In (111), we note that the back transformed image gradient is also commonly used for implementing standard DIC, and following the state-of-the-art implementation techniques, it was shown in, e.g., [Leclerc et al., 2009, Neggers et al., 2017, Réthoré, 2010], that $\nabla \hat{g}$ can be approximated by $\nabla f$ such that it does not need to be recomputed at each iteration. For application of Dirichlet boundary conditions, the simplest approach is to use the correlated displacement $u^{\text{obs}}$ from a DIC algorithm, as commonly done in the literature [Leclerc et al., 2009, Neggers et al., 2017].

## B Construction of Observation Noise Matrices

In this section, we reuse notations introduced in appendix A, and discuss the expression of the precision matrix used in the likelihood expression (31). As explained in section 5.2, it largely depends on the algorithm used to perform the correlation. An interesting case is the FE-DIC approach, in which the correlation is performed on a finite element mesh instead of a structured grid. For details about FE-DIC, we refer to [Besnard et al., 2006]. In this case, the displacement precision matrix can be derived analytically. Considering that the reference image $f$ is corrupt by a Gaussian white noise with variance $2\sigma_f^2$, the noise analysis performed in [Besnard et al., 2006, Réthoré, 2010] shows the following expression for the displacement precision matrix

$$ \Sigma_u^{-1} = \frac{1}{2\sigma_f^2} N^T (\nabla f)^T \nabla f N \quad (112) $$

where $N$ is a sparse matrix of size $(2N) \times N_V$ containing the evaluation of FEM basis functions at pixel values. Note that considering the notations of appendix A, $N = \frac{dU}{du}$. For a specific choice of finite elements, the marginal variance of the displacement noise can be further expanded (see [Besnard et al., 2006] for the demonstration with quadrilateral elements). In the case of multi-camera setups, such as in stereo-DIC, the same noise sensitivity analysis can be performed. When using the FE-stereo-DIC correlation algorithm [Pierré et al., 2016], the derivation of the noise precision can be found in equation 12 of [Hild and Roux, 2020]. The case of I-DIC is simpler in a multi camera setup as the gray level noise remains spatially uncorrelated, and the precision matrix can be derived easily from the gray level uncertainty of each camera.

In some contexts, a closed form expression may not be available, for instance when the DIC algorithm is unknown (e.g. when using commercial software that hides implementation details). An empirical approach may be used to construct the precision matrix numerically. First, a set of $n$ correlated displacements $u_1, ..., u_n$ is obtained by correlating the reference image $f$ with $f + \xi$ where $\xi \sim N(0, 2\sigma_f^2)$. The covariance matrix can then be estimated, with

$$ \Sigma_u \approx \sum_{k=1}^{n} (u_k - u^0)(u_k - u^0)^T, \quad (113) $$

$$ u^0 = \frac{1}{n} \sum_{k=1}^{n} u_k. \quad (114) $$

The resulting covariance matrix is dense and of rank $r \le n$. If $n < N_V$, $\Sigma_u$ is not invertible and thus cannot be used to compute the precision matrix. A common technique is to employ a $l^1$-penalized maximum likelihood to obtain an estimator

$$ \hat{Q} = \underset{Q}{\operatorname{argmin}} \left( \operatorname{tr}(\Sigma_u Q) - \log \det Q + \gamma \sum_{i \neq j} S(|Q_{ij}|) \right), \quad (115) $$

where $\gamma$ is a regularization parameter that favours sparse solutions and $S$ a shrinkage function. For details, we refer to [Fan et al., 2015].

## References

* Hugo Leclerc, Jean-Noël Périé, Stephane Roux, and François Hild. Integrated digital image correlation for the identification of mechanical properties. volume 5496, pages 161–171, 05 2009. ISBN 978-3-642-01810-7. doi: 10.1007/978-3-642-01811-4_15.
In (111), we note that the back transformed image gradient is also commonly used for implementing standard DIC, and following the state-of-the-art implementation techniques, it was shown in, e.g., [Leclerc et al., 2009, Neggers et al., 2017, Réthoré, 2010], that $\nabla \hat{g}$ can be approximated by $\nabla f$ such that it does not need to be recomputed at each iteration. For application of Dirichlet boundary conditions, the simplest approach is to use the correlated displacement $u^{\text{obs}}$ from a DIC algorithm, as commonly done in the literature [Leclerc et al., 2009, Neggers et al., 2017].

