📚 Year 12 SQA Geography: Formula and Theorem Quick Reference Handbook | 12年级 SQA 地理：公式定理速查手册
This handbook provides a concise reference for the key quantitative skills and theoretical models assessed in Year 12 SQA Geography. Whether you are analysing river data or explaining urban patterns, these formulas and theorems will help you secure higher marks in exam questions and assignments. All content is closely aligned with the SQA course specifications for physical and human environments, as well as geographical skills.
本手册为12年级SQA地理考核所涉关键量化技能与理论模型提供简明参考。无论是分析河流数据还是解释城市格局，这些公式与定理都将助你在考试和作业中斩获更高分数。所有内容紧扣SQA课程中自然与人文环境及地理技能的规范。
1. Discharge and Channel Geometry | 流量与河道几何
River discharge is the volume of water flowing through a channel cross‑section per unit time, measured in cubic metres per second (m³ s⁻¹). It is the fundamental metric for flood studies and water resource management.
Q = A × V
河流流量是单位时间内通过河道横截面的水量，以立方米每秒 (m³ s⁻¹) 为单位。它是洪水研究与水资源管理的基本指标。
Cross‑sectional area (A) is typically calculated by multiplying channel width by mean depth at a gauging station. Velocity (V) is measured using a flow meter or float method and averaged across the profile.
横截面积 (A) 通常按测站处河宽乘以平均水深计算。流速 (V) 用流速仪或浮标法测得，并求剖面平均值。
Accurate discharge estimates require multiple vertical sections and velocity readings to account for variation across the channel.
准确的流量估算需布设多条垂线并读取流速，以考虑河道横向变化。
2. Hydraulic Radius and Channel Efficiency | 水力半径与河道效率
Hydraulic radius (R) describes the efficiency of a river channel to convey water. It is the ratio of cross‑sectional area (A) to wetted perimeter (P).
R = A / P
水力半径 (R) 表征河道输送水流的效率，为横截面积 (A) 与湿周 (P) 之比。
A larger hydraulic radius indicates less frictional resistance and higher flow velocity for a given gradient. Along the long profile, R typically increases downstream as channels become deeper and wider.
水力半径越大，在给定比降下摩擦阻力越小、流速越高。沿长剖面而下，随着河道变深变宽，R 通常增大。
Wetted perimeter is the length of the channel bed and banks in contact with water, measured in metres. In flood conditions, wetted perimeter grows, which can reduce efficiency temporarily.
湿周是河床与河岸接触水体的长度，以米计。洪水期间湿周增加，可能暂时降低输送效率。
3. Drainage Basin Metrics | 流域度量
Drainage density (Dd) expresses the total length of stream channels per unit area of basin. It reveals how well a basin is drained and influences flood response.
Dd = L / A
排水密度 (Dd) 表示流域单位面积内河流总长度，反映流域排水程度并影响洪水响应。
Stream frequency (Fs) is the total number of stream segments divided by basin area. Together with drainage density, it helps classify basin texture.
Fs = N / A
河流频度 (Fs) 是流段总数除以流域面积，可与排水密度共同用于划分流域纹理。
Bifurcation ratio (Rb) compares the number of streams of one order to the number of the next higher order. It indicates the degree of branching in a network.
Rb = Nₙ / Nₙ₊₁
分叉率 (Rb) 将某一级河道数量与次一级相比较，显示河网分支程度。通常 Rb 值在 3 到 5 之间。
These metrics are commonly applied in Horton‑Strahler stream ordering exercises and help predict basin hydrology.
这些指标常用于霍顿‑斯特拉勒水系分级练习，并帮助预测流域水文特征。
4. Bradshaw Model | 布拉德肖模型
The Bradshaw model is a conceptual theorem describing how river characteristics change from source to mouth. It predicts, on average, that discharge, channel depth, width, and velocity all increase downstream, while bedload particle size and channel gradient decrease.
布拉德肖模型是一个概念性定理，描述了河流特征从源头到河口的变化。它预测，总体而言，流量、河深、河宽和流速均向下游增加，而河床载荷粒径与河道比降则减小。
This model is crucial for fieldwork comparisons: students measure variables at multiple sites and check whether data aligns with the expected downstream trends. Deviations often highlight local controls like tributary inputs or human intervention.
该模型对实地考察对比至关重要：学生测量多地点变量，并检验数据是否符合预期的下游变化趋势。偏差通常突显出支流汇入、人类干预等局地控制因素。
While not a mathematical formula, the Bradshaw model provides a powerful predictive framework that must be supported by annotated diagrams in SQA answers.
