Year 13 CCEA Geography: Quick Reference Formula & Principles Handbook | CCEA 地理公式定理速查手册

📚 Year 13 CCEA Geography: Quick Reference Formula & Principles Handbook | CCEA 地理公式定理速查手册

This handbook brings together the essential formulas, indices, and principles that underpin the Year 13 CCEA Geography specification. From river channel hydraulics to statistical tests, each entry is presented with clear notation and example application to support revision and fieldwork analysis.

本手册汇集了 CCEA 地理 Year 13 课程中支撑核心分析的公式、指数与定理。涵盖河流动水力学、海岸剖面测度、人口指标、统计检验等关键内容，每一个条目均以清晰的符号表达并配以应用示例，助力考前梳理与野外调查数据处理。

1. Drainage Basin Morphometry | 流域形态计量

Drainage density (D_d) is the total length of streams per unit area, indicating basin efficiency in removing water.

河网密度（D_d）指单位面积内水道总长度，反映流域排水效率。

D_d = ΣL / A

where ΣL is total stream length (km) and A is basin area (km²). A high D_d (>2.0) often correlates with impermeable geology and sparse vegetation.

式中 ΣL 为水道总长度（km），A 为流域面积（km²）。高河网密度（>2.0）常与不透水岩层和稀疏植被相关。

Stream frequency (F_s) complements D_d by counting the number of stream segments per unit area: F_s = N / A, where N is the total number of stream segments.

水道频度（F_s）通过统计单位面积内水道段数对河网密度进行补充：F_s = N / A，其中 N 为水道段总数。

Bifurcation ratio (R_b) measures the ratio of number of streams of one order to the next higher order, typically between 3 and 5 for natural networks.

分叉比（R_b）是某级水道数目与相邻更高级水道数目之比，天然水系通常落在 3 至 5 之间。

2. River Channel Hydraulics | 河道水力学

Discharge (Q) is the volume of water passing a cross‑section per second.

流量（Q）为单位时间通过断面的水体体积。

Q = A × v

where A is cross‑sectional area (m²) and v is mean velocity (m s⁻¹). Velocity is often measured by flowmeter at 0.6 depth.

其中 A 为过水断面面积（m²），v 为平均流速（m s⁻¹）。流速常用流速仪在 0.6 水深点测得。

Hydraulic radius (R) is used in the Manning equation to estimate velocity:

水力半径（R）用于曼宁公式估算流速：

R = A / P

with P being wetted perimeter (m). For wide rectangular channels, R ≈ depth.

式中 P 为湿周（m）；宽浅矩形断面下，R 近似于水深。

Manning’s equation for mean velocity:

曼宁公式计算平均流速：

v = (R^²/³ × S^½) / n

where S is channel slope (dimensionless) and n is Manning’s roughness coefficient. Typical n values: 0.025 for smooth concrete, 0.035 for earth‑bottom channels, 0.050‑0.100 for heavily vegetated floodplains.

式中 S 为水面比降（无量纲），n 为曼宁糙率系数。参考值：光滑混凝土 0.025，土质河底 0.035，植被茂密的洪泛滩地 0.050‑0.100。

3. Coastal Profiling and Sediment Analysis | 海岸剖面与沉积物分析

Beach gradient is calculated from the rise and run of the profile measured by tape and clinometer.

海滩坡度由卷尺和测斜仪测得的高差与水平距计算。

tan θ = (vertical rise) / (horizontal distance)

Often expressed as a ratio 1 : X, where X is the distance for 1 m fall. Wider beaches with finer sediment tend to have gentler gradients.

常以 1 : X 的比例表示，X 为下降 1 m 所需的水平距。沉积物较细的宽滩坡度通常更平缓。

Sediment roundness indices (Powers scale) and Cailleux roundness index are used for pebble analysis. For sediment size, phi (φ) scale converts particle diameter:

沉积物形状（帕沃斯圆度）和凯约圆度指数用于卵石分析。粒度可用 phi (φ) 标度转换：

φ = −log₂(D / D₀)

where D is grain diameter in mm and D₀ is 1 mm reference. Positive φ values indicate fine sediment, negative φ values coarse.

D 为颗粒直径（mm），D₀ 为 1 mm 基准。φ 正值表示细粒，负值表示粗粒。

Longshore drift volume can be estimated by integrating tracer movement over time, but a simple rate formula using a floating object is:

沿岸漂沙量可通过对示踪物过程积分估算，简易浮标法测定速率公式为：

Drift rate = distance (m) ÷ time (s)

4. Hillslope and Mass Movement | 坡面过程与块体运动

The factor of safety (F) for slope stability compares resisting forces to driving forces.

