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A bidirectional reflectance distribution function, or BRDF, describes how a surface scatters incident illumination into each outgoing direction. At a fixed surface point $p$ and wavelength, its inputs are two directions above the surface, ${\omega }_i$ toward the incident source and ${\omega }_o$ toward the observer; equivalently these are two pairs of angular coordinates. Its output is the ratio of differential reflected radiance to differential incident irradiance:Position and wavelength may be additional arguments when material properties vary. A colour implementation can return one value per colour channel. The BRDF is a material response, not the final outgoing light value: it must be combined with the incident illumination and the projected-angle factor.^[ref]

For incident radiance $L_i$, the reflection part of the rendering equation isEmission is added separately if the surface itself emits. This integral explains how a BRDF replaces the assumed material response of a local lighting model. For Lambertian reflectance, $f_r={\rho }_d/\pi$, so the cosine factor yields the diffuse term from part (a). A directional specular lobe supplies a highlight. The empirical Phong reflection model is an approximate angular-response model of this kind; its original direct specular term is not automatically a normalized, energy-conserving BRDF. The ambient term approximates missing illumination and is not itself a third direction-independent material scattering mechanism. More general BRDFs can reproduce measured surface behaviour, while illumination from other surfaces supplies indirect light in the rendering equation.

Each electromagnetic normal mode is a quantum harmonic oscillator with $E_n=(n+1/2)h\nu$. For $x=h\nu /(k_{B}T)$ its canonical partition function and occupation probabilities areThe geometric distribution is normalized and gives thermal mode energy $h\nu /(e^x-1)$. The zero-point term $h\nu /2$ is not part of the thermally emitted black-body intensity. The equilibrium photon chemical potential is zero because photon number is not conserved.

Counting two transverse polarizations gives $8\pi {\nu }^2/c^3$ modes per volume per frequency interval. Thus the spectral energy density is $u_{\nu }=8\pi h{\nu }^3/[c^3(e^x-1)]$. In an isotropic radiation field, spectral radiance is $B_{\nu }=c{u}_{\nu }/(4\pi )$. Transforming the spectral measure, $B_{\lambda }=B_{\nu }|d\nu /d\lambda |$, givesHere intensity means radiance per wavelength and solid angle; the hemispheric spectral exitance of a black surface is $M_{\lambda }=\pi B_{\lambda }$. The extra Jacobian is essential: the wavelength peak is not found by simply converting the frequency peak using $\lambda =c/\nu$.

At short wavelength the exponential suppresses radiation. At long wavelength the Rayleigh-Jeans limit is $B_{\lambda }\simeq 2c{k}_{B}T/{\lambda }^4$; using classical equipartition at every frequency instead would produce the ultraviolet catastrophe. The wavelength maximum solves $5(1-e^{-x})=x$, with nonzero root $x=4.9651$, yielding Wien's displacement law. For the specified temperature,Integrating the hemispheric flux gives the Stefan-Boltzmann law, $M=\sigma T^4\simeq 3.01\times 10^{-6}\mathrm{W}{\mathrm{m}}^{-2}$. The plot uses the specified ideal temperature, rather than substituting a later measured value. The physical black-body description of the cosmic microwave background is tested by comparing the sky spectrum with an absolute reference blackbody and separating instrumental and Galactic contributions.^[ref]

Detector noise and graphical reading. Noise-equivalent power specifies the incident optical power producing signal-to-noise ratio one, with the electrical bandwidth and averaging convention stated. A spectral NEP has units $\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$; a band-integrated NEP has units W. Specific detectivity normalizes the latter by detector area and noise bandwidth:Its usual unit, the Jones, is $\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}{\mathrm{W}}^{-1}$. Removing the expected square-root area and bandwidth dependence helps compare different detector sizes and readout bandwidths, provided noise conditions and spectral response are comparable. This does not normalize away every background, temperature or implementation difference.

