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REVIEW 3 major objections 4 minor 60 references

Adjoint-based Recovery of Thermal Fields from Displacement or Strain Measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Using only displacement or strain measurements, a finite-element adjoint optimizer reconstructs the structure's temperature field, and the sensors do not need to be inside the heated region.

desk verdict A useful engineering application of standard adjoint inversion, but the paper has a serious unresolved discrepancy between the printed thermal-stress equations and the two tested element types, plus an inverse-crime validation. read the letter →

arxiv 2411.18118 v1 pith:5FA5LYNF submitted 2024-11-27 math.OC

Talhah Shamshad Ali Ansari , Rainald Löhner , Roland Wüchner , Harbir Antil , Suneth Warnakulasuriya , Ihar Antonau , Facundo Airaudo This is my paper · ORCID
classification math.OC MSC 49K2065K1074F0574S05
keywords thermalfieldreconstructionadjointmethodstructuralhealthmonitoringfiniteelementinverseproblemvertexmorphingdisplacementsensorskriging
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes and tests a way to recover the temperature distribution of a structure from ordinary deformation measurements instead of temperature readings. It sets up an optimization problem: find the nodal temperature field whose thermal forces, under known mechanical loads, reproduce the displacements or strains recorded by sensors. The gradient of the mismatch is computed with an adjoint solve, and the field is updated by steepest descent with a Barzilai-Borwein step, with Vertex Morphing used to smooth ill-conditioned updates. On three synthetic examples (a plate with a hole, a truss bridge, and a simplified dam), the reconstructed fields localize the heated region even when no sensor lies inside it, and the method lowers root-mean-square error by up to 38.4%, 94%, and 40% compared with k-nearest-neighbor and kriging interpolation using temperature sensors in the same locations.

What carries the argument

The central object is the Lagrangian functional $\mathcal{L}(u,\Delta T,\tilde{u}) = I(u,\Delta T) + \tilde{u}^t (f_{\mathrm{ext}} + f_{\Delta T} - K u)$, which couples the measurement mismatch $I$ to the finite-element equilibrium equations. The stationarity conditions produce the forward problem $K u = f_{\mathrm{ext}} + f_{\Delta T}$, the adjoint problem $K^t \tilde{u} = \frac{\partial I}{\partial u}$ (with $K^t = K$ for structural systems), and the gradient formula for the thermal field. Vertex Morphing acts as the regularizer: a kernel operator maps physical nodal temperatures through a control field, smoothing the gradient and suppressing high-frequency artifacts in the ill-posed inversion.

What would settle it

Run the same inversion on a physical plate-with-a-hole specimen where the heated patch is known and measured with thermocouples, but feed the optimizer only strain-gauge readings that include realistic noise and unknown small load variations; if the recovered peak temperature stays far below the true value, the heated region is misplaced, or the root-mean-square error no longer beats kriging, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the inverse problem of thermal field identification is tractable from displacement or strain data alone: given the load cases and the sensor readings, minimizing the weighted mismatch between measured and computed displacements yields a spatial temperature distribution that approximates the target field. The derivation shows that the gradient of the cost function with respect to the nodal temperature changes requires only the forward solve and an adjoint solve with the same stiffness matrix, since the stiffness matrix is symmetric for these structural systems, and the gradient takes the form $\nabla I(\Delta T) = \frac{\partial I}{\partial \Delta T} + \tilde{u}^t \frac{\partial f_{\Delta T}}{\partial \Delta T}$.

Load-bearing premise

The demonstrations assume the displacement data come from the same noiseless linear finite-element model used in the inversion, with the loads, boundary conditions, and material properties known exactly.

