The wave equation is second order in time. Even one normal mode obeys a second-order ordinary differential equation, so its initial displacement and initial velocity are independent data. Physically they specify stored elastic deformation and the initial motion.

The heat equation is first order in time; its initial temperature already fixes the instantaneous time derivative through the spatial Laplacian. It needs one initial field. In either case spatial boundary conditions remain necessary and are not substitutes for initial data. Classical fixed-end string data must satisfy the appropriate endpoint compatibility, including $f(0)=f(L)=0$ and, when endpoint velocities exist, $g(0)=g(L)=0$.

Since tension is a force and ${\lambda }_0$ is a linear mass density,Thus $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$. It is the transverse wave speed of the ideal uniform string. The wave equation on a string assumes small transverse slopes, constant positive tension, positive uniform linear mass density, and negligible damping and bending stiffness.

The uniform source adds a background without changing the spatial gradients relative to the unforced solution:At $10\mathrm{s}$ the offset is $1-e^{-1}\simeq 0.6321$. On the corrected grid the central peak is therefore about $0.8321$, and regions far from the hot spot are near $0.6321$. The hot spot's broadening is unchanged; the source is spatially uniform. The spatial mean of a periodic heat equation rises by the same amount, tending to a total increment of one temperature unit as $t\to \infty$.

A spatially uniform source changes only the zero Fourier mode. Its exact integrated increment over a time step beginning at $t$ isThe unnormalised two-dimensional discrete Fourier transform stores a constant field $q$ as $n_x{n}_{y}q$ in element (1,1).^[ref] Keep the existing homogeneous exponential multiplication and add this source increment to that element:This replacement uses the original variable names and MATLAB indexing. The uniform forcing of a periodic heat equation is integrated exactly in time here. No source multiplier is applied to the other modes, and the zero-mode homogeneous multiplier equals one. Adding only dq to the unnormalised coefficient would underestimate the physical temperature increment by the factor $n_x{n}_y$.

Aliasing occurs because a sampled grid cannot distinguish certain different continuous Fourier modes. With spacing $h$, modes $k$ and $k+2\pi m/h$ agree at every sample point. Frequencies above the Nyquist frequency can therefore be misidentified as lower frequencies, changing the resolved spectrum and its subsequent diffusion rate.

For the corrected grid, $h=10/128\mathrm{cm}$ and the maximum resolved angular wavenumber is approximately $\pi /h\simeq 40.2\mathrm{c}{\mathrm{m}}^{-1}$. The initial Gaussian function has spectral envelope proportional to $e^{-a|k{|}^2/4}$. At that one-coordinate cutoff the factor is about $e^{-40.4}$, so the omitted high-frequency tail is tiny. Its continuation mismatch at the distant edges is also tiny. Significant aliasing is not expected for this resolved, smooth initial condition.

The linear constant-coefficient heat equation only damps existing modes; it does not generate higher frequencies through nonlinear products. An unresolved narrow peak, a sharp edge or a nonlinear source would require a separate resolution assessment. The duplicated-endpoint grid issue described above should not be diagnosed as high-frequency folding.

Diffusion smooths and broadens the hot spot while conserving its integral. For an isolated two-dimensional initial profile $u_0=e^{-r^2/a}$, the Gaussian hot spot under diffusion isHere $a=0.1\mathrm{c}{\mathrm{m}}^2$ and $t=10\mathrm{s}$, giving a width parameter of $0.5\mathrm{c}{\mathrm{m}}^2$ and a peak of $0.20$ in the initial temperature units. The periodic-grid calculation agrees to displayed precision because the periodic copies are far away. Running the endpoint-inclusive listing as written instead gives a sampled initial peak near $0.9695$ and final peak near $0.1963$; the difference is consistent with the coordinate-grid issue, rather than a different diffusion law.

Increasing positive thermal diffusivity at fixed time produces a wider, lower peak. Each nonconstant Fourier mode decays faster, while the spatial mean of a periodic heat equation remains fixed. The profiles approach a uniform mean at long times under these periodic boundary conditions.

The line references below use the original listing's line numbers.