Reading the native embedded voltage plot gives approximate means $V_C\simeq 0.331\mathrm{V}$, $V_S\simeq 0.406\mathrm{V}$ and $V_H\simeq 0.493\mathrm{V}$. The one-second root mean square scatter is about $0.006\mathrm{V}$; the plotted peak-to-peak excursions are larger and must not be substituted for an root mean square noise. Independent graphical digitization of the blue trace gives means near $0.331,0.406,0.493$ V and scatters about $0.006$–$0.007$ V, consistent with this estimate. These are readings from a drawing, not exact original data.

The supplied Planck approximation gives $B_C\simeq 0.08590$ and $B_H\simeq 0.12836$ in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}(\mathrm{c}{\mathrm{m}}^{-1}{)}^{-1}$. Thus the noise-equivalent radiance of one one-second sample isHere the calibrating means are used to determine the response slope; their own uncertainty is included explicitly in the final scene estimate below.

Optical throughput and detectivity. A paraxial $f/F$ cone has half-angle approximately $1/(2F)$ and projected solid angle approximately $\pi /(4F^2)$. With $F=2$ and $A_d=10^{-6}{\mathrm{m}}^2$,Using the exact cone angle $\tan \theta =1/(2F)$ gives projected etendue $A_d\pi {\sin }^2\theta =1.85\times 10^{-7}{\mathrm{m}}^2\mathrm{sr}$, within the intended approximation. The radiance noise corresponds to detector power noiseThe wavenumber bandwidth is $1\mathrm{c}{\mathrm{m}}^{-1}$ and must match the radiance's per-wavenumber unit; it is not an electrical noise bandwidth. For the common order-of-magnitude assignment $\Delta f\sim 1\mathrm{Hz}$ to a one-second sample,Area in the last expression is $1\mathrm{m}{\mathrm{m}}^2=0.01\mathrm{c}{\mathrm{m}}^2$. For a specified one-second boxcar with white noise, the one-sided equivalent noise bandwidth is instead $1/(2t)=0.5\mathrm{Hz}$. That convention gives spectral NEP $\simeq 2.5\times 10^{-10}\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$ and $D_{\ast }\simeq 3.9\times 10^8$ Jones. The source does not specify the electrical filter, so the two conventions should not be mixed or assigned spurious precision.

Two-point retrieval and its uncertainty. Assume linear, stable response for the three views. The radiometer two-point calibration isWith target radiances treated as known, differentiating givesNotice $b<0$ for an intermediate scene: raising the hot voltage while keeping the scene voltage fixed reduces the inferred scene radiance. Numerically $a\simeq 0.2621$, $b\simeq -0.1214$ in radiance units per volt.

The scene interpolation fraction is about $0.463$, giving $B_S\simeq 0.1056$ andFor independent white noise in each view, each hundred-sample mean has standard error ${\sigma }_{\bar{V}}=0.006/10=0.0006\mathrm{V}$. The radiance error isUsing the given temperature sensitivity,A scatter estimate closer to $0.007$ V changes this to roughly $0.14$ K. These are random standard errors under the averaging assumptions. Target-temperature/emissivity uncertainty, response drift and atmospheric or surface-emissivity corrections are separate: the retrieved quantity is a brightness temperature, not automatically the true sea-surface temperature.

For the single-line greenhouse deficit, use hydrostatic balance, fixed surface pressure $p_s$, constant gravity $g$, and the dry adiabatThe gas constant and specific heat refer to the background gas. Let ${\mu }_d=\cos \tilde{\theta }>0$ be the supplied effective ray cosine. The vertical optical depth and slant transmittance from level $p$ to space areThis permits pressure-dependent line widths. With local thermodynamic equilibrium, negligible infrared scattering, a black thermal surface, and no incident thermal radiation from space, the formal radiative transfer solution isThe weights add to one. Subtract the transparent-surface radiance and put $B(p)=B_{{\nu }_0}(T(p))$, using the narrow-line approximation only in the line deficit. Define its pressure-dependent equivalent widthApproximating each hemispheric flux by $\pi$ times the effective-ray radiance preserves the black-surface flux. The outgoing longwave flux is thereforeIntegration by parts gives the particularly transparent formBoth boundary terms vanish: $E(0,q)=0$ and $B(p_s)-B(p_s)=0$. The dry adiabat and monotonic Planck function give $dB/dp>0$. Atmospheric emission replaces part of the warm surface radiation by emission from cooler levels, reducing emission to space for a specified $T_s$. The inequality is strict when opacity occurs over a nonzero range with temperature contrast; an isothermal absorbing layer at the surface temperature would not produce such a contrast.