Editorial extensions

If this is right

  • Thermal fields can be reconstructed by deformation sensors alone, reducing reliance on temperature sensors in inaccessible or non-uniformly heated regions.
  • Sensor placement does not need to intersect the heated region; it only needs to be sensitive to the deformation that region produces.
  • Increasing sensor count and improving sensor placement sharpens both the magnitude and the localization of the recovered temperature field.
  • Vertex Morphing filtering consistently lowers reconstruction error, with root-mean-square error reductions of up to 16%, 24.4%, and 18% across the three examples.
  • At equal sensor counts, the proposed physics-based approach beats k-nearest-neighbor and kriging interpolation, with root-mean-square error reductions of up to 38.4%, 94%, and 40%.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the method's success likely depends on the accuracy of the finite-element model and on the assumption that the measured deformation is purely thermal-mechanical; realistic sensor noise, load uncertainty, and model error are listed by the authors as future work and would probably degrade the reconstruction.
  • Inference: combining displacement or strain sensors with temperature sensors in the same cost function could fill the low-sensitivity 'blind spots' the paper observes near corners and edges.
  • Inference: since thermal loads enter the equilibrium equations as equivalent nodal forces through $f_{\Delta T}$, the same adjoint machinery could be redirected to identify other distributed quantities, such as prestress or spatially varying material properties.
  • Inference: extending the cost function over time would allow recovery of transient thermal fields, at the cost of solving many more forward and adjoint problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a finite-element adjoint-based optimization procedure to reconstruct spatial temperature fields from displacement or strain measurements under known loads. The method models thermal strains as equivalent nodal loads, derives the adjoint gradient of a weighted least-squares misfit, and regularizes the ill-posed nodal temperature update with Vertex Morphing filtering. Three numerical examples are presented: a plate with a hole modeled with plane-stress triangles, a truss bridge, and a simplified Hoover Dam modeled with tetrahedra. In each case, a target temperature field is prescribed, displacement/strain data are generated synthetically, and the optimizer attempts to recover the field from sensor measurements. The results are compared with k-nearest-neighbor, ordinary kriging, and universal kriging interpolations using temperature sensors in identical configurations, with reported RMSE reductions up to 38.4%, 94%, and 40% for the three examples.

Significance. If the claims are borne out, the method would be a useful addition to structural health monitoring and digital twin workflows, since it would allow thermal fields to be reconstructed from deformation sensors alone, including scenarios where no temperature sensor is located in the heated region. The paper is clearly written and the optimization machinery is standard; the comparison with spatial interpolation methods is informative, and the authors are explicit about many limitations (e.g., noise, parameter uncertainty, sensor blind spots) in Section 5. The main concerns are that the printed thermal-load equations do not match the element types tested in Examples 1 and 2, and that the numerical validation uses an inverse-crime setup with noiseless synthetic data, so the stated claim of 'accurate reconstruction' is stronger than the evidence supports. These issues are fixable and do not invalidate the underlying idea.