For this linear homogeneous problem, the exponential update is exact for the retained modes, apart from roundoff error. It is not a forward Euler method step and does not require the explicit finite-difference diffusion stability restriction. Sampling, periodic modelling and spatial resolution still matter.

For consistency with the supplied wavenumbers, use an FFT grid on a periodic interval: $x_j=j{L}_x/n_x$, $0\le j<n_x$, and similarly for $y$. In MATLAB, replace the endpoint-inclusive coordinate definitions by x = (0:nx-1)*Lx/nx; and y = (0:ny-1)*Ly/ny;. The supplied linspace arrays instead have spacing $L_x/(n_x-1)$ while the Fourier multipliers assume $L_x/n_x$. Including both copies of the periodic endpoint creates a grid inconsistency, separate from high-frequency aliasing.

Let ${\hat{u}}_0$ be the Fourier transform of the initial temperature. Solving the scalar linear ordinary differential equation givesRecover $u$ using the Fourier inversion theorem, with inverse factor $(2\pi {)}^{-2}$ for this two-dimensional convention. Equivalently, take the convolution of the initial data with the two-dimensional heat kernel. On a periodic square, the same exponential applies to each discrete Fourier mode, with $k_x=2\pi m/L_x$ and $k_y=2\pi n/L_y$.

Apply the Fourier transform of a derivative twice in each spatial coordinate, and commute the time derivative with the spatial Fourier transform. For the Laplacian,ThusThe minus sign is essential: positive thermal diffusivity damps short spatial scales rather than amplifying them. The zero Fourier mode is unchanged in the absence of a source.

Assume sufficient decay and differentiability to justify integration by parts and exchange of integrals; Schwartz functions are one convenient class. With the stated Fourier transform convention, integrate first in $x$:The boundary contribution vanishes under the decay assumption. The Fourier transform of a derivative consequently turns constant-coefficient spatial differential operators into multipliers. Repeating it gives $\mathcal{F}[u_{xx}]=-k_x^2\hat{u}$, so the heat equation becomes an independent temporal ordinary differential equation at each wavevector.

The photic zone is the illuminated layer supporting photosynthesis; its euphotic depth is conventionally associated with about 1% of surface photosynthetically active radiation, integrated over approximately $400$–$700\mathrm{nm}$. This is a useful convention rather than a universal organism-specific compensation depth. Spectral filtering can be represented locally bywhere $K_d$ is the wavelength-dependent diffuse attenuation coefficient. It includes effects of absorption and scattering and is not simply equal to the absorption coefficient $a$.

In clear, low-chlorophyll water, pure water removes red light strongly, while weak blue pigment and CDOM absorption permits blue light, typically around $450$–$490\mathrm{nm}$, to penetrate furthest. The deepest usable light is consequently dim and blue-rich, and the euphotic zone can extend relatively deep. At high pigment concentrations, phytoplankton pigments preferentially remove blue and red light; associated detritus or CDOM can remove still more blue. The remaining transmission window commonly shifts toward blue-green or green wavelengths, roughly $500$–$570\mathrm{nm}$, and the total light falls off faster, so the euphotic depth is shallower. Exact window and depth depend on the full particle/dissolved mixture, not chlorophyll alone. The diagram normalizes the two deep spectra separately to show their different shapes, not equal absolute light at a common depth.

This confines photosynthetic growth to a thinner illuminated interval in high-chlorophyll water and allows a deeper potential habitat in clear low-chlorophyll water, subject also to nutrient supply, mixing and the organism's compensation requirement. Cells near the base benefit from low-light acclimation and pigment systems that capture the remaining wavelengths. In blue-rich deep ocean water, blue-absorbing pigments and some low-light-adapted picoplankton can be favored; where the surviving light is greener, groups with effective green-absorbing accessory pigments, such as certain phycobilins, gain an advantage. A pigment appears coloured by light it fails to absorb, so green-absorbing accessory pigments need not themselves look green. There is no universal deep species determined by chlorophyll concentration alone. A deep chlorophyll maximum can partly reflect increased pigment per cell rather than a proportional maximum in carbon biomass.^[ref]

Chlorophyll a occurs in nearly all oxygen-producing phytoplankton groups and is central to their photosynthesis. Its concentration can be measured reproducibly by pigment extraction or calibrated optical methods, and its absorption affects ocean colour, making it observable over large areas. It is therefore a practical common index of photosynthetic standing stock across diverse communities, unlike a pigment limited to one group.

