Thermographic Measurement Equation

• Thermographic measurement equation is a family of radiometric models that connect a target’s thermal state to infrared detector signals through self-emission, reflection, and atmospheric contributions.
• It provides both conceptual radiative balance and quantitative detector-level formulations to accurately relate optical-path geometry with measurable thermal flux.
• Practical applications include calibration techniques, dual- and multi-band thermography, and inversion models for determining parameters like emissivity, heat flux, and thermal conductivity.

The thermographic measurement equation denotes the class of radiometric and inversion relations that connect a target’s thermal state to the signal recorded by an infrared instrument. In the literature surveyed here, it appears both as a conceptual radiative balance and as a quantitative detector-level formulation. At its most general, it links self-emission, reflected ambient radiation, atmospheric or optical-path contributions, detector calibration, and scene–sensor geometry to measurable flux, radiance, intensity, or temperature-like observables. In applied settings, the same term also extends to inverse formulations that recover surface temperature, subsurface temperature, heat flux, emissivity, thermal conductivity, or related parameters from calibrated thermographic data (Sova et al., 15 Aug 2025).

1. Conceptual radiative balance

A widely used conceptual form writes the radiant flux reaching a detector pixel as the sum of attenuated object emission, attenuated reflected ambient radiation, and atmospheric emission: $$\Phi_{\text{total}}=\tau \cdot \left[ \varepsilon \,\Phi_{\text{obj}} + (1-\varepsilon)\Phi_{\text{ref}} \right] + (1-\tau)\Phi_{\text{atm}}.$$ Here, $\tau$ is the transmittance of the atmosphere or IR window, $\varepsilon$ is the emissivity of the surface, $\Phi_{\text{obj}}$ is the flux emitted by the object, $\Phi_{\text{ref}}$ is the reflected ambient flux, and $\Phi_{\text{atm}}$ is the atmospheric emission. This formulation identifies the three main radiative contributors in infrared thermography: self-emission, reflection, and atmospheric emission/attenuation (Sova et al., 15 Aug 2025).

A more elaborate image-formation model resolves additional optical stages. In one dual-band formulation, the radiation leaving the object is

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$

then, after the transmission path,

$$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$

and, after optics and narcissus-related terms,

$$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$

with the band-integrated flux

$$\Phi_{\text{tot}}=\int_{\lambda_{\min}}^{\lambda_{\max}}\Phi_C(\lambda)\,d\lambda.$$

In a flame-exposed two-color formulation, the same structure is expressed in spectral radiance form, with a surface term

$\tau$0

followed by atmospheric and filter transformations. These formulations make explicit that a thermographic measurement is not, in general, a direct temperature measurement; it is a radiative transport measurement whose interpretation depends on emissivity, reflectivity, transmission, and the optical train (Narayanan et al., 14 Sep 2025, Pelzmann et al., 2022).

2. Quantitative detector-level formulation

The transition from conceptual balance to quantitative measurement requires an explicit scene–detector coupling. One formulation introduces the optogeometric factor

$\tau$1

which, in the paraxial approximation, becomes

$\tau$2

A reduced form is defined as

$\tau$3

Its role is to convert a surface-exitance quantity in $\tau$4 into a pixel-level radiant flux in $\tau$5, thereby supplying the geometric factor absent from the purely conceptual balance (Sova et al., 15 Aug 2025).

In the simplest case, with atmosphere neglected, the quantitative thermography equation for a single pixel is

$\tau$6

Two practical paraxial approximations are also given: $\tau$7 where $\tau$8 is the entrance pupil diameter, $\tau$9 is the instantaneous field of view of a pixel, $\varepsilon$0 is the pixel pitch, and $\varepsilon$1 is the f-number. The same work emphasizes that the often-used temperature-based expression

$\varepsilon$2

is incomplete unless

$\varepsilon$3

is identified explicitly. A central implication is that quantitative thermography is not just a radiometric balance problem; it is also an imaging-geometry problem (Sova et al., 15 Aug 2025).

The same framework accommodates angular emissivity. For

$\varepsilon$4

the pixel flux becomes

$\varepsilon$5

This suggests that the thermographic measurement equation is naturally extensible to non-isotropic surfaces, provided the emissivity law and the geometric coupling are specified (Sova et al., 15 Aug 2025).

