
其中 E 为能量（焦耳），M 为震级（近似）。这表明震级每增加 1 单位，释放的能量约扩大 31.6 倍。
For volcanic explosivity, the VEI is a logarithmic scale, but you may need to calculate the total tephra volume ejected. No straightforward formula is required, but you should be able to interpret such calculations when given.
火山爆炸性指数（VEI）为对数尺度，可能需要计算喷出火山碎屑的总体积。虽无直接公式需要记忆，但当给定数据时应能够进行解释计算。
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