## B Construction of Observation Noise Matrices

In this section, we reuse notations introduced in appendix A, and discuss the expression of the precision matrix used in the likelihood expression (31). As explained in section 5.2, it largely depends on the algorithm used to perform the correlation. An interesting case is the FE-DIC approach, in which the correlation is performed on a finite element mesh instead of a structured grid. For details about FE-DIC, we refer to [Besnard et al., 2006]. In this case, the displacement precision matrix can be derived analytically. Considering that the reference image $f$ is corrupt by a Gaussian white noise with variance $2\sigma_f^2$, the noise analysis performed in [Besnard et al., 2006, Réthoré, 2010] shows the following expression for the displacement precision matrix

$$ \Sigma_u^{-1} = \frac{1}{2\sigma_f^2} N^T (\nabla f)^T \nabla f N \quad (112) $$

where $N$ is a sparse matrix of size $(2N) \times N_V$ containing the evaluation of FEM basis functions at pixel values. Note that considering the notations of appendix A, $N = \frac{dU}{du}$. For a specific choice of finite elements, the marginal variance of the displacement noise can be further expanded (see [Besnard et al., 2006] for the demonstration with quadrilateral elements). In the case of multi-camera setups, such as in stereo-DIC, the same noise sensitivity analysis can be performed. When using the FE-stereo-DIC correlation algorithm [Pierré et al., 2016], the derivation of the noise precision can be found in equation 12 of [Hild and Roux, 2020]. The case of I-DIC is simpler in a multi camera setup as the gray level noise remains spatially uncorrelated, and the precision matrix can be derived easily from the gray level uncertainty of each camera.

In some contexts, a closed form expression may not be available, for instance when the DIC algorithm is unknown (e.g. when using commercial software that hides implementation details). An empirical approach may be used to construct the precision matrix numerically. First, a set of $n$ correlated displacements $u_1, ..., u_n$ is obtained by correlating the reference image $f$ with $f + \xi$ where $\xi \sim N(0, 2\sigma_f^2)$. The covariance matrix can then be estimated, with

$$ \Sigma_u \approx \sum_{k=1}^{n} (u_k - u^0)(u_k - u^0)^T, \quad (113) $$

$$ u^0 = \frac{1}{n} \sum_{k=1}^{n} u_k. \quad (114) $$

The resulting covariance matrix is dense and of rank $r \le n$. If $n < N_V$, $\Sigma_u$ is not invertible and thus cannot be used to compute the precision matrix. A common technique is to employ a $l^1$-penalized maximum likelihood to obtain an estimator

$$ \hat{Q} = \underset{Q}{\operatorname{argmin}} \left( \operatorname{tr}(\Sigma_u Q) - \log \det Q + \gamma \sum_{i \neq j} S(|Q_{ij}|) \right), \quad (115) $$

where $\gamma$ is a regularization parameter that favours sparse solutions and $S$ a shrinkage function. For details, we refer to [Fan et al., 2015].

## References

Hugo Leclerc, Jean-Noël Périé, Stephane Roux, and François Hild. Integrated digital image correlation for the identification of mechanical properties. volume 5496, pages 161–171, 05 2009. ISBN 978-3-642-01810-7. doi: 10.1007/978-3-642-01811-4_15.
Julien Réthoré. A fully integrated noise robust strategy for the identification of constitutive laws from digital images. *International Journal for Numerical Methods in Engineering*, 84:631-660, 2010. doi: 10.1002/nme.2908. URL https://hal.archives-ouvertes.fr/hal-00938286.

Stéphane Avril, Marc Bonnet, Anne Sophie Bretelle, Michel Grediac, François Hild, Patrick Ienny, Félix Latourte, Didier Lemosse, Stéphane Pagano, Emmanuel Pagnacco, and Fabrice Pierron. Overview of identification methods of mechanical parameters based on full-field measurements. *Experimental Mechanics*, 48(4):381-402, August 2008. doi: 10.1007/s11340-008-9148-y. URL https://hal.archives-ouvertes.fr/hal-00274639.

J. Neggers, F. Mathieu, F. Hild, S. Roux, and N. Swiergiel. Improving full-field identification using progressive model enrichments. *International Journal of Solids and Structures*, 118-119:213-223, 2017. ISSN 0020-7683. doi: https://doi.org/10.1016/j.ijsolstr.2017.03.013. URL https://www.sciencedirect.com/science/article/pii/S0020768317301191.

