📚 Year 12 OCR Geography: Formula & Theorem Quick Reference Handbook | OCR 十二年级地理：公式定理速查手册
Welcome to your essential quick-reference guide for Year 12 OCR Geography. This handbook pulls together the key formulas, equations, indices, and conceptual theorems you need to apply across physical and human geography topics – from river discharge and hydraulic radius to population change, urban gravity models, and statistical tests for your independent investigation. All equations are presented using standard notation, accompanied by clear explanations to support both in-class exercises and exam preparation.
欢迎使用这份 OCR 十二年级地理必备公式定理速查手册。手册汇集了自然地理与人文地理课程中需要掌握的关键公式、方程、指数和概念性定理，从河流流量与水力半径，到人口变动、城市引力模型，再到独立调查所需的统计检验，所有方程均采用标准记法，并配有简明解释，帮助你应对课内练习和考试复习。
1. River Channel & Drainage Basin Formulas | 河流与流域公式
Discharge (Q) is the volume of water flowing past a cross-section per unit time, usually expressed in cubic metres per second (m³/s). It is the product of cross-sectional area (A) and mean velocity (v):
Q = A × v
流量(Q)是单位时间内通过某一断面的水体体积，通常以立方米/秒(m³/s)表示。它是过水断面面积(A)与平均流速(v)的乘积：
Q = A × v
Hydraulic radius (R) describes channel efficiency by relating cross-sectional area to wetted perimeter (P): ground-breakingindex++
R = A / P
Wetted perimeter is the length of the channel bed and banks in contact with water. A higher hydraulic radius generally indicates a more efficient channel with less friction.
水力半径(R)通过过水断面面积(A)与湿周(P)的比值反映河道输水效率：
R = A / P
湿周是水流与河道底面及岸壁接触的长度。水力半径越大，通常表示河道阻力越小、输水效率越高。
Stream gradient is the vertical fall over a given horizontal distance, often expressed as a ratio or percentage.
Gradient = Vertical drop / Horizontal distance
河流比降是一定水平距离上的垂直落差，常用比率或百分比表示：
比降 = 垂直落差 / 水平距离
The Bradshaw model is a conceptual theorem describing downstream changes in channel characteristics. It predicts that as one moves from source to mouth, discharge, mean velocity, and hydraulic radius increase, while bedload particle size and gradient decrease. This theorem guides many fieldwork hypotheses.
布拉德肖模型是描述河道特征沿程变化的概念性定理。该模型预测，从源头到河口，流量、平均流速和水力半径逐渐增大，而河床物质粒径和比降逐渐减小。这一定理为许多野外实习假设提供了指导。
2. Coastal Process & Erosion Formulas | 海岸过程与侵蚀公式
Wave energy (E) per unit length of wave crest is often simplified as proportional to the square of wave height (H). While full equations include water density (ρ) and gravitational acceleration (g), for comparative purposes E ∝ H² captures the steep energy gradient between constructive and destructive waves.
单位波峰长度的波浪能量(E)通常简化为与波高(H)的平方成正比。尽管完整公式包含水体密度(ρ)和重力加速度(g)，但在比较时常使用 E ∝ H² 来反映建设性海浪与破坏性海浪之间悬殊的能量差异。
The rate of longshore drift (LSD) transport can be estimated from tracer studies:
LSD rate = Distance moved ÷ Time (m/day or m/hour)
沿岸漂移物的输运速率可通过示踪物研究进行估算：
漂移速率 = 移动距离 ÷ 时间 (米/日或米/小时)
Fetch length (F) is the distance of open water over which the wind blows. Longer fetches generate higher energy waves. It is measured along a straight line on a chart, but effective fetch accounts for directional variability.