边坡稳定系数（F）为抗滑力与下滑力之比。

F = (shear strength × area) / (shear stress × area)

When F > 1.0, the slope is stable; when F < 1.0, failure occurs. Shear strength τ = c + σ tan φ, where c is cohesion, σ is normal stress, φ is angle of internal friction.

F > 1.0 为稳定，F < 1.0 发生失稳。抗剪强度 τ = c + σ tan φ，c 为粘聚力，σ 为正应力，φ 为内摩擦角。

5. Climate and Water Balance | 气候与水量平衡

The water balance equation links precipitation (P), evapotranspiration (E), runoff (Q) and changes in soil moisture storage (ΔS).

水量平衡方程关联降水（P）、蒸发散（E）、径流（Q）和土壤储水量变化（ΔS）。

P = Q + E + ΔS

For humid mid‑latitude catchments, actual evapotranspiration (AE) is often less than potential (PE) when soil moisture is limiting.

湿润中纬流域在土壤水分不足时，实际蒸发散（AE）常低于潜在蒸发散（PE）。

Aridity index (AI) indicates moisture availability:

干燥度指数（AI）指示水分供给状况：

AI = P / PE

where AI < 0.2 hyper‑arid, 0.2–0.5 semi‑arid, and > 0.65 humid.

AI < 0.2 极端干旱，0.2–0.5 半干旱，> 0.65 湿润。

6. Population and Demographic Indices | 人口与人口统计指数

Crude birth rate (CBR) and crude death rate (CDR) are per 1000 population:

粗出生率（CBR）与粗死亡率（CDR）按每千人计算：

CBR = (live births / total population) × 1000

CDR = (deaths / total population) × 1000

Rate of natural increase (RNI) = (CBR − CDR) / 10, expressed as a percentage.

自然增长率（RNI）= (CBR − CDR) / 10，以百分数表示。

Doubling time (years) under constant growth uses the Rule of 70:

在恒定增长率下，倍增年数用“70 法则”：

T_d = 70 / annual growth rate (%)

Dependency ratio highlights the proportion of non‑working population:

抚养比突出非劳动人口比例：

Dependency ratio = [(Pop 0‑14 + Pop 65+) / Pop 15‑64] × 100

Youth dependency and old‑age dependency can be separated to analyse demographic pressure.

可进一步拆分为少年抚养比与老年抚养比，以分析人口压力。

Sex ratio = (number of males / number of females) × 100. Values above 100 indicate male predominance.

性别比 =（男性人口数 ÷ 女性人口数）× 100，数值高于 100 表示男性偏多。

7. Migration and Population Projection | 迁移与人口预测

Net migration = immigrants − emigrants. Migration rate per 1000:

净迁移 = 迁入 − 迁出。每千人迁移率：

Net migration rate = [(net migration) / total population] × 1000

Population projection using the cohort‑component method estimates future population by updating each age‑sex group with fertility, mortality and migration rates.

队列要素法通过生育率、死亡率和迁移率逐组推演未来人口。

Stepwise gravity model for migration flows between i and j:

迁移引力模型的分步公式：

M_ij = k × (P_i × P_j) / d_ij^b

where M_ij is migration from i to j, P are populations, d is distance, b is distance exponent (often 2), and k a constant.

M_ij 为 i 地至 j 地迁移量，P 为两地人口，d 为距离，b 为距离指数（常取 2），k 为常数。

8. Spatial Statistics: Nearest Neighbour and Location Quotient | 空间统计：最近邻与区位商

Nearest neighbour index (R_n) quantifies the degree of clustering or dispersion of points.

最近邻指数（R_n）用于量化点状要素的聚集或离散程度。

R_n = D_obs / D_ran

where D_obs is mean observed nearest neighbour distance and D_ran = 1 / (2√ρ), with ρ being point density (points per unit area). R_n = 1 random, < 1 clustered, > 1 dispersed.

其中 D_obs 为实测平均最近邻距离，D_ran = 1 / (2√ρ)，ρ 为点密度（单位面积点数）。R_n = 1 随机，< 1 聚集，> 1 离散。

Location quotient (LQ) measures the concentration of an industry in a region relative to the nation:

区位商（LQ）衡量某区域特定产业相对于全国的集中度：

LQ = (e_i / e) / (E_i / E)

where e_i = local employment in industry i, e = total local employment, E_i = national employment in i, E = total national employment. LQ > 1 indicates export‑oriented specialisation.

e_i 为地方 i 产业就业，e 为地方总就业，E_i 为全国 i 产业就业，E 为全国总就业。LQ > 1 暗示外向型专业化。

9. Inequality and Development Indices | 不平等与发展指数

Gini coefficient is derived from the Lorenz curve and summarises income or land distribution.

