EOFk 空间型
不随时间变化的“典型地图”，回答信号在哪里：哪些区域同位相、哪些区域反位相。EOFk spatial pattern
A time-invariant “typical map” answering where the signal is: which regions are in phase and which are out of phase.
课程导航Course Navigation
第一部分 · 七问精解：沿“混合层（Mixed Layer）—温跃层（Thermocline）—深层水”知识链，掌握能量来源（风、冷却、太阳辐射）→ 垂直结构（温/盐/密）→ 水平差异（纬度带、洋盆对比）→ 维持机制（风生环流＋温盐环流）。第二部分 · EOF 方法：学习如何把海量的海洋时空观测场压缩成少数几个“空间型＋时间序列”，并识别 ENSO 等主导模态。Part 1 · Seven Key Questions: following the chain “Mixed Layer — Thermocline — Deep Water”, master the energy sources (wind, cooling, solar radiation) → vertical structure (T/S/density) → horizontal contrasts (latitude belts, basin comparisons) → maintenance mechanisms (wind-driven and thermohaline circulation). Part 2 · The EOF Method: learn how to compress vast oceanic spatiotemporal observation fields into a few “spatial patterns + time series”, and identify leading modes such as ENSO.
第一部分 · 七问精解Part 1 · Seven Key Questions
Part 1 Mixed Layer · Thermocline · Water Masses —— 7 个核心问题（英文原题＋中文解答＋示意图）Part 1 Mixed Layer · Thermocline · Water Masses — 7 core questions (original questions + answers + schematic figures)
1
How deep is a typical mixed layer if mixed by wind? How deep can it reach if driven by cooling?
风混合形成的混合层通常多深？由冷却驱动时又能达到多深？
解答要点Key Points
▸ 风致混合（Wind-driven mixing）▸ Wind-driven mixing
风通过风应力、波浪破碎和 Ekman 层湍流把热量、动量和浮力向下搅拌。典型混合层深度为 20～200 m，中低纬度平均约 50～100 m（夏季更浅，常仅 10～30 m）。它受层化限制——一旦遇到强密度跃层，风能输入便难以继续向下穿透。Wind stirs heat, momentum and buoyancy downward through wind stress, wave breaking and Ekman-layer turbulence. A typical wind-mixed layer is 20–200 m deep, averaging about 50–100 m at low–mid latitudes (even shallower in summer, often only 10–30 m). It is limited by stratification—once a strong pycnocline is met, wind energy can hardly penetrate farther down.
▸ 冷却驱动的对流混合（Cooling-driven convection）▸ Cooling-driven convection
冬季海面失热（加上强风、盐析）使表层水变冷变密，产生浮力对流（自由对流），混合深度远大于纯风混合：Winter surface heat loss (together with strong winds and brine rejection) makes surface water colder and denser, driving buoyant (free) convection, which reaches far deeper than wind mixing alone:
- 亚热带冬季：一般可达 200～500 m；Subtropical winters: generally 200–500 m;
- 亚极地海区（拉布拉多海、格陵兰海）及地中海、威德尔海/罗斯海等深对流区：可穿透 1000～2000 m 以上，个别地点直至海底，形成深层/底层水。In the subpolar seas (Labrador Sea, Greenland Sea), the Mediterranean and the Weddell/Ross Seas—deep-convection regions—it can penetrate 1000–2000 m or more, locally reaching the seafloor and forming deep/bottom water.
记忆口诀：风混合“百米量级”，冷却对流“数百米至数千米”——机械能搅拌受层化阻挡，而密度对流是整层“自上而下”的倾覆。Rule of thumb: wind mixing is “order 100 m”, cooling convection “hundreds to thousands of metres”—mechanical stirring is blocked by stratification, whereas density convection overturns the whole layer top-down.
2
What are the typical vertical temperature and salinity profiles in the subtropical and subpolar regions of the North Pacific?