major comments (3)
  1. [Section 3.1, Eqs. (3) and (7)] Equation (3) gives sigma_ij = -alpha E/(1-2nu) Delta T delta_ij, which is the triaxially constrained 3D thermal stress, after the text introduces the equation as the thermal stress for a 'one-dimensional rod'. The uniaxial rod stress would be -alpha E Delta T. The same coefficient is carried into the finite-element thermal nodal load in Eq. (7). This is inconsistent with the examples: Section 4.1 uses plane-stress triangular elements, where the correct thermal nodal-load coefficient is alpha E/(1-nu), and Section 4.2 uses truss elements, where the axial coefficient is alpha E. Only the tetrahedral dam example of Section 4.3 matches the printed E/(1-2nu) coefficient. Since the paper describes an 'FE-method-dependent' procedure and all simulations were run in Kratos, Eqs. (3)-(7) should describe the implemented model; as printed, they do not, so a reader cannot reproduce the reported reconstructions from the paper. Please correct the equations or state clearly how the thermal-load coefficient is adapted for each element formulation.
  2. [Section 4, opening; Section 5] All 'measured' data are generated by the same noiseless linear finite-element model that is used for the inversion, with perfectly known external loads, boundary conditions, and material properties. This is an inverse-crime setup. The abstract's conclusion that the approach 'can accurately reconstruct the target thermal distribution' is therefore not established for realistic measurements containing sensor noise, load uncertainty, or model error; these effects are listed as future work in Section 5. Please either add numerical experiments with additive sensor noise, perturbed material/load parameters, or a mismatched mesh/model, or qualify the central claim so that it is limited to the noiseless synthetic setting.
  3. [Section 4.4.3, Tables 1-3] The headline RMSE reductions (up to 38.4%, 94%, and 40%) compare the proposed displacement-based adjoint method with interpolation methods that use temperature sensors only. In configurations where no temperature sensor detects the heated region (e.g., the 6-sensor plate case in Figure 21a), the interpolation baseline returns essentially Delta T = 0 by construction, so large percentage improvements are expected and do not indicate superiority in a like-for-like information setup. The comparison is still meaningful for the claimed 'sensors not in the region of interest' scenario, but the abstract and Section 4.4.3 should state this asymmetry explicitly rather than presenting the percentages as a general accuracy comparison between reconstruction methods.
minor comments (4)
  1. [Section 3.2, Eq. (8)] The strain misfit term in Eq. (8) uses the notation ums_ij, but the text defines the measured strain as sms_ij; please make the notation consistent.
  2. [After Eq. (12)] The symbol Js_ij is introduced to denote the displacement-strain relationship, but the cost function in Eq. (8) uses Is_ij; please align or define both symbols.
  3. [Section 4.1] The Young's modulus for the plate is given as E = 2e12 Pa, which is an order of magnitude above typical structural steel; if this value is intentional, please state so, otherwise it appears to be a typo.
  4. [Tables 1-3] Several table captions refer to a single figure number (e.g., Table 2 refers to 'Figure 8') when the target distribution is only the bottom half of the displayed figure; please make the cross-references more precise.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the adjoint recovery of the thermal field is validated against externally prescribed target distributions, and the self-citations present are not load-bearing.

full rationale

The central derivation chain is not circular. The forward problem (Eqs. 7 and 9a) maps a nodal temperature field to displacements through the thermal load vector and stiffness matrix; the cost function (Eq. 8) compares computed displacements or strains with measured values; and the adjoint gradient (Eqs. 13-15) is derived explicitly from the Lagrangian (Eq. 11) within the paper itself, so no step of the derivation is defined in terms of the predicted quantity. The validation is against externally prescribed target fields (Section 4), with RMSE tables (Section 4.4.3) computed against those targets; no parameter is fitted to the target to make the reconstruction succeed, since the Barzilai-Borwein step (Eq. 16) and the Vertex Morphing radius are chosen a priori and results are reported both with and without filtering (e.g., Figures 5, 11, 17). Self-citations (Antil et al. 2018 for the Lagrangian functional; Antonau 2023 for the BB method; Antonau et al. 2022 and Hojjat et al. 2014 for Vertex Morphing; Airaudo, Lohner, and Warnakulasuriya et al. for the analogous weakness-identification approach) are present but not load-bearing: the Lagrangian derivation is reproduced in full, the BB update is stated explicitly, and Vertex Morphing is shown to be a smoothing convenience rather than a requirement for reconstruction. Two genuine caveats belong to correctness and validity risk rather than circularity. First, the displacement measurements are synthetic, generated by the same finite-element model used for the inversion, which is an inverse-crime consistency test with noiseless data; the paper itself lists sensor noise, parameter uncertainty, and time lag as open future-work items in Section 5, and that limitation is weighed here as a modeling caveat, not as a logical reduction of the prediction to its inputs. Second, the printed thermal-stress coefficient alpha*E/(1-2*nu) in Eqs. (3) and (7) matches the 3D tetrahedral dam example but is inconsistent with the plane-stress plate (alpha*E/(1-nu)) and truss bridge (alpha*E) elements used in Sections 4.1 and 4.2, which is a reproducibility gap between the stated equations and the tested Kratos models; again, this concerns whether the paper describes the model it ran, not whether the recovered fields reduce to their inputs.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard linear thermoelastic FEM, perfectly known loads and material data, and a hand-chosen smoothness prior from Vertex Morphing. There are no invented physical entities. The main unverified inputs are the model fidelity and the regularization choices, both of which are ad hoc to the numerical demonstration.

