📚 AS CIE Geography: Formulas & Theorems Quick Reference | AS CIE 地理：公式定理速查手册
Mastering the essential formulas and conceptual theorems is crucial for success in AS CIE Geography. This quick reference handbook compiles the key quantitative tools and models you need to apply in hydrological, demographic, and urban contexts. Keep this guide close when analyzing river discharge, interpreting population pyramids, or evaluating urban land-use patterns.
掌握核心公式与概念定理是顺利通过 AS CIE 地理考试的关键。这本速查手册汇集了在水文学、人口学和城市地理情境下必须运用的主要定量工具与模型。在分析河流流量、解读人口金字塔或评估城市土地利用格局时，请随时参考本指南。
1. River Discharge & Hydraulic Radius | 河流流量与水力半径
River discharge (Q) quantifies the volume of water passing a given cross-section per unit time. It is calculated by multiplying the cross-sectional area (A) by the mean flow velocity (V). Hydraulic radius (R) measures the efficiency of a channel in conveying water and is defined as the ratio of the cross-sectional area to the wetted perimeter (P).
Q = A × V R = A / P
A larger hydraulic radius typically indicates a more efficient channel, where less energy is lost overcoming friction. For a given discharge, an increase in hydraulic radius reduces the velocity required, influencing erosion and sediment transport capacity. The wetted perimeter must always be measured along the bed and banks in contact with water.
河流流量 (Q) 定量描述单位时间内通过某一断面的水体体积，由断面面积 (A) 乘以平均流速 (V) 求得。水力半径 (R) 衡量河道输水效率，定义为断面面积与湿周 (P) 的比值。
Q = A × V R = A / P
水力半径越大通常意味着河道效率越高，克服摩擦力所损耗的能量越少。对于给定的流量，水力半径增大可降低所需流速，进而影响侵蚀与输沙能力。湿周须始终沿水面以下的河床与岸壁测量。
2. Manning’s Equation for Flow Velocity | 曼宁流速方程
Manning’s equation estimates the mean velocity (V) of water in an open channel under uniform flow. It links hydraulic radius (R), channel slope (S), and a roughness coefficient (n) representing the channel’s bed and bank irregularities.
V = (1 / n) × R2/3 × S1/2
In this formula, a smaller Manning’s n denotes a smoother channel (e.g., concrete-lined), resulting in higher velocities. A steeper slope (S) and a larger hydraulic radius both increase velocity. Geographers use Manning’s equation to predict how channel modifications or changes in stage affect flow conditions and flood risk.
曼宁方程用于估算均匀流条件下明渠的平均流速 (V)，它将水力半径 (R)、河道比降 (S) 以及代表河床与岸壁不规则性的糙率系数 (n) 联系起来。
V = (1 / n) × R2/3 × S1/2
该式中，曼宁系数 n 越小表示河道越光滑（如混凝土衬砌），流速越高；比降越大、水力半径越大，流速也越大。地理学者利用曼宁方程预测河道改造或水位变化如何影响水流状况与洪水风险。
3. Bradshaw Model of River Characteristics | 布拉德肖河流特征模型
The Bradshaw model summarizes downstream changes in channel and flow variables along a river’s longitudinal profile. As one moves from source to mouth, width, depth, mean velocity, discharge, and the amount of bedload all increase, while particle size decreases and channel roughness falls.
| Variable | Downstream Change |
|---|---|
| Channel width & depth | Increase |
| Mean velocity & discharge | Increase |
| Bedload particle size | Decrease |
| Channel roughness & slope | Decrease |
This model is a helpful checklist for fieldwork investigations. Exceptions can occur where tributaries or human interference alter the pattern. The Bradshaw model enables students to formulate hypotheses and link observed changes to processes like attrition and downstream fining.