北太平洋亚热带与亚极地海区典型的温度、盐度垂直剖面各是什么样？
解答要点Key Points
| 亚热带北太平洋（约 20–35°N）Subtropical N. Pacific (~20–35°N) | 亚极地/副极地北太平洋（约 45–60°N，亚北极环流区）Subpolar N. Pacific (~45–60°N, subarctic gyre) | |
|---|---|---|
| 温度 TTemp. T | 表层暖而均匀（SST 约 20–28 ℃）；其下为强而持久的永久温跃层（约 100–800 m），至 1000 m 降至约 4–5 ℃；深层约 1–2 ℃。夏季还叠加浅的季节温跃层。Warm, uniform surface (SST about 20–28 ℃); below lies a strong, persistent permanent thermocline (roughly 100–800 m), cooling to about 4–5 ℃ by 1000 m and 1–2 ℃ in the deep ocean. A shallow seasonal thermocline is superimposed in summer. | 表层全年冷（冬季约 0–5 ℃，夏季约 8–12 ℃）；无强永久温跃层。约 70–150 m 处出现冷水核心（dichothermal layer 冷温层），其下 300–500 m 有微弱的次表层暖水（mesothermal）；夏季形成明显季节温跃层。Cold surface year-round (about 0–5 ℃ in winter, 8–12 ℃ in summer); no strong permanent thermocline. A cold core (the dichothermal layer) occurs at about 70–150 m, with weak subsurface warm water (mesothermal) at 300–500 m; a pronounced seasonal thermocline forms in summer. |
| 盐度 SSalinity S | 表层盐度较高（约 34.8–35.3，蒸发>降水）；100–200 m 有次表层盐度极大值（副热带次表层水）；500–800 m 出现盐度极小值——北太平洋中层水 NPIW（约 34.2–34.5）；深层趋于均匀（约 34.65–34.7）。Relatively high surface salinity (about 34.8–35.3; E > P); a subsurface salinity maximum at 100–200 m (subtropical subsurface water); a salinity minimum at 500–800 m—North Pacific Intermediate Water, NPIW (about 34.2–34.5); the deep ocean is nearly uniform (about 34.65–34.7). | 表层显著低盐（约 32.5–33.3），来自强降水与陆地径流；100–200 m 处是强而持久的盐跃层（halocline），盐度随深度迅速升至约 34.3，深层约 34.6。A markedly fresh surface (about 32.5–33.3) from heavy precipitation and land runoff; a strong, persistent halocline at 100–200 m where salinity rises rapidly with depth to about 34.3, and about 34.6 at depth. |
关键对比：亚热带是“暖表水＋强温跃层＋盐度先极大后极小”；亚极地是“冷而淡的表层盖层＋强盐跃层＋弱温跃层”。正是低盐表层与强盐跃层抑制了深对流——北太平洋因此不形成深层水。Key contrast: the subtropics show “warm surface water + strong thermocline + a salinity maximum then minimum”; the subpolar region shows a “cold, fresh surface lid + strong halocline + weak thermocline”. The fresh surface layer and strong halocline suppress deep convection—so the North Pacific forms no deep water.
3
What are the dominant regions of net evaporation in the ocean?
海洋中净蒸发占主导的区域是哪些？
解答要点Key Points
净蒸发区（E − P > 0，蒸发量大于降水量）集中在南北纬约 15°–35° 的副热带——这里是哈得来环流的下沉支、副热带高压控制区，晴朗少雨、信风强盛。主要海域：Net-evaporation regions (E − P > 0, evaporation exceeding precipitation) are concentrated in the subtropics at about 15°–35° N/S—the descending branch of the Hadley cell under subtropical highs, with clear skies, little rain and strong trade winds. Main areas:
- 北大西洋副热带—热带信风带（马尾藻海及非洲西北侧外海）：全球最强的净蒸发中心；Subtropical–tropical trade-wind belt of the North Atlantic (the Sargasso Sea and waters off NW Africa): the strongest net-evaporation centre on Earth;
- 南大西洋副热带：就整个洋盆而言，大西洋是唯一净蒸发大于净降水的大洋；Subtropical South Atlantic: basin-wide, the Atlantic is the only ocean whose net evaporation exceeds net precipitation;
- 阿拉伯海、红海、波斯湾、地中海：被干旱陆地环绕，蒸发极强（红海盐度 >40，地中海约 38）；Arabian Sea, Red Sea, Persian Gulf and Mediterranean Sea: enclosed by arid land, evaporation is extremely strong (Red Sea salinity >40, Mediterranean about 38);
- 南、北太平洋与南印度洋的副热带环流中心亦为净蒸发，但强度较弱。The centres of the subtropical gyres in the North/South Pacific and South Indian Ocean are also net-evaporation regions, though weaker.