free parameters (6)
  • Vertex Morphing filter radius r = 5 (Plate), 6 (Bridge), 50 (Dam)
    Chosen by rule of thumb as 2-3 element side lengths (Sections 4.1-4.3). It controls the smoothness of the recovered field and changes RMSE by up to 24.4% when enabled, so the central demonstration depends on this hand-chosen regularization scale.
  • Maximum gradient-descent stepsize = 2.5e-3 (Plate), 2.5e-2 (Bridge), 1e-1 (Dam)
    Set to prevent huge updates (Sections 4.1-4.3); affects the convergence trajectory, though the constant-step comparison gave nearly identical final fields.
  • Convergence criterion = 1e-5 times initial cost function
    Stopping after 5 orders of magnitude cost reduction determines the final RMSE and iteration counts; no physical basis is given.
  • Cost function weights w_md and w_ms = unspecified (assumed equal)
    Eq. (8) defines weighted displacement and strain terms, but the actual weights are never stated in the paper; the recovered field depends on this choice.
  • kNN hyperparameters = k=3 (Plate/Bridge), k=5 (Dam), inverse distance weighting
    Used for the baseline interpolation comparison; chosen without systematic tuning.
  • Kriging variogram and trend = Gaussian variogram, regional linear trend for universal kriging
    Baseline interpolation settings; different variograms can change interpolation results.
assumptions (4)
  • domain assumption Linear isotropic thermoelasticity with constant Young's modulus (Eqs. 2-5, Section 3.1).
    The thermal force depends on alpha, E, nu, and Delta T; E is assumed temperature-independent because temperature differences are small (Section 3.2).
  • domain assumption The finite element model is an exact representation of the physical structure, including known external loads and boundary conditions (Eq. 9, Section 4).
    All measurements are generated by simulating this model; no model error or load uncertainty is considered.
  • standard math The reduced optimization problem is convex enough for steepest descent to reach a meaningful minimum (Section 3.3).
    With constant K and linear temperature-to-force mapping, the cost is quadratic in Delta T, but the paper does not analyze conditioning or non-uniqueness; the Vertex Morphing filter is introduced to handle ill-conditioning.
  • ad hoc to paper Smoothness imposed by Vertex Morphing is an appropriate prior for the unknown temperature field (Section 3.3.1).
    The filter radius and sigmoid projection are heuristic regularization choices; they can suppress sharp thermal fronts and bias the solution.

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Cite this review

Pith. "Pith review of Adjoint-based Recovery of Thermal Fields from Displacement or Strain Measurements." pith.science (2026). https://pith.science/paper/5FA5LYNF

@misc{pith2026241118118,
  author       = {Pith},
  title        = {Pith review of: Adjoint-based Recovery of Thermal Fields from Displacement or Strain Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FA5LYNF}},
  note         = {Machine review of arXiv:2411.18118}
}
read the original abstract

A finite-element method dependant adjoint-based procedure to determine the temperature field of structures based on measured displacements or strains and a set of standard loads is developed and tested. Given a series of force and deformation measurements, the temperature field is obtained by minimizing the adequately weighted differences between the measured and computed values. Three numerical examples - a Plate With a Hole, a Bridge, and a Hoover Dam example - each with multiple sensors distributed in different configurations, demonstrate the procedure's capabilities. A target temperature distribution is prescribed in all cases, and the displacement sensor data is recorded. The optimization algorithm (here, steepest descent with Barzilai-Borwein step) uses this data to optimize the temperatures such that the same deformation is obtained at the sensor locations. Vertex Morphing is used as a filter to mitigate the ill-conditioning. Results show that the proposed approach can accurately reconstruct the target thermal distribution, especially when more sensors are used. Additionally, it is observed that the sensors do not need to be positioned in the region of interest; the method remains effective as long as the sensors can detect changes related to that area. A comparison with standard spatial interpolation techniques, namely, k-nearest neighbors and ordinary and universal kriging, is performed using temperature sensors in the same configurations. The proposed approach performs remarkably better than the interpolation techniques with a reduction in the root-mean-squared error of up to 38.4%, 94%, and 40%, for the Plate With a Hole, the Bridge, and the Dam examples, respectively.