3. Calibration, linearization, and band-limited forms

In practical systems, the measurement equation is frequently expressed in calibrated signal coordinates rather than directly in flux or radiance. In lock-in thermography for the spin Peltier effect, the pixelwise first-harmonic temperature modulation is obtained from the first-harmonic infrared intensity modulation through

$\varepsilon$6

where the calibration factor is derived from steady-state $\varepsilon$7-versus-$\varepsilon$8 data. The same framework yields amplitude $\varepsilon$9 and phase $\Phi_{\text{obj}}$0 at the modulation frequency, and its stated validity condition is that surface emissivity is high and uniform, so reflection/transmission are negligible (Daimon et al., 2017).

A near-ambient dual-band formulation uses a radiometric mapping $\Phi_{\text{obj}}$1 from blackbody temperature to camera counts and writes

$\Phi_{\text{obj}}$2

Around ambient temperatures, the response is approximated as linear,

$\Phi_{\text{obj}}$3

which leads, for band $\Phi_{\text{obj}}$4, to

$\Phi_{\text{obj}}$5

For two bands,

$\Phi_{\text{obj}}$6

When $\Phi_{\text{obj}}$7 and $\Phi_{\text{obj}}$8 are known, the system admits closed-form recovery of $\Phi_{\text{obj}}$9 and $\Phi_{\text{ref}}$0; when they are unknown, temporal constraints on foreground and background variation are used to infer emissivities (Narayanan et al., 14 Sep 2025).

Two-color IR thermography expresses the same idea through in-band radiances. For filter bands $\Phi_{\text{ref}}$1 and $\Phi_{\text{ref}}$2,

$\Phi_{\text{ref}}$3

and, under the assumptions that the sample behaves as a gray body within each narrow band and that emissivity is the same in both bands, the ratio removes emissivity. Once $\Phi_{\text{ref}}$4 is obtained from the ratio, emissivity in a band is recovered as

$\Phi_{\text{ref}}$5

This ratio-based form is especially prominent when the optical configuration is calibrated radiometrically and the experiment is engineered to keep atmospheric and filter self-emission negligible (Pelzmann et al., 2022).

4. Semitransparent and subsurface formulations

For semitransparent media, a surface-emission equation is often inadequate. Infrared thermotransmittance instead assumes that, at a fixed IR wavelength, the transmitted infrared intensity varies linearly with temperature variation: $\Phi_{\text{ref}}$6 The measured camera signal is decomposed as

$\Phi_{\text{ref}}$7

where $\Phi_{\text{ref}}$8 is proper emission and $\Phi_{\text{ref}}$9 is transmitted illumination; chopping yields

$\Phi_{\text{atm}}$0

Using lock-in demodulation, the amplitude and phase of the thermotransmitted signal are retrieved, and a detection limit of $\Phi_{\text{atm}}$1 is estimated for the reported implementation. This formulation is explicitly presented as an alternative for semitransparent media, where classical thermography techniques based on proper material emission cannot be used (Bourges et al., 2022).

Depth thermography extends the measurement equation further by modeling thermal emission as a depth integral rather than a surface term. The experimentally recorded spectrum is

$\Phi_{\text{atm}}$2

and, in semitransparent media, the emitted spectrum is written in continuous form as

$\Phi_{\text{atm}}$3

or, in a layered discretization,

$\Phi_{\text{atm}}$4

The opaque-surface limit is recovered when

$\Phi_{\text{atm}}$5

This formulation makes the measurement equation an inverse spectral problem for $\Phi_{\text{atm}}$6, rather than a direct estimate of surface temperature (Xiao et al., 2019).

These semitransparent formulations differ from the surface-balance equation in a structural way. The unknown is not simply a single $\Phi_{\text{atm}}$7 or $\Phi_{\text{atm}}$8, but a temperature field that modulates transmission or depth-resolved emission. A plausible implication is that “thermographic measurement equation” names a broader family of optical-thermal forward models whose specific state variable depends on whether the medium is opaque, semitransparent, or spectrally depth-sensitive (Bourges et al., 2022, Xiao et al., 2019).

5. Thermographic inversion and parameter identification

In many applications, the thermographic measurement equation serves as the front end of an inverse heat-transfer model. In ST40, the infrared camera measures a divertor surface temperature field,

$\Phi_{\text{atm}}$9

which is mapped into CAD space and imposed as a Dirichlet boundary condition on a 2D tile model: $$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$0 The internal temperature is then evolved with

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$1

and the plasma-perpendicular surface heat flux density is recovered from Fourier’s law,

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$2

Here the thermographic measurement equation is inseparable from the inversion equation: temperature is measured, but heat flux is the target quantity (Moscheni et al., 2024).