▸ 对照：净降水区（P > E）▸ Contrast: net-precipitation regions (P > E)
赤道辐合带（ITCZ）与西太平洋暖池、南太平洋辐合带（SPCZ）、亚极地北大西洋/北太平洋，以及 45°–60°S 的南大洋“咆哮西风带”。The Intertropical Convergence Zone (ITCZ) and the western Pacific warm pool, the South Pacific Convergence Zone (SPCZ), the subpolar North Atlantic/North Pacific, and the “Roaring Forties/Fifties” of the Southern Ocean at 45°–60°S.
规律：蒸发—降水的纬度分布呈“两峰一谷”——两个副热带蒸发峰夹着赤道降水谷，高纬再度转为净降水。Pattern: the latitudinal E–P distribution shows “two peaks and one trough”—two subtropical evaporation peaks flanking the equatorial precipitation trough, with net precipitation again at high latitudes.
4
What are the key components of the Mixed Layer heat budget?
混合层热收支的关键组成项有哪些？
解答要点Key Points
设混合层厚 h、温度 T，单位面积热收支可写为：For a mixed layer of depth h and temperature T, the heat budget per unit area can be written as:
ρ cp h · ∂T/∂t = Qnet + Qadv + Qentr + Qdiff
- 热含量变化（储能项）左项：混合层升温/降温速率，是各项平衡的结果；Heat-content change (storage) — the left-hand term: the warming/cooling rate of the mixed layer, the net result of all terms;
- 海气净热通量 Qnet：
Qnet = QSW − QLW − QE − QH
即吸收的太阳短波辐射，减去长波回辐射、潜热通量（蒸发耗热）与感热通量（导热/对流）。部分短波可穿透混合层底部；Net air–sea heat flux Qnet:
Qnet = QSW − QLW − QE − QH
absorbed solar shortwave radiation minus outgoing longwave radiation, latent heat flux (evaporative cooling) and sensible heat flux (conduction/convection). Some shortwave radiation penetrates below the mixed-layer base; - 水平平流 Qadv：地转流与 Ekman 流的暖/冷水输运及暖水辐聚（如副热带辐聚、西边界流区）；Horizontal advection Qadv: transport of warm/cold water by geostrophic and Ekman currents and convergence of warm water (e.g. subtropical convergence, western-boundary-current regions);
- 夹卷/卷夹 Qentr：混合层冬季加深时把温跃层冷水卷入（降温项）；夏季变浅时发生“潜沉 detrainment”；Entrainment Qentr: as the mixed layer deepens in winter it entrains cold thermocline water (a cooling term); when it shallows in summer, detrainment occurs;
- 扩散项 Qdiff：跨混合层底的湍流/分子热扩散，量级通常较小；海冰生消、融冰淡水在高纬另需考虑。Diffusion Qdiff: turbulent/molecular heat diffusion across the mixed-layer base, usually small; sea-ice growth/melt and meltwater must be considered additionally at high latitudes.
季节性直觉：夏季 QSW 主导 → 混合层升温、变浅；冬季 QE、QH、QLW 净失热 + 夹卷冷水 → 混合层降温、加深。Seasonal intuition: in summer QSW dominates → the mixed layer warms and shallows; in winter net heat loss via QE, QH and QLW plus entrainment of cold water → it cools and deepens.
5
Is the North Atlantic Ocean saltier or fresher on average than the Pacific? Why?
北大西洋平均盐度比太平洋更高还是更低？为什么？
解答要点Key Points
北大西洋更咸（saltier）。北大西洋平均盐度约 34.9，副热带表层可超过 36.5–37；北太平洋平均约 34.6，亚极地表层低至 32.5–33.3。主要原因：The North Atlantic is saltier. Its mean salinity is about 34.9, and subtropical surface waters can exceed 36.5–37; the North Pacific averages about 34.6, with subpolar surface values as low as 32.5–33.3. The main reasons:
- ① 淡水收支相反：大西洋整体 E − P > 0（净蒸发洋盆），太平洋整体 P − E > 0（净降水洋盆），赤道西太平洋暖池与 ITCZ/SPCZ 降雨极强；① Opposite freshwater budgets: the Atlantic as a whole has E − P > 0 (a net-evaporating basin), while the Pacific has P − E > 0 (a net-precipitating basin), with extremely heavy rainfall over the western Pacific warm pool and the ITCZ/SPCZ;
- ② 大气水汽跨洋盆输送：东北信风把大西洋上空的水汽带过狭窄的中美洲地峡（巴拿马）输入太平洋，通量约 0.2–0.3 Sv，使大西洋“失水变咸”、太平洋“得水变淡”；② Atmospheric water-vapor transport across basins: the northeast trade winds carry water vapor from the Atlantic across the narrow Central American Isthmus (Panama) into the Pacific, a flux of about 0.2–0.3 Sv, making the Atlantic saltier through water loss and the Pacific fresher through water gain;
- ③ 高盐水团注入：地中海出流水（S≈38.4）、红海/波斯湾高盐水抬升大西洋—印度洋的中层盐度；③ Input of highly saline water masses: Mediterranean outflow water (S≈38.4) and Red Sea/Persian Gulf hypersaline waters raise the intermediate-water salinity of the Atlantic–Indian Ocean system;
- ④ 太平洋几何与径流：太平洋在多雨的赤道带展布更宽，且接受大量陆地淡水（如鄂霍次克海、白令海及众多大河的影响），北太平洋亚极地形成持久低盐表层盖层。④ Pacific geometry and runoff: the Pacific spans a wider extent across the rainy equatorial belt and receives large amounts of continental freshwater (e.g., via the Sea of Okhotsk, the Bering Sea, and many great rivers), so a persistent low-salinity surface cap forms in the subpolar North Pacific.