For a spherical planet the mean absorbed instellation is $F_{\ast }=(1-\alpha )L_{⊛}/4$. The required global energy balance isConsequently every equilibrium of this model has $T_s\ge T_{\mathrm{bare}}=[(1-\alpha )L_{⊛}/(4{\sigma }_{\mathrm{SB}}){]}^{1/4}$, with strict warming when $D>0$. At fixed temperature profile, increasing $q$ increases $E$ and hence $D$. This argument does not assume that the greenhouse gas changes the stellar-band albedo, surface pressure, or background thermodynamics; those are held fixed. For sufficiently broad or overlapping saturated wings the full frequency-dependent Planck function must replace the narrow-line approximation.

Assume a plane-parallel atmosphere in local thermodynamic equilibrium, negligible infrared scattering, a black surface, and surface temperature equal to the adjacent air temperature. For a specified ray, let $\mathfrak{T}$ be its transmittance from a level to space, increasing from ${\mathfrak{T}}_0$ at the surface to one at the top. The formal radiative transfer solution isUsing ${\mathfrak{T}}_0+{\int }_{{\mathfrak{T}}_0}^{1}d\mathfrak{T}=1$ yieldsThis expresses absorption and re-emission together: opacity does not simply remove surface radiation without adding atmospheric radiation. An absorber close to a thermally coupled surface has nearly the same Planck function as that surface, so $B-B_0$ is small. Its emission nearly compensates the surface radiation it absorbs, giving weak radiance sensitivity to near-surface concentration. In an exactly isothermal atmosphere at surface temperature, $L=B_0$ for every optical depth, a concrete example of complete concentration ambiguity in this thermal measurement. Finite surface emissivity, reflection, scattering or a surface-air temperature discontinuity would require additional terms.

A hydrostatic pressure interval contains $dp/(gM)$ moles of background gas per area. Its absorbing column is $x(p)dp/(gM)$, so for a nadir rayThe assumed Planck-temperature power and dry adiabat give $B(p)/B_0=[T(p)/T_0{]}^{3.5}=p/p_0$. To first order in optical depth, $-d\mathfrak{T}\simeq d\tau$, and the radiance solution becomesFor a uniform mixing ratio this also evaluates to $L\simeq B_0[1-k_{0}x{p}_0/(6gM)]$. Its total optical depth is ${\tau }_0=k_{0}x{p}_0/(2gM)$, so the deficit is $B_0{\tau }_0/3$.

A pressure-resolved perturbation defines the infrared absorber sensitivity kernel by $\delta L=\int K_p(p)\delta x(p)dp$, withIt is zero at the top, negative in the interior, and zero again at the surface. The magnitude is largest where $d{K}_p/dp=0$, namelySign and coordinate qualification. Literally the maximum of the signed kernel is zero at the endpoints; $p_0/2$ is its minimum and the maximum of its absolute value. The derivative with respect to a single vertically uniform $x$ is the integrated constant $-k_0{B}_0{p}_0/(6gM)$ and has no pressure location. For equal-width logarithmic pressure perturbations the kernel is instead $K_{\log p}=p{K}_p$ and its magnitude peaks at $2p_0/3$. The stated pressure-interval convention is therefore essential.

The low top-of-atmosphere sensitivity reflects the small pressure-broadened absorption coefficient and small molecular column per $dp$ there; the low surface sensitivity reflects small thermal contrast. The printed phrase that temperature decreases with pressure reverses the actual stipulated relation: with positive exponent, temperature decreases as pressure decreases upward. The derivation uses the explicit adiabatic formula.

$I_{+}$ and $I_{-}$ are the positive upward and downward hemispherically integrated infrared fluxes, in $\mathrm{W}{\mathrm{m}}^{-2}$, rather than directional specific intensities. The source is the blackbody hemispheric flux $\sigma T^4$. Absorption removes a fraction proportional to the flux along a ray, and thermal emission replaces it according to Kirchhoff's law. Local thermodynamic equilibrium sets that emission source to the local Planck function.