风区长度(F)是风在水面上吹过的开阔水域的距离。风区越长，产生的波浪能量越高。通常在海图上沿直线量测，但有效风区长度还需考虑风向变化。
Beach sediment analysis often uses the phi scale (φ) to standardise particle size, where diameter (d) is in millimetres:
φ = –log₂(d)
海滩沉积物分析常使用φ标度统一表示粒径，其中d为粒径（毫米）：
φ = –log₂(d)
3. Water Cycle Stores & Fluxes | 水循环贮量与通量计算
The water balance equation for a drainage basin over a given time period is a fundamental theorem linking inputs, outputs, and storage change (ΔS):
P = Q + E + ΔS
流域在某一时段内的水量平衡方程是联结输入、输出和贮量变化(ΔS)的基本定理：
P = Q + E + ΔS
where P = precipitation, Q = runoff, E = evapotranspiration. This expresses the conservation of mass.
其中 P 为降水量，Q 为径流量，E 为蒸散发量。该方程表达了质量守恒原理。
Residence time (T) indicates how long a water molecule, on average, remains in a store:
T = Volume of store / Flux in or out
停留时间(T)表示一个水分子在某个库中平均停留的时长：
T = 贮量 / 通量
Runoff coefficient (C) measures the proportion of precipitation that becomes direct runoff. It helps compare catchment responses.
C = Runoff / Precipitation
径流系数(C)衡量降水量中转化为直接径流的比例，有助于比较不同流域的响应：
C = 径流量 / 降水量
Soil moisture deficit is the amount of water needed to bring soil back to field capacity, often calculated as field capacity minus current soil moisture.
土壤水分亏缺是使土壤恢复到田间持水量所需的水量，通常用田间持水量减去当前土壤含水量计算。
4. Carbon Cycle Quantities | 碳循环量化公式
Net Primary Productivity (NPP) is the rate at which plants accumulate carbon after accounting for respiration (R) from Gross Primary Productivity (GPP):
NPP = GPP − R
净初级生产力(NPP)是植物在扣除呼吸消耗(R)后，由总初级生产力(GPP)所积累碳的速率：
NPP = GPP − R
NPP is usually expressed in grams of carbon per square metre per year (gC m⁻² yr⁻¹). This metric reveals how effectively ecosystems sequester carbon.
NPP 通常以每年每平方米碳克数(gC m⁻² yr⁻¹)表示，这一指标反映出生态系统固碳的效率。
Carbon flux (F) between stores, such as atmosphere to biosphere via photosynthesis, is calculated as:
F = Mass of carbon moved / Time (GtC yr⁻¹)
碳通量(F)指碳在库之间的迁移量，如通过光合作用从大气进入生物圈，其计算为：
F = 迁移碳质量 / 时间 (GtC/年)
Residence time for carbon pools follows the same principle as water: pool size divided by total flux in or out. The atmosphere’s residence time for CO₂, for example, is approximately 5–15 years when considering rapid exchanges with oceans and biosphere.
碳库的停留时间与水体相同：库大小除以流入或流出总通量。例如，大气中CO₂停留时间，就快速与海洋和生物圈交换而言，约为5–15年。
5. Population & Migration Formulas | 人口与迁移公式
Crude Birth Rate (CBR) and Crude Death Rate (CDR) are expressed per 1,000 population per year:
CBR = (Number of live births / Total population) × 1,000
CDR = (Number of deaths / Total population) × 1,000
粗出生率(CBR)和粗死亡率(CDR)通常表示为每年每千人之比：
CBR = (活产数 / 总人口) × 1,000
CDR = (死亡数 / 总人口) × 1,000
Rate of Natural Increase (RNI) can be given as a percentage or per 1,000:
RNI (per 1,000) = CBR − CDR
RNI (%) = (CBR − CDR) / 10
自然增长率(RNI)可以千分比或百分比表示：
RNI (‰) = CBR − CDR
RNI (%) = (CBR − CDR) / 10
Net Migration Rate (NMR) quantifies the balance of international or internal movement:
NMR = ((Immigration − Emigration) / Total population) × 1,000
净迁移率(NMR)用于量化国际或国内迁移净差额：
NMR = ((迁入 − 迁出) / 总人口) × 1,000
Overall population change over a period is sum of natural change and net migration:
ΔP = (Births − Deaths) + (Immigration − Emigration)
某一时期总人口变动为自然变动与净迁移之和：
ΔP = (出生−死亡) + (迁入−迁出)
Dependency ratio measures the pressure on the productive population:
Dependency Ratio = [(Population 0–14 + Population 65+) / Population 15–64] × 100
赡养比衡量劳动年龄人口所承受的抚养压力：
赡养比 = [(0–14岁人口 + 65岁以上人口) / 15–64岁人口] × 100
Doubling time approximates the number of years for a population to double at a constant growth rate (r%):
Td ≈ 70 / r (%)
倍增时间用于估算在恒定增长率(r%)下人口翻一番所需的年数：
Td ≈ 70 / r (%)
6. Urban Models & Spatial Interaction | 城市模型与空间相互作用
The gravity model is used to estimate the interaction (I) between two places based on their population sizes (P₁, P₂) and distance (d):
I = P₁ × P₂ / d²
引力模型根据两地人口规模(P₁, P₂)和距离(d)来估算它们之间的相互作用(I)：
I = P₁ × P₂ / d²
Reilly’s law of retail gravitation determines the breakpoint (BP) between two centres where customers are equally likely to travel to either:
BP = d / (1 + √(P₂ / P₁))
赖利的零售引力法则用于计算两个中心之间的断裂点(BP)，在该点消费者前往任一中心的概率相等：
BP = d / (1 + √(P₂ / P₁))
The rank-size rule describes the relationship between city size and rank: the population of a city (Pₙ) is inversely proportional to its rank (n). If the largest city has population P₁, then the nth largest city is Pₙ = P₁ / n. Deviations from this pattern suggest primacy or conformity to Zipf’s law.
位序-规模法则描述城市规模与位序之间的关系：第n大的城市的人口(Pₙ)与其位序(n)成反比。若首位城市人口为P₁，则 Pₙ = P₁ / n。偏离该模式可能意味着首位度较高或符合齐普夫定律。
Location Quotient (LQ) measures the concentration of an industry in a local area compared to a wider reference region:
LQ = (Local employment in sector / Total local employment) ÷ (National employment in sector / Total national employment)
区位商(LQ)用于衡量某产业在地方相对于更广参照区的集中程度：
LQ = (地方该行业就业人数 / 地方总就业) ÷ (全国该行业就业人数 / 全国总就业)
An LQ > 1 indicates a spatial concentration and potential export orientation.
LQ 大于1表明存在空间集中和可能的输出导向。
7. Statistical Tests for Fieldwork | 地理野外实习统计检验
Spearman’s rank correlation coefficient (rₛ) tests the strength and direction of association between two ranked variables. Differences (D) are the discrepancies between rank pairs.
rₛ = 1 − (6 Σ D²) / (n(n² − 1))
斯皮尔曼等级相关系数(rₛ)用于检验两组等级变量之间的关联强度和方向。D 是每对等级的差值。
rₛ = 1 − (6 Σ D²) / (n(n² − 1))
Values close to +1 indicate a strong positive correlation; values close to −1 indicate a strong negative correlation.
数值接近+1表示强正相关，接近−1表示强负相关。
Chi-squared test (χ²) assesses whether there is a significant difference between observed (O) and expected (E) frequencies in categorical data.
χ² = Σ (O − E)² / E
卡方检验(χ²)用于评估分类数据中观测频数(O)与期望频数(E)是否存在显著差异：
χ² = Σ (O − E)² / E
Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). Compare the calculated χ² to a critical value to determine significance.
自由度(df) = (行数−1) × (列数−1)。将计算出的 χ² 与临界值比较以判断显著性。
Mann-Whitney U test compares two independent samples. The U statistic is calculated for each sample, and the smaller U is used for hypothesis testing.
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁
曼-惠特尼U检验用于比较两个独立样本。每个样本的U统计量按上述公式计算，取较小的U值进行假设检验：
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁
where R₁ is the sum of ranks for sample 1.
其中 R₁ 为样本1的秩和。
8. Fieldwork Sampling & Measurement Techniques | 野外采样与测量技术
Systematic sampling selects points at regular intervals (e.g., every 10 metres along a transect). The sampling fraction is n/N, where n is sample size and N is total possible points. This ensures even coverage and reduces bias in many
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