深远影响：高盐使北大西洋表层水冬季冷却后密度足够大，能够下沉形成北大西洋深层水（NADW），驱动全球温盐环流；北太平洋盐度不足，故无深水形成。Far-reaching consequence: high salinity gives North Atlantic surface waters, after winter cooling, a density high enough to sink and form North Atlantic Deep Water (NADW), driving the global thermohaline circulation; the North Pacific is not salty enough, so no deep water forms there.
6
What regions of the ocean are characterized by large differences from summer to winter? What regions have the least seasonal variability?
哪些海区夏冬差异最大？哪些海区季节变化最小？
解答要点Key Points
▸ 季节变化最大：中纬度（约 30°–50°，尤其洋盆西北部）▸ Largest seasonal changes: mid-latitudes (about 30°–50°, especially the northwestern parts of basins)
- 西北大西洋（纽芬兰—湾流外侧）与西北太平洋（黑潮—亲潮交汇带）：海表温度年较差可达 8–15 ℃；Northwest Atlantic (Newfoundland–off the Gulf Stream) and Northwest Pacific (Kuroshio–Oyashio confluence): the annual SST range can reach 8–15 °C;
- 混合层深度从夏季的一二十米增至冬季的 200–500 m，温跃层、热含量乃至生态过程（浮游植物勃发）均随之季节反转；mixed-layer depth deepens from 10–20 m in summer to 200–500 m in winter; the thermocline, heat content, and even ecological processes (phytoplankton blooms) undergo seasonal reversals accordingly;
- 季风区（阿拉伯海、孟加拉湾）与高纬海冰区（混合层、海冰覆盖）季节反差也很强。monsoon regions (Arabian Sea, Bay of Bengal) and high-latitude sea-ice regions (mixed layer, ice cover) also show strong seasonal contrasts.
▸ 季节变化最小：赤道带（约 10°N–10°S）▸ Least seasonal variability: the equatorial belt (about 10°N–10°S)
- 太阳高度年变化小，西太平洋—东印度洋暖池 SST 终年约 27–29 ℃，温盐结构最稳定；the annual variation of solar elevation is small; in the western Pacific–eastern Indian Ocean warm pool, SST stays around 27–29 °C year-round and the thermohaline structure is most stable;
- 极地近岸海冰下 SST 终年贴近冰点，温度年较差也小（但海冰与盐度层结季节变化大）。under nearshore polar sea ice, SST stays near the freezing point all year, so the annual temperature range is also small (although sea ice and salinity stratification vary strongly with the seasons).
成因：季节信号的强度取决于太阳辐射年变幅与表层热含量的惯性——中纬大陆边缘既受寒暖气团交替影响，又有强冬季冷却与强夏季增温，故年循环最强。Causes: the strength of the seasonal signal depends on the annual range of solar radiation and the inertia of surface heat content — the mid-latitude continental margins are affected by alternating cold and warm air masses and experience both strong winter cooling and strong summer warming, giving the strongest annual cycle.
7
Please describe the two processes that maintain the thermocline.