A plane-parallel, non-scattering, gray atmosphere with a two-stream/angular closure reduces the angle- and frequency-dependent transfer equation to these two flux equations. The angular path factors are absorbed into the effective optical depth. Since the supplied coordinate increases upwards, the upward flux loses $I_{+}d\tau$ while gaining $\sigma T^{4}d\tau$ in moving upward; the downward ray traverses the coordinate in the opposite direction, reversing the differential signs in its equation. The equations therefore encode emission as well as absorption and are consistent with an isothermal source approaching thermal radiation in an optically thick medium. The source's use of “radiance” for these integrated quantities should not introduce an extra factor of $\pi$.

For a non-scattering atmosphere above a blackbody surface, the difference between the two curves is the transmitted surface term $B_{\lambda }(T_s){\mathcal{T}}_{\lambda ,s}$. The atmospheric-only curve is an integral of local thermal emission weighted by transmission to space. Molecular absorption blocks some wavelengths and permits narrow windows at others. In a transparent window, adding the hot surface substantially increases the emerging radiance; in an opaque interval, the two curves nearly coincide because the surface contribution is extinguished.

The original graph shows the clearest surface-dominated window around $0.99$–$1.04\mu \mathrm{m}$, with the blue curve near $5\mu \mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mu {\mathrm{m}}^{-1}$ at its maximum and the atmospheric-only contribution small. Weaker possibilities are around $0.85$–$0.92\mu \mathrm{m}$ and near $0.8\mu \mathrm{m}$, with less available signal. The $1.17$–$1.20\mu \mathrm{m}$ peak contains a measurable surface contribution but a larger atmospheric background. The near-equality around $1.27$–$1.29\mu \mathrm{m}$ makes that peak a poor surface thermometer despite its appreciable total radiance. These are approximate readings, not precisely tabulated band edges.

At $T_s\simeq 766\mathrm{K}$, the Wien displacement law puts the thermal wavelength peak near $3.8\mu \mathrm{m}$. A $1\mu \mathrm{m}$ window is still in the Wien approximation regime, but its radiance is exponentially larger than for an Earth-like $300\mathrm{K}$ or colder Martian surface. For example, at $1\mu \mathrm{m}$ the Planck function ratio $B_{\lambda }(766)/B_{\lambda }(300)$ is of order $10^{12}$. A modest hot-spot temperature increase likewise gives a relatively large fractional radiance change. Inverting this signal for surface temperature requires an atmospheric transmission and emission model, not just identifying a peak.

The etendue is the area–solid-angle acceptance of a ray bundle, more generally $\mathcal{G}=n_{mopt}^2\int \cos \theta dAd\Omega$. In an ideal lossless optical system it is conserved. Increasing pupil area increases accepted power only together with the appropriate field and detector cone; passive optics cannot arbitrarily increase radiance. For a small nadir pixel here, $A\Omega$ denotes entrance-pupil area times the solid angle of its surface footprint.

Choose $1.00$–$1.03\mu \mathrm{m}$ within the surface-sensitive window. A representative band-mean total radiance read from the original blue curve is ${\bar{L}}_{\lambda }\simeq 4\mu \mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mu {\mathrm{m}}^{-1}$, uncertain at roughly the tens-of-per-cent level. Its band integral isThe provided SNR formula depends on $L\Delta \nu$ only through this band integral. If $\nu$ is wavenumber, the selected band spans about $291\mathrm{c}{\mathrm{m}}^{-1}$ and $L_{\nu }d\nu =L_{\lambda }|d\lambda |$; numerical radiance densities in the two coordinates must not be mixed. Convert the specific detectivity to SI: $D^{\ast }=5\times 10^9\mathrm{m}\mathrm{H}{\mathrm{z}}^{1/2}{\mathrm{W}}^{-1}$ and $\sqrt{a}=2\times 10^{-4}\mathrm{m}$. Using the footprint dwell time from part (c),A square $200\mathrm{m}$ footprint at altitude $259\mathrm{km}$ subtends $\Omega \simeq (200/259000{)}^2=5.96\times 10^{-7}\mathrm{sr}$. Thus the collecting area and circular aperture diameter areThe Rayleigh criterion independently requires only $D\gtrsim 1.22\lambda /(200/h)\simeq 1.6\mathrm{mm}$ at $1.015\mu \mathrm{m}$, so the noise requirement dominates diffraction. The $200\mu \mathrm{m}$ detector size corresponds to focal length about $0.259\mathrm{m}$ for this pixel field. This assumes one detector footprint per stated feature size; stricter spatial sampling would need smaller footprints and a revised photon budget.