基尼系数由洛伦兹曲线导出，度量收入或土地分布的不均等程度。

G = A / (A + B)

where A is area between line of equality and Lorenz curve, B is area under Lorenz curve. G ranges 0 (perfect equality) to 1 (maximum inequality).

A 为绝对平等线与洛伦兹曲线间的面积，B 为洛伦兹曲线下方面积。G 值介于 0（完全平等）至 1（极端不平等）之间。

Human Development Index (HDI) aggregates life expectancy, education (mean & expected years) and GNI per capita using geometric mean:

人类发展指数（HDI）采用几何平均合成预期寿命、教育（平均/预期受教育年限）和人均国民收入：

HDI = (I_health × I_education × I_income)^⅓

Gender Inequality Index (GII) reflects reproductive health, empowerment and labour market participation. Lower values indicate higher equality.

性别不平等指数（GII）反映生殖健康、赋权和劳动市场参与，数值越低越平等。

10. Correlation and Statistical Testing for Fieldwork | 野外数据相关性及统计检验

Spearman’s rank correlation coefficient (r_s) tests the strength and direction of a monotonic relationship between two ordinal variables.

斯皮尔曼等级相关系数（r_s）检验两个顺序变量间的单调关系强度与方向。

r_s = 1 − [ 6Σd² / (n³ − n) ]

where d is the difference in ranks for each pair and n is the number of data pairs. r_s values close to +1 or −1 indicate a strong correlation; significance is checked against a critical value table under a chosen confidence level (e.g. 0.05).

d 为每对数据的等级差，n 为数据对数。r_s 接近 ±1 表示强相关；显著性需对照临界值表（如 0.05 置信度）。

Chi‑squared test (χ²) assesses the association between categorical variables:

卡方检验（χ²）评估分类变量间的关联性：

χ² = Σ [ (O − E)² / E ]

where O = observed frequency, E = expected frequency. Degrees of freedom = (rows − 1) × (columns − 1). A high χ² value (above critical) means a significant relationship exists.

O 为观测频数，E 为期望频数。自由度 = (行数−1) × (列数−1)。χ² 值高于临界值时表示存在显著关系。

Student’s t‑test for comparing two sample means:

独立样本 t 检验用于比较两组均值：

t = (x̄₁ − x̄₂) / √[ (s₁²/n₁) + (s₂²/n₂) ]

where x̄ are means, s² variances and n sample sizes. The calculated t is compared with the critical t to assess whether any difference is statistically significant at a given p‑value.

其中 x̄ 为均值，s² 为方差，n 为样本量。计算所得 t 值与临界 t 比较，以判断给定 p 值水平下差异是否显著。

11. Manning, Velocity and Discharge in Fieldwork | 野外流速流量与曼宁公式应用

When collecting primary data for a river fieldwork enquiry, wetted perimeter is often measured with a chain taut across the channel bed. Cross‑sectional area is calculated by dividing the channel into segments (trapezoidal rule).

在河流野外调查中，湿周常用链条紧贴河床量测；过水断面面积通过将断面划分为多个梯形计算。

Trapezoidal rule for cross‑section area:

断面面积的梯形法则：

A = Σ [ ½ (d_i + d_i+1) × w ]

where d_i are depths at regular width intervals w.

d_i 为固定水平间距 w 的各点水深。

Combining Manning’s n with measured slope and hydraulic radius yields a predicted velocity, which can be compared with the actual velocity recorded by a flowmeter to comment on channel roughness and sediment transport.

将曼宁系数 n 结合实测比降和水力半径可求出预测流速，与流速仪实测值比较，用以分析河床糙率与输沙特征。

12. Gradient, Stream Order and Longitudinal Profile | 比降、水道分级与纵剖面

Channel slope (S) between two points is calculated as:

两点间的河槽比降（S）计算公式为：

S = (elevation change) / (horizontal distance along channel)

Stream ordering (Strahler) rules: first‑order streams have no tributaries; when two first‑order streams join they form a second‑order; when two second‑order streams join they form a third‑order, etc. The law of stream length often shows that mean length increases with order.

斯特拉勒水道分级法：一级水道无支流；两条一级水道交汇形成二级；两条二级交汇形成三级，依此类推。平均长度随级别升高而增大的规律常见。

Longitudinal profile concavity index can be described by the stream length‑gradient index (SL) for identifying knickpoints:

纵剖面凹度可用河段比降指数（SL）描述，用于识别裂点：

SL = (ΔH / ΔL) × L

where ΔH is elevation change over reach, ΔL is reach length, and L is total distance from source. Elevated SL indicates possible tectonic or lithological disturbance.

ΔH 为河段落差，ΔL 为河段长，L 为距源头总长。SL 值偏高暗示可能受构造或岩性扰动。

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