尽管不是数学公式，但布拉德肖模型提供了强大的预测框架，在 SQA 答题中须辅以标注图示加以说明。
5. Population Change Formulas | 人口变化公式
Crude Birth Rate (CBR) is the number of live births per 1000 people in a given year. Crude Death Rate (CDR) follows the same logic.
CBR = (live births / total population) × 1000
CDR = (deaths / total population) × 1000
粗出生率 (CBR) 指某年每千人的活产数。粗死亡率 (CDR) 同理。
Rate of Natural Increase (RNI) is derived by subtracting CDR from CBR and is often expressed as a percentage or per 1000. The annual population growth rate adds net migration.
RNI = CBR – CDR
自然增长率 (RNI) 由 CBR 减去 CDR 得出，常以百分比或千分比表示。年人口增长率还需加入净迁移。
Population density measures how crowded an area is, calculated as population divided by land area (persons per km²). Physiological density relates population to arable land only.
Population Density = Population / Area
人口密度衡量区域拥挤程度，等于人口除以土地面积（人每 km²）。生理密度仅考虑人均耕地。
Doubling time estimates how many years it takes for a population to double at current growth rate. The 'Rule of 70' gives a quick approximation.
Doubling Time ≈ 70 / annual growth rate (%)
倍增时间估算按当前增长率人口翻倍所需年数。“70法则” 提供快速近似。
6. Demographic Transition Model (DTM) | 人口过渡模型
The Demographic Transition Model is a theorem linking population change to economic development. It consists of five stages: high stationary (stage 1), early expanding (2), late expanding (3), low stationary (4), and potential decline (5).
人口过渡模型是将人口变化与经济发展联系起来的定理，包含五个阶段：高位静止（阶段1）、早期扩张（2）、后期扩张（3）、低位静止（4）及潜在衰退（5）。
In stage 2, death rates fall but birth rates remain high, causing rapid population growth. By stage 4, both rates are low and the population stabilises. SQA often requires linking DTM stages to specific countries and population pyramids.
阶段2中死亡率下降而出生率仍高，导致人口快速增长。到阶段4，两率均低，人口趋于稳定。SQA 常要求将 DTM 各阶段与特定国家和人口金字塔关联。
Understanding DTM helps explain historical and contemporary population dynamics without a single mathematical formula, yet it is indispensable for human geography essays.
理解 DTM 有助于在缺少单一数学公式的情况下解释历史和当代人口动态，然而它在人文地理论述中不可或缺。
7. Urban Land Use Models | 城市土地利用模型
The Burgess Concentric Zone Model (1925) theorises that cities grow in a series of concentric rings around a central business district (CBD), with transition zones of industry and residential classes moving outward.
伯吉斯同心圆模型 (1925) 提出城市以中央商务区 (CBD) 为中心，形成一圈圈同心环带，向外依次为工业过渡带和不同等级的居住区。
Hoyt’s Sector Model (1939) modifies this by suggesting that land uses develop in wedges along transport routes, not continuous rings. High‑income residential stretches outward along the best transport axes.
霍伊特扇形模型 (1939) 修正了这一点，认为土地利用沿交通线呈楔形发展，而非连续环带。高收入住宅沿最优交通轴线向外延伸。
The Harris‑Ullman Multiple Nuclei Model (1945) posits that cities have several specialised centres, such as industrial parks, outlying shopping malls, or university districts, reducing the dominance of a single CBD.
哈里斯‑厄尔曼多核心模型 (1945) 假设城市拥有多个专业中心，如工业园区、郊区购物中心或大学区，削弱了单一 CBD 的主导地位。
The Bid‑Rent curve explains how land values and rents decline with distance from the CBD. Retail bids highest near the centre, while manufacturing and residential occupy lower‑rent zones further out.
竞租曲线说明地价和租金如何随距 CBD 的距离而下降。零售业在中心附近竞租最高，制造业与居住则占据更外围的低租金区域。
8. Von Thünen’s Agricultural Land Use Model | 冯·屠能农业土地利用模型
Von Thünen’s theorem (1826) predicts concentric rings of agricultural activity around a central market town. Perishable and heavy products like dairy and market gardening locate closest to the market, while grains and livestock rearing occur further away.