The estimate assumes a clear, non-scattering atmosphere as in the graph, a night-side or otherwise isolated thermal signal, negligible extra stray-light and read-noise penalties beyond the supplied detectivity, and transmission $0.7$ for the instrument alone. Atmospheric attenuation is already included in the plotted incoming radiance. The selected band has little atmospheric-only signal, but actual hot-spot detection concerns contrast relative to the background: SNR $50$ on total radiance is not a guarantee for an arbitrarily small temperature anomaly. An illustrative radiance range $3$–$5$ in the plotted units would give diameters about $0.20$–$0.15\mathrm{m}$.

A calibrated multispectral filter radiometer is a natural choice for rapid thermal mapping: a few selected bands provide higher photon throughput and faster imaging than spreading the light over many spectrometer channels. Several bands can help separate temperature, dust emissivity and atmospheric effects. A spectrometer would be justified if identifying composition or resolving spectral structure were a central additional objective; one broadband radiance alone cannot generally determine both temperature and unknown emissivity.

Thermal infrared radiation is preferable to reflected visible light for measuring temperature. The Rayleigh criterion setsA band near $10\mu \mathrm{m}$ is therefore a reasonable compromise between useful planetary thermal emission and the required diffraction-limited resolution. At that wavelength, the nominal surface Rayleigh criterion separation is about $415\mathrm{m}$; at $15\mu \mathrm{m}$ it is about $623\mathrm{m}$ and fails the target. The exact bands must avoid or model strong atmospheric absorption and account for the dust spectrum. The $500\mathrm{m}$ sampling pitch is still not a guarantee of faithful reconstruction of arbitrary $500\mathrm{m}$ features: detector sampling and the point spread function must be considered together.

Write ${\tau }_0={\tau }_{\infty }$, $n=2a$, and $t={\tau }_0(p/p_s{)}^2$. The integrated Planck function is $B(T)=\sigma T^4/\pi$ by the Stefan–Boltzmann law, andThe formal radiative transfer equation solution for upward radiance at the top isThe first term is transmitted surface emission; the second is emission from the atmosphere. Integrating radiance over the upward hemisphere gives the outgoing radiant flux per unit area,Here $E_j$ is an exponential integral. For fixed $a$ and ${\tau }_0\to \infty$, surface transmission is negligible and the upper limit can be extended to infinity. Positivity permits interchanging the integrals; putting $t=\mu x$ givesConsequently the exact hemispheric optically thick result isThis is a dimensional scale times a dimensionless definite integral depending only on $a$. In the commonly used vertical-ray approximation, define $F_{\mathrm{v}}=\pi I(0,1)$. ThenA vertical-ray approximation therefore has the same scaling but omits the angular factor $1/(a+1)$. Stating which quantity is used prevents a purely angular normalization discrepancy.

The research challenge is to retain causal control while measuring appearance under the coupled spectral, spatial and temporal variations of actual scenes. Simply replacing a flat patch with a photograph does not guarantee that either the stimulus or the task is ecologically informative.

A display's RGB values do not specify the spectrum reaching a participant's cone cells. Primary spectra, viewing geometry, calibration, luminance range and the observer's lens and pigment sensitivities matter. A photograph can clip highlights, alter white balance and compress dynamic range; a display cannot necessarily reproduce the original spectrum, brightness or angular light field. A researcher studying spectral constancy should measure radiance and illuminants or specify the limits of a display simulation, rather than treating an uncalibrated image as physical ground truth.