Figures

Figures reproduced from arXiv: 2411.18118 by the authors.

Figure 1
Figure 1. Temperature variation with depth according to the Waka Kotahi NZ Transport Agency (2013). From the Waka Kotahi NZ Transport Agency Bridge Manual SP/M/022, Third Edition, Amendment 4. Used with permission. 3/40 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Vertical thermal design gradient according to the American Association of State Highway and Transportation Officials (2020). From AASHTO LRFD Bridge Design Specifications, 2020, published by the American Association of State Highway and Transportation Officials, Washington, D.C., USA. Used with permission. distribution on the surface due to shadow and irregular irradiation. Santillán et al. (2015) introduced a 1-D a… view at source ↗
Figure 3
Figure 3. Plate With a Hole: Mesh used (left) and the target temperature distribution (right). in a haphazard manner, resulting in a rough surface or temperature field with local and sharply changing temperatures. To remedy this and obtain smooth design updates, Vertex Morphing works as a smoothing operator that performs a convolution in a specified radius, thereby resulting in smoother results. An in-depth explanation of Ver… view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: Plate With a Hole: 6 (left) and 14 (right) sensors configurations and the temperature distribution at optimization start ∆T = 0. 10/40 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Plate With a Hole, 6 sensors configuration: Temperature distribution obtained without and with Vertex Morphing (VM) filtering at different iterations and the cost function convergence plots. in each iteration. The optimization starts with the presumption that the ∆T = …
Figure 6
Figure 6. Figure 6: Plate With a Hole, 14 sensors configuration: Temperature distribution obtained without and with Vertex Morphing (VM) filtering at different iterations and the cost function convergence plots. The ’with’ Vertex Morphing case is noted to require a higher number of iterat…
Figure 7
Figure 7. Figure 7: Plate With a Hole, 6 and 14 sensors configurations: Temperature distribution obtained with Vertex Morphing (radius = 5) filtering using steepest descent and constant step of 1 · 10−4 at final iterations and the cost function convergence plots. Figures 5c,5f, is due to …
Figure 8
Figure 8. Figure 8: Bridge example: Mesh used (top) and the target temperature distribution (bottom). 14/40 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Bridge example: Sensors distribution for the 8 (top) and 20 (bottom) sensors configurations. obtained, especially in reducing the sharp peak temperatures. The temperature distribution in the case without Vertex Morphing is in the range [−0.54,10.24] ◦C, whereas it is i…
Figure 10
Figure 10. Figure 10: Bridge example: Temperature Distribution at Optimization Start ∆T = 0. 15/40 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Bridge example, 8 sensors configuration: Temperature distributions (left) and the cost function convergence plots (right) obtained without and with Vertex Morphing filtering. The heated region is identified and localized fairly quickly within 20% of the total iteratio…
Figure 12
Figure 12. Figure 12: Bridge example, 20 sensors configuration: Temperature distributions (left) and the cost function convergence plots (right) obtained without and with Vertex Morphing filtering. 17/40 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Bridge example, 8 and 20 sensors configurations: Intermediate temperature distributions obtained at (approx.) 20% of the optimization progress i.e., at Iteration = 20% * (Iterations to converge) with Vertex Morphing (radius = 6) filtering. 18/40 [PITH_FULL_IMAGE:figu…
Figure 14