请描述维持温跃层的两个过程。
解答要点Key Points
永久温跃层是暖表层水与冷深层水之间陡峭的温度梯度，它并非静止存在，而是由两类过程的动态平衡维持：The permanent thermocline is the steep temperature gradient between warm surface water and cold deep water. It does not exist statically but is maintained by the dynamic balance of two kinds of processes:
▸ 过程一：热量向下扩散 ＋ 暖水持续供给▸ Process 1: downward heat diffusion ＋ continuous supply of warm water
太阳把热量注入表层，湍流混合（内波破碎、双扩散、涡旋）使热量不断沿温跃层向下扩散，倾向于抹平温度梯度；同时风生副热带环流通过 Ekman 抽吸使表层暖水在副热带辐聚、潜沉（subduction），持续把暖水“灌入”温跃层上部。The Sun deposits heat in the surface layer, and turbulent mixing (breaking internal waves, double diffusion, eddies) continually diffuses heat downward through the thermocline, tending to smooth out the temperature gradient; at the same time, the wind-driven subtropical gyre causes surface warm water to converge in the subtropics via Ekman pumping and to subduct, continuously “pumping” warm water into the upper thermocline.
▸ 过程二：冷水的大尺度上升补偿（温盐环流）▸ Process 2: large-scale upward compensation of cold water (thermohaline circulation)
高纬海区冷却下沉形成的深层水（NADW、AABW）在洋盆中缓慢流动并在广阔区域上升（upwelling），把冷水持续送到温跃层下方，抵消下扩的热量。Deep water formed by cooling and sinking at high latitudes (NADW, AABW) flows slowly through the basins and upwells over broad regions, continually delivering cold water beneath the thermocline and offsetting the downward-diffusing heat.
w · ∂T/∂z = Kv · ∂²T/∂z²
Munk (1966) “Abyssal Recipes”：冷水上升（w ≈ 10⁻⁷ m·s⁻¹，约 1 cm/天）与向下涡动扩散（Kv ≈ 10⁻⁴ m²·s⁻¹）相平衡。Munk (1966) “Abyssal Recipes”: upwelling of cold water (w ≈ 10⁻⁷ m·s⁻¹, about 1 cm/day) balances downward eddy diffusion (Kv ≈ 10⁻⁴ m²·s⁻¹).
一句话总结：温跃层靠“风生环流把暖水维持在上层并下压”与“温盐环流把冷水维持在下层并上抬”这两个过程共同维持；任一过程停止，扩散最终都会把温跃层夷平。In one sentence: the thermocline is maintained jointly by the wind-driven circulation keeping warm water in the upper layer and pushing it down and the thermohaline circulation keeping cold water in the deep layer and lifting it up; if either process stopped, diffusion would eventually flatten the thermocline.
第二部分 · EOF 经验正交函数Part 2 · Empirical Orthogonal Function (EOF)
Part 2 Empirical Orthogonal Function —— 把随时间变化的海洋场，分解为少数几个“空间型 × 时间序列”Part 2 Empirical Orthogonal Function — decomposing time-varying ocean fields into a few “spatial patterns × time series”
E1
What is the Empirical Orthogonal Function (EOF)?
什么是经验正交函数（EOF）？
核心定义Core definition
EOF（Empirical Orthogonal Function，经验正交函数）是一种把随时间演变的空间场（如逐月海表温度距平 SSTA、海平面高度异常、风应力场）分解为若干互不相关的典型空间型及其对应时间系数的统计方法。EOF (Empirical Orthogonal Function) is a statistical method that decomposes a spatial field evolving in time (such as monthly sea-surface temperature anomalies SSTA, sea-level height anomalies, or wind-stress fields) into a set of mutually uncorrelated typical spatial patterns and their corresponding time coefficients.
一句话理解：EOF 直接从观测数据本身出发（“经验”），找出一组正交（互相垂直、彼此不相关）的典型空间振荡型（“函数”），并按各自解释的方差从大到小排列——通常前几个模态就能刻画整个场的主要变化，实现“海量格点数据，几条曲线看懂”。In one sentence: starting directly from the observed data (“empirical”), EOF finds a set of orthogonal (mutually perpendicular, uncorrelated) typical spatial oscillation patterns (“functions”), ordered from the most to the least variance explained—usually the first few modes capture the field's main variability, so that “huge gridded datasets can be understood through a few curves”.
▸ 时空分解的思想▸ The idea of space–time decomposition
任意时刻的距平场，都可写成若干张固定“地图”的线性组合：The anomaly field at any time can be written as a linear combination of several fixed “maps”:
X(x, y, t) ≈ Σk EOFk(x, y) · PCk(t)
PCk 主成分（时间系数）
该空间型在每个时刻的强弱与正负，回答信号何时强、何时反相。PCk principal component (time coefficient)
The strength and sign of that pattern at each time, answering when the signal is strong and when it reverses.