布拉德肖模型总结了河流纵剖面沿线河道及水流变量的下游变化。从源头向河口移动时，河宽、水深、平均流速、流量与推移质数量均增加，而粒径减小、河床糙率下降。
| 变量 | 下游变化 |
|---|---|
| 河宽与水深 | 增大 |
| 平均流速与流量 | 增大 |
| 推移质粒径 | 减小 |
| 河床糙率与比降 | 减小 |
该模型为野外考察提供了有用的核查清单。当支流汇入或人类活动干预时可能出现例外。布拉德肖模型帮助学生提出假说，并将观察到的变化与磨蚀、下游细化等过程联系起来。
4. Hjulström Curve: Erosion, Transport, and Deposition | 夏普斯特曲线：侵蚀、搬运与沉积
The Hjulström curve illustrates the critical flow velocities required to erode, transport, and deposit particles of different grain sizes. Coarse sands and gravels require the highest velocities for erosion because their weight and interlocking make them hard to entrain; silts and clays, though tiny, also need relatively high velocities for erosion because their cohesive nature resists detachment.
Curve interpretation: the erosion line curves upward for very fine and very coarse sediment. Once in motion, particles are transported at lower velocities than needed for erosion. Deposition occurs when the velocity falls below the falling limb of the curve. The transport zone lies between the erosion and deposition lines.
夏普斯特曲线展示了侵蚀、搬运和沉积不同粒径颗粒所需的临界流速。粗砂与砾石因重量大且颗粒嵌合紧密，需要最高的侵蚀流速；粉砂与黏土粒径虽小，但因黏聚力抵抗分离，同样需要较高的侵蚀流速。
曲线解读：极细与极粗颗粒的侵蚀线均向上弯曲。颗粒一旦启动，维持搬运所需的流速低于侵蚀流速；当流速降至沉降线以下时则发生沉积。侵蚀线与沉积线之间为搬运带。
5. Basic Demographic Formulas | 基础人口统计公式
Demographic rates are the foundation of population geography. The crude birth rate (CBR) and crude death rate (CDR) are expressed per 1000 population, providing standardized measures for comparison across societies.
CBR = (Number of live births / Total population) × 1000
CDR = (Number of deaths / Total population) × 1000
Natural Increase = CBR – CDR (per 1000)
The natural increase rate can be converted into a percentage for long-term projections. These raw numbers ignore age structure, so they can mask underlying demographic dynamics; nevertheless, they remain essential inputs for calculating population doubling time and for understanding stage classifications in the demographic transition model.
人口率是人口地理学的基础。粗出生率 (CBR) 与粗死亡率 (CDR) 均以千分比表示，为跨地区比较提供了标准化量度。
CBR = (活产婴儿数 / 总人口) × 1000
CDR = (死亡人数 / 总人口) × 1000
自然增长率 = CBR – CDR（每千人）
自然增长率可换算为百分比用于长期预测。这些原始数据忽略了年龄结构，可能掩盖潜在的人口动态变化；但作为计算人口倍增时间、理解人口转变模型中阶段分类的基础，它们仍是不可或缺的输入指标。
6. Population Growth Rate & Migration Rate | 人口增长率与迁移率
Overall population change combines natural increase and net migration. The total growth rate is usually expressed as a percentage of the baseline population over a specific period.
Population Growth Rate = [(Births – Deaths) + (Immigrants – Emigrants)] / Initial population × 100
Net migration rate is similarly calculated per 1000 people. Migration efficiency indices, such as the index of migration effectiveness, help geographers assess whether movements are redistributive or neutralizing. These metrics are fundamental when evaluating the demographic consequences of push-pull factors, government policies, and regional inequalities.
总体人口变化由自然增长与净迁移两部分组成。总增长率通常以特定时期内占基期人口的百分比表示。
人口增长率 = [(出生 – 死亡) + (迁入 – 迁出)] / 期初人口 × 100
净迁移率同样以千分比计算。迁移效率指标（如迁移效果指数）有助于地理学者判断人口移动是加剧还是中和空间再分配。在评估推拉因素、政府政策及区域不平等的后果时，这些计量手段至关重要。
7. Dependency Ratio | 抚养比
The dependency ratio assesses the economic pressure on the working-age population by comparing the typically non-working (young and elderly) to those of working age. It is a vital indicator for planning health care, education, and pension systems.