Steady-state infrared thermography of suspended thin films provides another canonical inversion. For a circular hole geometry, homogeneous absorbed power density $$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$3, film thickness $$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$4, and in-plane conductivity $$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$5, the steady temperature field is

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$6

or, equivalently,

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$7

The curvature of the thermographic temperature map therefore yields

$$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$8

when $$\Phi_A(\lambda)=\epsilon_s(\lambda)\Phi_s(\lambda)+(1-\epsilon_s(\lambda))\Phi_b(\lambda),$$9 is fitted. In this case, the measurement equation is a geometry-specific steady-state solution of the heat equation (Greppmair et al., 2016).

A simpler measurement-and-model chain appears in the study of a heated metallic tube. The thermal image is converted to a temperature array through a linear color-to-temperature transformation,

$$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$0

and the extracted temperature field is interpreted with a stationary conduction–convection equation,

$$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$1

together with a cooling law for the average temperature,

$$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$2

This sequence shows a common pattern: image intensity is first mapped to temperature, and temperature is then embedded in a thermal transport model to infer physically meaningful coefficients (Jankových et al., 2019).

6. Assumptions, failure modes, and common misconceptions

A persistent misconception is that thermography reduces to emissivity multiplied by a blackbody law. The surveyed formulations repeatedly reject that simplification. Reflected ambient radiation, atmospheric terms, optics, and geometric coupling all enter the measured signal, and omission of the scene–detector coupling factor renders a pixel-level flux equation dimensionally and physically incomplete. The same point reappears in dual-band video thermography, where emitted and reflected/transmitted background components are explicitly separated only after calibration and model-based decomposition (Sova et al., 15 Aug 2025, Narayanan et al., 14 Sep 2025).

A second misconception is that two-color or dual-band methods are automatically emissivity-independent. The reported formulations require narrow adjacent bands, approximately equal emissivity in the two bands, careful radiometric calibration, and close correspondence between the two frames or channels. The reported error analysis at about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$3 states that camera noise per channel gives a temperature error of about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$4, parallax error contributes about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$5, camera calibration accuracy contributes about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$6, geometric correction contributes about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$7, emissivity assumption contributes about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$8, and frame-rate related error contributes about $$\Phi_B(\lambda)=\tau_t(\lambda)\Phi_A(\lambda)+(1-\tau_t(\lambda))\Phi_t(\lambda),$$9, for a total standard uncertainty of $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$0 and an expanded uncertainty $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$1 with $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$2 of $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$3. The same study further reports that, at $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$4, a 1% error in the FW#7 spectral band can cause $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$5 underestimation, a 3% error can cause $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$6 underestimation, and assuming emissivity $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$7 for a sample whose emissivity is $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$8 can underestimate temperature by about $$\Phi_C(\lambda)=\tau_o(\lambda)\Phi_B(\lambda)+\epsilon_o(\lambda)\Phi_o(\lambda)+r_o(\lambda)\Phi_i(\lambda),$$9 (Pelzmann et al., 2022).

A third misconception is that lock-in thermography returns an undifferentiated thermal response. In the reported spin Peltier measurements, the signal of interest is linear in current,

$$\Phi_{\text{tot}}=\int_{\lambda_{\min}}^{\lambda_{\max}}\Phi_C(\lambda)\,d\lambda.$$0

whereas Joule heating scales as

$$\Phi_{\text{tot}}=\int_{\lambda_{\min}}^{\lambda_{\max}}\Phi_C(\lambda)\,d\lambda.$$1

With a symmetric square-wave current and no DC offset, Joule heating does not appear in the first-harmonic lock-in response, while the spin Peltier contribution does. This separation is specific to the excitation symmetry and the harmonic chosen; it is not a generic property of all thermographic measurements (Daimon et al., 2017).

A fourth misconception is that the classical surface-emission equation is sufficient for semitransparent media. The thermotransmittance and depth-thermography formulations show otherwise. In one case, the measured quantity is a transmitted intensity perturbation proportional to temperature variation; in the other, it is a depth-integrated spectrum whose kernel depends on optical attenuation and local emissivity density. This suggests that the thermographic measurement equation is best understood not as a single immutable formula but as a family of forward models tailored to the radiative physics, the optical geometry, and the inversion target of a given experiment (Bourges et al., 2022, Xiao et al., 2019).