It is a proxy, not a fixed conversion to phytoplankton biomass or primary production. The carbon-to-chlorophyll ratio varies with species, light acclimation, nutrient supply and physiological condition; low-light cells often contain more pigment per unit carbon. Dead pigment and methodological differences can also affect interpretation, and nonphotosynthetic plankton are not represented. A pigment increase need not mean the same proportional increase in living carbon biomass.^[ref]

Many open-ocean chlorophyll retrieval algorithms infer chlorophyll a from the blue-to-green remote-sensing reflectance ratio. Increasing pigment generally depresses blue reflectance relative to green. In coastal waters, however, coloured dissolved organic matter and non-algal particles can independently absorb blue light, as in the sketches, and therefore mimic an increase in pigment. Their concentrations need not covary with phytoplankton biomass: river runoff and sediment resuspension can change them separately.

Moreover, the satellite measures emerging radiance, not absorption directly. Schematically,where $b_b$ is the backscattering coefficient; this is an explanatory approximation, not a complete retrieval formula. Suspended particles change both absorption and backscattering, and different mixtures can produce similar band ratios. Shallow-bottom reflection, land-adjacency effects and more difficult atmospheric correction introduce further coastal errors. Consequently a simple blue/green ratio can overestimate chlorophyll a when excess CDOM removes blue light, although sediment scattering and other effects mean the bias is not universally positive. Multiband or hyperspectral methods that separately constrain pigment, dissolved absorption and particles, supported by local observations, are preferable in optically complex water.^[ref]

The principal non-water visible-light absorbers can be grouped into phytoplankton pigments, coloured dissolved organic matter (CDOM), and non-algal particles (NAP), including organic detritus and absorbing mineral material. A useful additive decomposition of the absorption coefficient isThe supplied sketch separates the three non-water contributions; its amplitudes are arbitrary and are not concentrations or measured spectra.

Phytoplankton pigments give prominent absorption maxima around $440$ and $675\mathrm{nm}$, associated with chlorophyll a, and a relative minimum in the green. Accessory pigments, including carotenoids and phycobilins, broaden or add absorption bands, so an assemblage is not just a pure chlorophyll-a solution. Intracellular packaging and cell size also modify the shape and absorption per unit pigment.

Coloured dissolved organic matter consists of dissolved light-absorbing organic compounds, often including humic material from terrestrial runoff and marine degradation. Its absorption is strongest at the blue/ultraviolet end and declines approximately exponentially toward red, without the narrow chlorophyll red peak. A common local parameterization is $a_g(\lambda )=a_g({\lambda }_0)e^{-S_g(\lambda -{\lambda }_0)}$ with positive spectral slope $S_g$; here $S_g$ is an optical slope, not gas saturation.

Non-algal particles include nonliving organic detritus, pigment degradation products and absorbing suspended mineral particles. Their aggregate absorption often also decreases approximately exponentially from blue to red, with variable slope and mineral-dependent departures. Suspended sediments can strongly scatter as well as absorb; bright sediment-laden water is not evidence of a red absorption peak. Chlorophyll-containing living cells belong in $a_{\mathrm{ph}}$, so these categories should not be double counted.^[ref]

Healthy green vegetation has strong visible absorption in the blue, around $0.43$–$0.47\mu \mathrm{m}$, and red, around $0.65$–$0.68\mu \mathrm{m}$, mainly from chlorophylls and other photosynthetic pigments. Weaker absorption around green wavelengths produces a relative reflectance maximum near $0.55\mu \mathrm{m}$ and the familiar green appearance. The red edge is the steep reflectance rise around $0.68$–$0.75\mu \mathrm{m}$. In the near-infrared, weak pigment absorption and multiple scattering at internal leaf tissue–air interfaces produce high reflectance, commonly a broad plateau. This plateau is primarily a structural scattering effect, rather than chlorophyll reflecting the light.^[ref]

Leaf water produces absorption troughs in the shortwave infrared, especially near $1.4$ and $1.9\mu \mathrm{m}$, with weaker features near $0.97$ and $1.2\mu \mathrm{m}$. Atmospheric water vapor also strongly absorbs near $1.4$ and $1.9\mu \mathrm{m}$, so satellite bands usually avoid the centres of those deep troughs. Drying, senescence, pigment loss and changes in canopy structure alter the spectrum; measured canopy reflectance also includes shadows and the soil background.