Dependency Ratio = [(Population aged 0–14 + Population aged 65+) / Population aged 15–64] × 100
A high dependency ratio signals that a large share of the population relies on the productive output of a relatively small workforce. Countries in the early stages of demographic transition often have high youth dependency; ageing societies face growing old-age dependency. The formula can be decomposed into youth and old-age dependency ratios to separate the two policy challenges.
抚养比通过将通常不从事经济活动的人口（少儿与老人）与劳动年龄人口比较，衡量经济负担压力，是规划医疗、教育和养老金体系的关键指标。
抚养比 = [(0–14 岁人口 + 65 岁及以上人口) / 15–64 岁人口] × 100
高抚养比意味着大量人口依赖相对较少的劳动力的产出。处于人口转变初期的国家少儿抚养比往往偏高，老龄化社会则面临日益增长的老年抚养负担。该公式可分解为少儿抚养比与老年抚养比，以区分两类政策挑战。
8. Burgess Concentric Zone Model | 伯吉斯同心圆模型
Ernest Burgess’s 1925 model describes urban social structures as a set of concentric rings expanding outward from the central business district (CBD). Zone I is the CBD; Zone II, the transition zone, mixes light industry with lower-quality housing; Zone III houses workers in modest terraced homes; Zone IV is the middle-class residential zone; and Zone V constitutes the commuter belt of higher-quality suburban dwellings.
The underlying theorem assumes a process of invasion and succession: as the city grows, inner rings expand outward, displacing residents in the next ring. Although based on 1920s Chicago, the model captures the broad tendency of land values and social status to rise with distance from the CBD in many North American cities. Its limitations include ignoring physical barriers, transport corridors, and modern decentralized development.
伯吉斯 1925 年的模型将城市社会结构描述为从中心商务区 (CBD) 向外扩张的一系列同心圆环带。I 环为 CBD；II 环为过渡带，轻工业与低质量住房混杂；III 环为工人住宅区，分布着朴素排房；IV 环为中产阶层居住区；V 环为郊区别墅构成的通勤带。
该理论假定“侵入–演替”过程：随着城市增长，内环向外扩张，迫使下一环带的居民向外迁移。虽然基于 1920 年代的芝加哥，该模型捕捉了许多北美城市中地价和社会地位随距 CBD 距离增加而上升的总体趋势。其局限在于忽略了自然障碍、交通走廊及现代多中心发展格局。
9. Hoyt Sector Model | 霍伊特扇形模型
Homer Hoyt’s sector model (1939) proposes that cities develop in wedge-shaped sectors radiating from the CBD along major transport routes. High-rent residential areas, for instance, extend outward along desirable axes, while industrial zones follow railways or rivers. Lower-income housing tends to cluster adjacent to industrial belts.
Hoyt’s theorem emphasizes the primacy of transport arteries and that once a land use is established in a sector, it will extend outward as the city grows, rather than being squeezed into a new ring. This theory better accounts for the elongated shape of many urban land-use patterns and the persistence of high-status areas along specific corridors. However, it, too, struggles with contemporary polycentric megacities and the influence of zoning regulations.
霍伊特的扇形模型（1939 年）提出，城市沿主要交通线路呈楔形扇区从 CBD 向外辐射发展。例如，高租金居住区沿优选方向轴向外延伸，而工业区则沿铁路或河流布局。低收入住房往往集中在工业带周边。
霍伊特的理论强调交通动脉的首要性，并认为某种用地一旦在扇区内形成，就会随着城市增长向外延伸，而非被挤入新环带。该理论较好地解释了诸多城市土地用途的狭长形状以及高尚住区沿特定走廊持续存在的现象，但仍难以解释当代多中心巨型城市及分区规章的影响。
10. Harris–Ullman Multiple Nuclei Model | 哈里斯–乌尔曼多核心模型
The multiple nuclei model (1945) argues that cities develop around several discrete nodes rather than a single CBD. Different activities—retail, port, manufacturing, university—attract complementary land uses and repel incompatible ones, creating mini-centers. Nuclei arise from historical accident, agglomeration economies, or specific locational advantages.