该空间型在每个时刻的强弱与正负，回答信号何时强、何时反相。PCk principal component (time coefficient)
The strength and sign of that pattern at each time, answering when the signal is strong and when it reverses.
▸ EOF 与 PCA / SVD 的关系▸ Relation of EOF to PCA / SVD
EOF 是气象与海洋学界的习惯叫法，数学上与统计学中的 PCA（Principal Component Analysis，主成分分析）完全等价；计算上通过数据矩阵的 SVD（奇异值分解）或协方差矩阵的特征值分解实现。该思想由 Lorenz (1956) 引入天气预报、Kutzbach (1967) 首次用于气候场分析。EOF is the conventional name in meteorology and oceanography; mathematically it is exactly equivalent to PCA (Principal Component Analysis) in statistics, and is computed via SVD (singular value decomposition) of the data matrix or eigen-decomposition of the covariance matrix. The approach was introduced to weather forecasting by Lorenz (1956) and first applied to climate fields by Kutzbach (1967).
“正交”意味着各空间型之间、各 PC 之间两两不相关，所以第 k 个模态独立解释 λk 份方差，解释量可直接相加。“Orthogonal” means the spatial patterns—and the PCs—are pairwise uncorrelated, so mode k independently accounts for λk of the variance, and the explained amounts can be added directly.
E2
Mathematical formulation: covariance matrix, eigen-decomposition and variance
数学原理：数据矩阵、协方差、特征值分解与方差贡献率
五步数学推导Five-step mathematical derivation
① 构造数据矩阵并去距平① Build the data matrix and remove the mean
把 n 个空间格点、t 个时次的观测排成矩阵 X ∈ ℝn×t（每列为一个时刻的全场快照）。先扣除每个格点的气候平均，必要时再去线性趋势：Arrange observations at n spatial grid points and t times into a matrix X ∈ ℝn×t (each column is a full-field snapshot at one time). First subtract the climatological mean at each grid point, and remove a linear trend if necessary:
X′(i, t) = X(i, t) − X̄(i)
② 加面积权重（格点数据）② Apply area weights (gridded data)
经纬网格在高纬变密，需给每一格点乘 √cosφ（φ 为纬度），使每个格点代表相同的地球表面积：X′w = diag(√cosφ) · X′。A lon–lat grid becomes denser at high latitudes, so each grid point is multiplied by √cosφ (φ = latitude) so that every point represents the same Earth surface area: X′w = diag(√cosφ) · X′.
③ 求空间协方差矩阵③ Compute the spatial covariance matrix
C = 1 / (t − 1) · X′w X′wT （n × n 实对称矩阵）(an n × n real symmetric matrix)
④ 特征值分解（等价于对 X′ 做 SVD）④ Eigen-decomposition (equivalent to SVD of X′)
C · ek = λk · ek ⇒ 特征向量 ek = EOFk（空间型）eigenvectors ek = EOFk (spatial patterns)
把每个空间型投影到原始距平场，即得该模态的时间系数：Projecting each spatial pattern onto the original anomaly field gives that mode's time coefficient:
PCk(t) = ekT · X′w(:, t)
⑤ 方差贡献率与场重构⑤ Explained-variance fraction and field reconstruction
第 k 模态方差贡献率Mode-k variance fraction = λk / Σj λj 重构reconstruction: X′ = Σk EOFk · PCk
λₖ（特征值）
第 k 模态所解释的方差量，λ 越大越“主导”，故模态按 λ 降序排列。λₖ (eigenvalue)
The amount of variance explained by mode k; the larger λ, the more “dominant” the mode, so modes are sorted by descending λ.
第 k 模态所解释的方差量，λ 越大越“主导”，故模态按 λ 降序排列。λₖ (eigenvalue)
The amount of variance explained by mode k; the larger λ, the more “dominant” the mode, so modes are sorted by descending λ.
EOFₖ（特征向量）
单位长度的空间型，只定“形状”；异常的量级由 PC 振幅承载。EOFₖ (eigenvector)
A unit-length spatial pattern defining only the “shape”; anomaly magnitude is carried by PC amplitude.
单位长度的空间型，只定“形状”；异常的量级由 PC 振幅承载。EOFₖ (eigenvector)
A unit-length spatial pattern defining only the “shape”; anomaly magnitude is carried by PC amplitude.