Real materials also vary in more than diffuse reflectance. Shape, roughness, specularity, translucency and surrounding illumination jointly determine an image. Changing one rendering parameter can unintentionally change highlight area, contrast and apparent shape. Conversely, holding an image statistic constant can produce an unnatural stimulus. Experiments need independent manipulation of candidate causes and a clear statement of which physical scene properties are represented by the apparatus.

James M. Kraft and David H. Brainard demonstrated a productive compromise with a chamber of actual surfaces. They independently controlled illumination and surface reflectances and used a projection colorimeter to let observers adjust a test patch to appear achromatic. Manipulations could hold candidate image statistics approximately fixed while changing the illumination. The results rejected explanations based solely on local adaptation, the image mean or the most intense region.^[ref] The achievement was not realism alone: it was enough realism combined with manipulations that made competing mechanisms disagree. These results do not show that local adaptation never contributes.

For material appearance, Roland W. Fleming and colleagues used rendered objects illuminated by natural illumination distributions and systematically altered those distributions. Reflectance matching was more reliable under realistic illumination statistics than under some altered statistics.^[ref] This makes illumination an experimental variable, not merely a nuisance background. It also warns that a model tested only under a small set of studio lights can exploit cues unavailable or misleading elsewhere.

The dependent variable needs equal care. Selecting the same paint, matching apparent color, adjusting an achromatic point, detecting an illuminant change and naming a material do not measure the same outcome. A person may correctly infer that two patches are the same material while still noticing an appearance difference. Instructions, response format, adaptation time and available context can alter the task. Eye and head movements change retinal exposure, so fixation constraints should be chosen deliberately and their cost to natural viewing acknowledged.

A strong program therefore combines calibrated simplified stimuli, controlled real scenes, validated rendering and tests on previously unseen objects and illumination. It should sample more than one observer and record relevant visual variation, report both accuracy and uncertainty, and test model generalization across scenes rather than only across random frames of the same scene. Proposed interventions can vary geometry while preserving spectral input, or alter illumination while suppressing a particular cue. Their outcomes must be measured, not assumed. The goal is interpretable generalization: experimental control sufficient to distinguish mechanisms and scene diversity sufficient to test whether those mechanisms remain useful.

A blackbody is an ideal body that absorbs all incident electromagnetic radiation, at every frequency and direction. Its absorptivity is one, so Kirchhoff's law of thermal radiation makes its equilibrium emissivity one as well. At a specified temperature its thermal spectrum is universal: it depends on temperature, not material or shape. This does not mean it looks black when hot; it can glow brightly while remaining a perfect absorber.

A small hole in a nearly isothermal opaque cavity approximates a blackbody, because radiation entering the hole undergoes many interactions before it could escape. In thermal equilibrium the cavity contains an approximately isotropic Planck spectrum. The emitted radiance follows Planck's law, its integrated outward radiant exitance follows the Stefan–Boltzmann law, and its wavelength peak follows Wien's displacement law. A real surface can have wavelength- and direction-dependent emissivity below this ideal limit.

Use calibrated multispectral imaging of reflected sunlight, with visible, near-infrared and short-wave-infrared bands. A satellite records radiance in several passbands for each ground pixel. Perform geometric registration and atmospheric correction to estimate spectral reflectance, then account for illumination differences on slopes and mask cloud, water and dense vegetation. A false-colour composite, diagnostic band ratios or a trained statistical classification can separate surface lithologies; field observations and representative spectra validate any mapped contact.

Different minerals have different absorption features: iron-bearing materials affect visible/near-infrared spectral reflectance, while hydroxyl-bearing clay minerals and carbonate-bearing rocks can differ in short-wave-infrared bands. Adjacent patches with reproducibly different spectra and coherent geological boundaries are candidates for lithological contacts. This is particularly useful in dry, exposed terrain. Vegetation, soil cover, similar rock compositions, weathering and topographic shadow can hide or mimic a contact; a colour boundary alone is insufficient.

Choose spatial resolution appreciably finer than the narrowest unit being mapped, with several pixels across that unit. Approximately $10$–$30\mathrm{m}$ is suitable for regional contacts separated by hundreds of metres; metre-scale units require finer imagery or field work. Sentinel-2 provides $10$–$20\mathrm{m}$ bands useful for this purpose, while the reflected-light multispectral bands of Landsat are typically $30\mathrm{m}$. Broad multispectral passbands distinguish many rock assemblages but cannot resolve every narrow mineral absorption feature; hyperspectral imaging is preferable when those features are essential.