This theorem explains the fragmented land-use mosaic of modern sprawling cities. It highlights the role of specialization (e.g., an airport-centered business cluster) and the influence of both centripetal and centrifugal forces. While more flexible than preceding models, the multiple nuclei concept requires careful identification of the dominant nuclei and their shifting relationships over time.
多核心模型（1945 年）认为，城市围绕着若干离散节点而非单一 CBD 发展。不同活动——零售、港口、制造、大学——吸引互补用地并排斥不相容用途，形成多个次级中心。核心源于历史偶然、集聚经济或特定区位优势。
该理论解释了现代蔓延城市破碎化的用地镶嵌格局，突出了专业化（如机场商务集群）的作用以及向心力与离心力的双重影响。尽管比之前的模型更灵活，多核心概念要求仔细甄别主导核心及其随时间变化的关系。
11. Demographic Transition Model (DTM) | 人口转变模型 (DTM)
The Demographic Transition Model describes the historical shift of populations from high birth and death rates to low birth and death rates as countries industrialize and urbanize. It consists of five stages: Stage 1 (high stationary – both rates high and fluctuating); Stage 2 (early expanding – death rate falls, birth rate remains high); Stage 3 (late expanding – birth rate begins to decline); Stage 4 (low stationary – both rates low); and Stage 5 (declining – death rate slightly exceeds birth rate, leading to natural decrease).
The model serves to interpret population pyramids and fertility patterns holistically. Exceptions (e.g., some African countries moving through Stage 2 due to delayed health transitions) and criticisms (Eurocentric assumptions, overlooking migration) must be noted. Nevertheless, DTM remains a foundational theorem for classifying populations and anticipating demographic challenges.
人口转变模型描述了随着工业化和城市化，人口从高出生率、高死亡率向低出生率、低死亡率转变的历史过程。它包含五个阶段：第一阶段（高位静止——两率均高且波动）；第二阶段（早期扩张——死亡率下降，出生率居高）；第三阶段（晚期扩张——出生率开始下降）；第四阶段（低位静止——两率均低）；第五阶段（衰减——死亡率略超出生率，导致自然减少）。
该模型用于整体解读人口金字塔和生育模式。须注意例外情况（如部分非洲国家因卫生转型滞后处于第二阶段）及批判（欧洲中心假设、忽视迁移）。然而，DTM 仍是划分人口类型、预判人口挑战的基础定理。
12. Strahler Stream Order & Bifurcation Ratio | 斯特拉勒河流等级与分叉比
Strahler’s method assigns a numerical order to stream segments: fingertip tributaries are order 1; when two streams of the same order join, the downstream segment increases by one order. The bifurcation ratio (Rb) is calculated by dividing the number of streams of a given order by the number of streams of the next higher order.
Rb = Nu / Nu+1
Typically, Rb ranges between 3 and 5. High values indicate a landscape with many low-order tributaries relative to higher-order channels, often associated with steep topography or strong fluvial dissection. A low bifurcation ratio points to a less branched network, which can affect flood response: higher Rb can lead to rapid concentration of runoff. This theorem underpins drainage basin morphometry and flood hazard assessment in AS Geography.
斯特拉勒法对河段进行数值分级：最末级细沟为 1 级；两条同级河流汇合后，下游河段级别加 1。分叉比 (Rb) 即某一级别河流数量除以下一高级别河流数量。
Rb = Nu / Nu+1
分叉比一般在 3 到 5 之间。高值意味着相对于高级别河道有大量低级别支流，常与陡峭地形或强烈流水切割相关；低分叉比则表示河网分支较少，可影响洪水响应——高 Rb 可能导致径流快速汇集。这条定理是 AS 地理中流域形态计量和洪水灾害评估的基础。
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