PCₖ（主成分）
时间序列，均值为 0、各 PC 互不相关；其方差正好等于 λₖ。PCₖ (principal component)
A zero-mean time series, uncorrelated with the other PCs; its variance equals λₖ.
时间序列，均值为 0、各 PC 互不相关；其方差正好等于 λₖ。PCₖ (principal component)
A zero-mean time series, uncorrelated with the other PCs; its variance equals λₖ.
SVD 视角
X′ = UΣVᵀ 中 U 列向量＝EOF，VΣ 的列＝PC；n≫t 时改在 t×t 矩阵上算更省时。SVD view
In X′ = UΣVᵀ, columns of U are the EOFs and columns of VΣ are the PCs; when n≫t it is cheaper to work with the t×t matrix.
X′ = UΣVᵀ 中 U 列向量＝EOF，VΣ 的列＝PC；n≫t 时改在 t×t 矩阵上算更省时。SVD view
In X′ = UΣVᵀ, columns of U are the EOFs and columns of VΣ are the PCs; when n≫t it is cheaper to work with the t×t matrix.
规范化约定：文献中有“EOF 单位化、PC 带振幅”与“PC 标准化、EOF 带振幅”两种写法（及载荷 loading），解释方差完全相同，比较振幅时需留意采用哪种约定。Normalization conventions: the literature uses both “unit EOFs, PCs carry the amplitude” and “standardized PCs, EOFs carry the amplitude” (plus loadings); the explained variance is identical, but note which convention is used when comparing amplitudes.
E3
How to perform an EOF analysis in oceanography — step by step
海洋学中如何做 EOF 分析：标准七步流程
标准流程Standard procedure
- 选定要素、区域与时段：如热带太平洋逐月 SSTA（1950–2024），统一到同一网格，缺测格点插值或掩膜。Choose the variable, domain and period: e.g. monthly SSTA over the tropical Pacific (1950–2024); interpolate onto a common grid and interpolate or mask missing points.
- 去气候平均（距平）：扣除逐月气候态 X̄，消除季节循环；按需要去线性趋势，或带通滤波（如保留 2–7 年的 ENSO 频段）。Remove the climatological mean (anomalies): subtract the monthly climatology X̄ to remove the seasonal cycle; detrend or band-pass filter as needed (e.g. retain the 2–7-year ENSO band).
- 格点面积权重：每点乘 √(cos 纬度)，防止高纬密集格点被“重复计数”。Apply grid-area weights: multiply each point by √(cos latitude) to prevent dense high-latitude grid points from being “double-counted”.
- 构造协方差矩阵 / 直接 SVD：C = X′X′ᵀ/(t−1)，或直接对 X′ 做奇异值分解，二者等价。Build the covariance matrix / use SVD directly: C = X′X′ᵀ/(t−1), or apply singular value decomposition directly to X′; the two are equivalent.
- 求解 EOF 与 PC：得到按 λₖ 降序排列的空间型与时间系数，计算各模态方差贡献率与累计贡献率。Solve for the EOFs and PCs: obtain spatial patterns and time coefficients sorted by descending λₖ, and compute each mode's explained-variance fraction and the cumulative fraction.
- 显著性与可分性检验：用 North et al. (1982) 准则判断相邻模态是否可分（特征值误差棒不重叠），辅以 Monte Carlo / Rule N 检验，避免把“噪声模态”当信号。Significance and separability tests: use the North et al. (1982) rule to judge whether adjacent modes are separable (non-overlapping eigenvalue error bars), supplemented by Monte Carlo / Rule N tests, to avoid mistaking “noise modes” for signal.
- 物理解释与重构：把 PC 回归到原场得到异常量级，结合动力机制命名模态（ENSO、PDO、AO/NAM…）；用前 K 个模态重构 X′ 即完成降维。Physical interpretation and reconstruction: regress the PCs onto the original field to obtain anomaly magnitudes, and name the modes using dynamical mechanisms (ENSO, PDO, AO/NAM…); reconstructing X′ with the first K modes completes the dimensionality reduction.
使用注意：① EOF 是统计正交模态，并不自动对应某种动力机制，物理解释需谨慎；② 模态符号可整体翻转；③ 分析区域（domain）的选择会改变模态形状；④ 旋转 EOF（REOF）结构更“局地”但失去正交性；⑤ 存在强年周期信号时应先去季节循环，否则 EOF₁ 只会是季节模。Caveats: ① EOFs are statistically orthogonal modes and do not automatically correspond to a dynamical mechanism, so physical interpretation requires care; ② mode signs can be flipped as a whole; ③ the choice of analysis domain changes the pattern shapes; ④ rotated EOFs (REOF) give more “localized” structures but lose orthogonality; ⑤ if a strong annual-cycle signal exists, remove the seasonal cycle first, or EOF₁ will simply be the seasonal mode.