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 projected area of a planar patch viewed from a direction is the area of its perpendicular shadow onto a plane normal to that direction. If $\mathbf{n}$ and $\mathbf{s}$ are its unit normal and viewing unit vector, then $A_{\perp }=A|\mathbf{n}\cdot \mathbf{s}|$ by the dot product. On the outward side, this is $A\cos \theta$. For a rectangle tilted around one edge, the width perpendicular to that edge contracts by $\cos \theta$ and the edge length stays fixed, proving the formula. At grazing view $A_{\perp }=0$; interpret the radiance definition by a limit rather than dividing by zero.

Radiance resolves both position and direction: it is radiant flux per projected area and per solid angle. For a surface element of area $dA$, and a small outgoing cone $d\Omega$ at angle $0\le \theta <\pi /2$ to its outward unit normal,Here $\mathbf{x}$ locates the patch and $\mathbf{s}$ is a unit vector along the outgoing direction. The factor $dA\cos \theta$ is the patch's apparent area viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive projected area for the crossing direction. ^[ref].

For a planar patch of area $A$ with spatially uniform, direction-independent radiance $L$, integrating over the patch givesTake $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ and $A=0.020{\mathrm{m}}^2$. The radiant intensities are $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and $I(60^{\circ })=0.10\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. A sufficiently narrow $0.010\mathrm{sr}$ cone therefore carries approximately $2.0\mathrm{mW}$ or $1.0\mathrm{mW}$, respectively. Both views have the same radiance $L$: the oblique view presents half the projected area. This is Lambert's cosine law, not a loss of radiance with viewing angle.

Radiance is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives radiant exitance; integrating incoming radiance at a receiver gives irradiance. In the stationary vacuum setting of vacuum conservation of specific intensity, a resolved uniform source retains its radiance along each light ray, while the radiant flux received from the whole source can decrease as its apparent solid angle shrinks.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

In geometrical optics, suppose radiation has a nonnegative radiance $L(\mathbf{x},\mathbf{s})$ whose directional contributions add as power. Here $\mathbf{s}$ is the actual propagation unit vector, $d\Omega$ is a solid angle element, and the collecting oriented surface $A$ has chosen unit normal $\mathbf{n}(\mathbf{x})$. Assume the following integrals exist. The outward and inward radiant fluxes areBoth quantities are nonnegative; their difference is the signed net radiant flux. The dot product supplies the cosine of the angle with the unit normal, hence the projected area. For incoming radiation the outward unit normal gives a negative dot product; the minus sign makes the collected power positive. Measuring the incoming angle from a source-facing unit normal instead gives the familiar positive $\cos \theta$. The units check is $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\times {\mathrm{m}}^2\times \mathrm{sr}=\mathrm{W}$.

Equivalently, the incoming surface integral is ${\Phi }_{-}={\int }_{A}EdA$ with local irradiance $E$; the emitted part of ${\Phi }_{+}$ is ${\int }_{A}MdA$ with radiant exitance $M$. For a small flat patch and a narrow accepted cone, nearly constant radiance and angle give $\Phi \simeq LA\cos \theta \Delta \Omega$. This approximation requires uniformity across the patch and cone. Direction-independent outgoing radiance over a whole hemisphere gives the different exact result ${\Phi }_{+}=\pi LA$, by Lambert's cosine law and the spherical coordinates calculation in radiant exitance. Replacing the cosine-weighted integral by the hemisphere's $2\pi$ solid angle would double the answer.^[ref]

These radiance formulas presume a description in additive directional power; they do not reconstruct arbitrary coherent wave interference from separate ray contributions. For general electromagnetic fields, the signed instantaneous power crossing an oriented surface is ${\int }_A\mathbf{S}\cdot \mathbf{n}dA$, using the Poynting vector $\mathbf{S}$; optical measurements commonly average over many wave periods. A zero net radiant flux can conceal equal nonzero incoming and outgoing radiant fluxes.