A buried heat source produces a surface pattern after heat travels through the surrounding structure. Depth, source size, power, interlayer thermal contact and surface cooling can all shape that pattern. Several combinations can reproduce a similar accessible temperature map.
An inverse calculation that returns one depth is therefore incomplete unless it shows that alternative depths and powers are inconsistent with the observations. A good fit is a necessary check on the model, not proof of a unique internal source.
Start with a forward comparison
Calculate or measure representative surface responses for known source positions and strengths using justified boundaries. Compare the spatial pattern and its time evolution. Retain the uncertainties in heat transfer, layer geometry and radiometric measurement. [1]
The purpose is to discover which observations distinguish the source variables. If changing depth and power together leaves the predicted data nearly unchanged, those variables are coupled in the inverse problem.
Compare more than one time
Early response, later spreading and the response after a controlled source change can contain different information. A single steady map may hide distinctions visible in a time trace. The source history must be known well enough to interpret those differences.
Compare more than one accessible region
Several views or mapped surface regions can add spatial information. Register them to a common geometry and account for differing cooling and emissivity. More pixels from the same uninformative view do not necessarily add independent information.
Compare candidate solutions explicitly
Retain combinations of depth, power and thermal parameters that reproduce the data within justified uncertainty. Report their range or joint relationship. Do not collapse the result to one favored solution solely because an optimizer returns it.
Verify the thermal boundary
Interlaminar contact and directional heat transfer can affect the source-to-surface path. A homogeneous block model may fit a broad pattern while assigning the wrong depth. Check its adequacy on representative known sources before using it to identify unknown layers.
Surface reflections, camera averaging and inaccessible regions also belong in the uncertainty. A predicted narrow hotspot cannot be compared directly with a pixel that averages a larger area without accounting for that observation process.
Ask which additional observation changes the decision
Use sensitivity studies to choose a new time interval, viewing region or independent electromagnetic measurement that separates plausible sources. The best next measurement is the one that reduces uncertainty relevant to the repair decision, rather than the one that merely produces a sharper image.
If the data identify a heated region but not its layer or source power, report that useful partial result. The investigation can then target the region with a complementary method. A surface thermal map supports internal localization only to the extent demonstrated by the forward comparisons and the surviving range of inverse solutions.
Use layer depth to choose informative observation times
With catalogue nominal thickness 0.23 mm [2], steel-only depths of 20 and 40 sheets are 4.60 and 9.20 mm. Assume a homogeneous medium with thermal diffusivity α = 1.0 × 10⁻⁶ m²/s solely for sensitivity planning. The diffusion time scale z²/α is 21.16 s for 4.60 mm and 84.64 s for 9.20 mm, a fourfold difference. These are characteristic scales, not first-arrival times or calibrated CRGO depth estimates. Collect observations across both time ranges and compare plausible source powers; a single steady image discards that discrimination. Include actual coatings, interlayer contact and boundary uncertainty before inverse localization [3].
Manufacturing controls. Layer spacing, coating, clamp pressure, source depth, cooling-duct placement and surface contact govern the directional thermal path. Nominal steel thickness excludes gaps; determine the real depth and effective cross-stack transport on representative assemblies.
References
- Deisenroth, D. C. et al., Measurement Uncertainty of Surface Temperature Distributions for Laser Powder Bed Fusion Processes, Journal of Research of NIST 126, 126013, 2021, sections 1.5 and 2.1; DOI: 10.6028/jres.126.013.
- Nippon Steel Corporation, Grain-Oriented Electrical Steel Sheets, catalogue D004je_03_202004, 2020, Specifications; Size and Form Tolerances.
- JCGM, JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, 2008.