E4
EOF in oceanography: ENSO and other leading modes of variability
海洋学应用：EOF1 与 ENSO，以及其他主导变化模态
经典案例：热带太平洋 SSTA 的 EOF₁ ≈ ENSOClassic case: EOF₁ of tropical Pacific SSTA ≈ ENSO
对热带太平洋月平均海表温度距平（SSTA）场做 EOF，第一模态通常解释 50%–70% 的方差：其空间型表现为赤道中、东太平洋一致的暖异常（向西延伸的“舌状”结构）与西太平洋暖池周边的反向冷异常（马蹄形）；其 PC₁ 曲线在 El Niño 年出现强正峰、La Niña 年出现深负谷——PC₁ 与常用的 Niño3.4 指数相关系数常超过 0.95，二者几乎是同一条曲线。Applying EOF to monthly mean sea-surface temperature anomaly (SSTA) fields over the tropical Pacific, the first mode typically explains 50%–70% of the variance: its spatial pattern shows a coherent warm anomaly across the central and eastern equatorial Pacific (a “tongue” extending westward) and an opposite cold anomaly around the western Pacific warm pool (horseshoe-shaped); its PC₁ shows strong positive peaks in El Niño years and deep negative troughs in La Niña years—PC₁ often correlates with the standard Niño3.4 index at above 0.95, and the two are nearly the same curve.
▸ 其他常见应用▸ Other common applications
- SST / SSTA 场：ENSO（2–7 年）、北太平洋 PDO（年代际，20–60 年）、印度洋海盆一致模（IOB）与偶极子（IOD）。SST / SSTA fields: ENSO (2–7 years), the North Pacific PDO (decadal, 20–60 years), the Indian Ocean basin mode (IOB) and dipole (IOD).
- 卫星高度计海平面高度异常（SSHA）：识别大尺度环流调整、涡动能量分布与海平面长期变化型。Satellite-altimeter sea-surface height anomalies (SSHA): identifying large-scale circulation adjustments, eddy-energy distributions and long-term sea-level change patterns.
- 温盐剖面、海流、风应力、海冰密集度等场的主导变化型提取。Extracting the leading patterns of temperature/salinity profiles, currents, wind stress and sea-ice concentration fields.
- 数据压缩与模式评估：用前几个模态重构场、对比观测与气候模式的模态结构，亦是统计预测（降维、可预报性）的标准前处理。Data compression and model evaluation: reconstructing fields with the first few modes and comparing modal structures between observations and climate models; it is also standard preprocessing for statistical prediction (dimensionality reduction, predictability).
看 EOF（地图）
ENSO 的“暖舌”在哪里、哪些区域反位相 → 空间型回答机制作用范围。Read the EOF (map)
Where the ENSO “warm tongue” sits and which regions are out of phase → the pattern answers where the mechanism acts.
ENSO 的“暖舌”在哪里、哪些区域反位相 → 空间型回答机制作用范围。Read the EOF (map)
Where the ENSO “warm tongue” sits and which regions are out of phase → the pattern answers where the mechanism acts.
看 PC（曲线）
1997/98、2015/16 为强正峰（El Niño），1999–2001 持续负值（La Niña）→ 时间系数回答事件年与强度。Read the PC (curve)
Strong positive peaks in 1997/98 and 2015/16 (El Niño), persistent negative values in 1999–2001 (La Niña) → the time coefficient answers event years and strength.
1997/98、2015/16 为强正峰（El Niño），1999–2001 持续负值（La Niña）→ 时间系数回答事件年与强度。Read the PC (curve)
Strong positive peaks in 1997/98 and 2015/16 (El Niño), persistent negative values in 1999–2001 (La Niña) → the time coefficient answers event years and strength.
本质上，Niño3.4 等“气候指数”就是 EOF 思想的手工版：选取一个典型空间区域（指数），追踪其平均异常随时间的演变（PC）；EOF 则让数据自己“找”出最优的空间型。In essence, “climate indices” such as Niño3.4 are a manual version of the EOF idea: choose a typical spatial region (the index) and track the evolution of its mean anomaly over time (the PC); EOF instead lets the data itself “find” the optimal spatial pattern.