冯·屠能定理 (1826) 预测围绕中心集镇形成一圈圈同心农业活动带。易腐、体积重量大的产品如乳制品与商品园艺距市场最近，谷物和畜牧业则安置在更远处。
The key variable is transport cost per tonne‑kilometre. The model assumes an isotropic plain with uniform climate and fertility, and a single mode of transport. These simplifications help isolate distance as the primary factor.
关键变量是每吨公里运输成本。模型假设均质平原、气候与肥力均匀、单一运输方式——这些简化有助于隔离距离作为主导因素。
In SQA exams, students must explain how modern refrigerated transport and motorways have distorted the original rings but the core principle of land‑use intensity declining with distance still holds.
在 SQA 考试中，学生须解释现代冷藏运输与高速公路如何扭曲了原始环带，但土地利用强度随距离衰减的核心原理仍然成立。
9. Measures of Central Tendency and Dispersion | 集中趋势与离散度量
Mean, median, and mode are essential statistics for summarising geographical data sets. The mean is the arithmetic average, median is the middle value, and mode is the most frequent value.
Mean (x̄) = Σx / n
平均数、中位数和众数是总结地理数据集的基本统计量。平均数为算术平均值，中位数是中间值，众数是最频值。
ge is the difference between the maximum and minimum values. Interquartile range (IQR = Q₃ – Q₁) measures the spread of the middle 50% and is more resistant to outliers.
Range = Xₘₐₓ – Xₘᵢₙ
IQR = Q₃ – Q₁
全距是最大值与最小值之差。四分位距 (IQR = Q₃ – Q₁) 衡量中间 50% 的离散程度，且对异常值更具抗性。
These measures underpin the construction of box‑and‑whisker plots, which are frequently required in graphical skills questions. Always choose the appropriate measure based on data distribution.
这些度量为箱线图的构建奠定基础，常出现在图形技能题中。请始终依据数据分布选择合适的度量。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank (ρ) tests the strength and direction of a monotonic relationship between two ranked variables. It is the most common inferential statistic in SQA geographical investigation.
ρ = 1 − (6 Σ d²) / (n (n² − 1))
斯皮尔曼等级相关系数 (ρ) 检验两个排序变量之间的单调关系强度与方向，是 SQA 地理探究中最常用的推断统计量。
Here, d is the difference between ranks of each paired observation, and n is the number of pairs. The result ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). A value near zero suggests no relationship.
式中 d 为每对观测量等级之差，n 为对数。结果范围从 -1（完全负相关）到 +1（完全正相关），接近零表示无关系。
Calculate by ranking each variable separately, finding the difference in ranks for each case, squaring them, summing the squared differences, and substituting into the formula.
计算步骤：分别对每个变量排秩，求各案例的秩差，平方后求和，再代入公式。
Always state a null hypothesis (e.g. 'there is no correlation between river depth and velocity') and compare the calculated ρ to a critical value at the 0.05 significance level for the given n.
始终提出零假设（如“河流水深与流速无相关性”），并将计算出的 ρ 与给定 n 在 0.05 显著性水平下的临界值比较。
11. Gravity Model and Spatial Interaction | 重力模型与空间互动
The gravity model predicts the interaction (I) between two places based on their population sizes (P₁, P₂) and the distance (D) between them. It is derived from Newton’s law of gravitation and applied to migration, trade, and traffic flows.
I = (P₁ × P₂) / D²
重力模型根据两地的人口规模 (P₁, P₂) 和它们之间的距离 (D) 预测互动量 (I)。它源自牛顿万有引力定律，应用于迁移、贸易和交通流分析。
The squaring of distance reflects the friction of distance: interaction declines sharply as separation increases. Some versions use a variable exponent or add a constant k to calibrate for specific regions.
距离平方反映了距离摩擦效应：互动量随距离增加而急剧下降。个别版本使用可变指数或添加常数 k 以针对特定区域校准。
In SQA contexts, this model is useful when evaluating the sphere of influence of settlements or explaining commuter flows between an urban centre and its suburbs.
在 SQA 情境中，该模型用于评估聚落的影响范围，或解释市中心与郊区之间的通勤流。
12. Location Quotient and Economic Base | 区位商与经济基础
Location quotient (LQ) measures the concentration of a particular industry in a local area compared to a larger reference area, such as the nation. It helps identify export‑oriented basic employment.
LQ = (eᵢ / eₜ) / (Eᵢ / Eₜ)
区位商 (LQ) 衡量某特定产业在地方相比于更大参考区（如全国）的集中度，有助于识别出口导向
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