To highlight vegetation, assign the near-infrared band to the red display channel, red to green and green to blue: vigorously vegetated areas appear bright red in this conventional false-colour composite. Alternatively calculate the normalized difference vegetation index from atmospherically corrected, spatially registered red and near-infrared reflectances:Large positive values indicate a strong vegetation-like red/NIR contrast. For example, ${\rho }_{\mathrm{NIR}}=0.50$ and ${\rho }_{\mathrm{red}}=0.05$ give about $0.82$. Mask clouds, their shadows and invalid low-signal pixels. Bare soil usually has a smaller contrast, while water often has negative values; fixed universal thresholds are inappropriate, and dense vegetation can saturate the index. A band ratio improves contrast but is not a direct count of plants or a unique measure of physiological health.^[ref]

For a surface-breaking reverse earthquake, expect a fault scarp or disrupted belt with the hanging wall displaced upward and commonly over the footwall. Small thrust faults, splays and backthrusts may repeat or override shallow deposits. Shortened soil, roads or other markers may buckle, while soft sediment can form pressure ridges and compression folds. Near the scarp, gravitational collapse and local bending can also produce tensile fissures and small normal faults; a reverse event need not create compression everywhere at the surface.

For a blind reverse rupture, expect coherent uplift and warping, newly tilted surfaces, or a fold scarp without a throughgoing primary fault offset. Small flexural-slip surfaces along bedding, secondary thrust faults or backthrusts, and opening cracks or small normal faults over the bent outer arc of the anticline can accompany the folding. Deformation may be distributed or localized at fold hinges. Thus a blind main fault does not imply an entirely unbroken ground surface, and secondary cracking must be distinguished from the principal rupture by its geometry and displacement pattern. Shaking-triggered landslides or soil liquefaction can accompany either case and should not automatically be mapped as primary tectonic rupture.^[ref]

Interferometric synthetic-aperture radar compares the phase of coherent radar returns acquired before and after the earthquake from closely repeated viewing geometry. After image registration and removal of orbital and topographic phase, the residual interferometric phase constrains relative ground displacement along the radar line of sight. Defining $u_{\mathrm{LOS}}$ positive toward the satellite and choosing the interferogram's phase order accordingly,One complete colour cycle therefore represents $\lambda /2$ of relative line-of-sight displacement. Reversing the image order reverses the displayed phase sign. This observes a projection of both horizontal and vertical motion, not uplift alone.^[ref]

If the main reverse rupture reaches the surface, coherent areas on opposite sides can show a displacement step: fringes become offset, truncated or abruptly juxtaposed along a trace. A narrow belt can lose interferometric coherence where cracking, intense gradients or changed scatterers prevent reliable phase comparison. If the rupture remains blind, a simple elastic model instead predicts a continuous surface displacement field, often a broad asymmetric lobe with continuous curved or closed fringes and a smooth displacement gradient across the projected tip.

These are diagnostic tendencies, not a unique test. An ordinary wrapped-phase boundary is just a $2\pi$ colour reset and is not itself a rupture. A step of an integer number of cycles can be hidden in wrapped phase; loss of interferometric coherence can hide either kind of near-field deformation. Vegetation, atmosphere and viewing direction also affect the pattern, and a displacement perpendicular to the radar line of sight is poorly sensed. Inspect coherence and unwrapped displacement, compare ascending and descending views where possible, and test a fault model against field observations. A sharp fold scarp above a blind rupture or a secondary shallow fault can also create localized gradients, so smooth fringes do not prove a buried tip and broken fringes alone do not prove a surface-breaking main rupture.