Figure 14. Figure 14: Dam example: Mesh used (left), the target temperature distribution (middle), and a threshold distribution of ∆T > 2 ◦C (right). 4.3 Simplified Hoover Dam A simplified Finite-Element model of the Hoover dam is shown in [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Dam example: Sensors distribution for the 27, 36, and 59 sensors configuration (left to right). 19/40 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Dam example: Temperature distribution at optimization start ∆T = 0. distributions shown in this paper display an element if at least one of its nodes satisfies the threshold condition. Due to this, the presented threshold distributions are not an exact representation …
Figure 17
Figure 17. Figure 17: Dam example, 27 sensors configuration: Temperature distributions obtained (leftmost) along with the ∆T > 2 ◦C thresholds, the ∆T < −1 ◦C thresholds, and the cost function convergence plots (right) without and with Vertex Morphing filtering. 21/40 [PITH_FULL_IMAGE:fig…
Figure 18
Figure 18. Figure 18: Dam example, 36 sensors configuration: Temperature distributions obtained (leftmost) along with the ∆T > 2 ◦C thresholds, the ∆T < −1 ◦C thresholds, and the cost function convergence plots (right) without and with Vertex Morphing filtering. 22/40 [PITH_FULL_IMAGE:fig…
Figure 19
Figure 19. Figure 19: Dam example, 59 sensors configuration: Temperature distributions obtained (leftmost) along with the ∆T > 2 ◦C thresholds, the ∆T < −1 ◦C thresholds, and the cost function convergence plots (right) without and with Vertex Morphing filtering. 23/40 [PITH_FULL_IMAGE:fig…
Figure 20
Figure 20. Figure 20: Dam example, 27, 36, and 59 sensors configurations: Intermediate temperature distributions obtained at (approx.) 20% of the optimization progress i.e., at Iteration = 20% * (Iterations to converge) with Vertex Morphing filtering. 24/40 [PITH_FULL_IMAGE:figures/full_f…
Figure 21
Figure 21. Figure 21: Plate With a Hole, 6 and 14 temperature sensors configurations: k-Nearest Neighbor interpolation (kNN) with k = 3 and inverse distance weighting. (a) kNN: 8 temperature sensors configuration (b) kNN: 20 temperature sensors configuration [PITH_FULL_IMAGE:figures/full_…
Figure 22
Figure 22. Figure 22: Bridge example, 8 and 20 temperature sensors configurations: k-Nearest Neighbor interpolation (kNN) with k = 3 and inverse distance weighting. 25/40 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: Dam example with 27, 36, and 59 temperature sensors configurations: k-Nearest Neighbor interpolation (kNN) with k = 5 and inverse distance weighting: Interpolated temperature distributions (left) along with the ∆T > 2 ◦C thresholds (middle), and the ∆T < −1 ◦C thresho…
Figure 24
Figure 24. Figure 24: Plate With a Hole, 6 and 14 temperature sensors configurations: Ordinary kriging (OK) interpolation with Gaussian variogram. 27/40 [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 25
Figure 25. Figure 25: Plate With a Hole, 6 and 14 temperature sensors configurations: Universal kriging (UK) interpolation with Gaussian variogram and ’regional linear’ trend. same prescribed temperature distribution specified in Section 4.1 (shown in [PITH_FULL_IMAGE:figures/full_fig_p02…
Figure 26
Figure 26. Figure 26: Bridge example, 8 and 20 temperature sensors configurations: Ordinary kriging (OK) interpolation with Gaussian variogram. 28/40 [PITH_FULL_IMAGE:figures/full_fig_p028_26.png]
Figure 27
Figure 27. Figure 27: Bridge example, 8 and 20 temperature sensors configurations: Universal kriging (UK) interpolation with Gaussian variogram and ’regional linear’ trend. ordinary kriging and universal kriging interpolated temperature distributions, respectively. For both, ordinary and u…
Figure 28
Figure 28. Figure 28: Dam example with 27, 36, and 59 temperature sensors configurations: Ordinary kriging (OK) interpolation with Gaussian variogram: Interpolated temperature distributions (left) along with the ∆T > 2 ◦C thresholds (middle), and the ∆T < −1 ◦C thresholds (right). 30/40 […
Figure 29
Figure 29. Figure 29: Dam example with 27, 36, and 59 temperature sensors configurations: Universal kriging (UK) interpolation with Gaussian variogram and ’regional linear’ trend: Interpolated temperature distributions (left) along with the ∆T > 2 ◦C thresholds (middle), and the ∆T < −1 ◦C…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.