Luca M. Martulli, Leen Muyshondt, Martin Kerschbaum, Soraia Pimenta, Stepan V. Lomov, and Yentl Wolfs. Carbon fibre sheet moulding compounds with high in-mould flow: Linking morphology to tensile and compressive properties. *Composites Part A: Applied Science and Manufacturing*, 126:105600, 2019. ISSN 1359-835X. doi: https://doi.org/10.1016/j.compositesa.2019.105600. URL https://www.sciencedirect.com/science/article/pii/S1359835X19303495.

Marco Alves, Yizhuo Li, and Soraia Pimenta. Spatial variability and characteristic length-scales of strain fields in tow-based discontinuous composites: Characterisation and modelling. *Composites Part B: Engineering*, 262:110789, 2023. ISSN 1359-8368. doi: https://doi.org/10.1016/j.compositesb.2023.110789. URL https://www.sciencedirect.com/science/article/pii/S1359836823002925.

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Yoshimi Watanabe, Hisashi Sato, and Yasuyoshi Fukui. Wear properties of intermetallic compound reinforced functionally graded materials fabricated by centrifugal solid-particle and <i>in-situ</i> methods. *Journal of Solid Mechanics and Materials Engineering*, 2(7):842-853, 2008. doi: 10.1299/jmmp.2.842.

M. T. Nair. On morozov's discrepancy principle for nonlinear ill-posed equations. *Bulletin of the Australian Mathematical Society*, 79(2):337-342, 2009. doi: 10.1017/S0004972708001342.

Per Christian Hansen. Analysis of discrete ill-posed problems by means of the l-curve. *SIAM Review*, 34(4):561-580, 1992. doi: 10.1137/1034115. URL https://doi.org/10.1137/1034115.

Finn Lindgren, Håvard Rue, and Johan Lindström. An explicit link between gaussian fields and gaussian markov random fields: the stochastic partial differential equation approach. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 73(4):423-498, 2011. doi: https://doi.org/10.1111/j.1467-9868.2011.00777.x. URL https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2011.00777.x.

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Julien Réthoré. A fully integrated noise robust strategy for the identification of constitutive laws from digital images. *International Journal for Numerical Methods in Engineering*, 84:631-660, 2010. doi: 10.1002/nme.2908. URL https://hal.archives-ouvertes.fr/hal-00938286.

Stéphane Avril, Marc Bonnet, Anne Sophie Bretelle, Michel Grediac, François Hild, Patrick Ienny, Félix Latourte, Didier Lemosse, Stéphane Pagano, Emmanuel Pagnacco, and Fabrice Pierron. Overview of identification methods of mechanical parameters based on full-field measurements. *Experimental Mechanics*, 48(4):381-402, August 2008. doi: 10.1007/s11340-008-9148-y. URL https://hal.archives-ouvertes.fr/hal-00274639.

J. Neggers, F. Mathieu, F. Hild, S. Roux, and N. Swiergiel. Improving full-field identification using progressive model enrichments. *International Journal of Solids and Structures*, 118-119:213-223, 2017. ISSN 0020-7683. doi: https://doi.org/10.1016/j.ijsolstr.2017.03.013. URL https://www.sciencedirect.com/science/article/pii/S0020768317301191.

Luca M. Martulli, Leen Muyshondt, Martin Kerschbaum, Soraia Pimenta, Stepan V. Lomov, and Yentl Wolfs. Carbon fibre sheet moulding compounds with high in-mould flow: Linking morphology to tensile and compressive properties. *Composites Part A: Applied Science and Manufacturing*, 126:105600, 2019. ISSN 1359-835X. doi: https://doi.org/10.1016/j.compositesa.2019.105600. URL https://www.sciencedirect.com/science/article/pii/S1359835X19303495.

Marco Alves, Yizhuo Li, and Soraia Pimenta. Spatial variability and characteristic length-scales of strain fields in tow-based discontinuous composites: Characterisation and modelling. *Composites Part B: Engineering*, 262:110789, 2023. ISSN 1359-8368. doi: https://doi.org/10.1016/j.compositesb.2023.110789. URL https://www.sciencedirect.com/science/article/pii/S1359836823002925.

Sebastian Geyer, Iason Papaioannou, and Daniel Straub. Spatial modeling of concrete strength based on data. *Structural Safety*, 103:102345, 2023. ISSN 0167-4730. doi: https://doi.org/10.1016/j.strusafe.2023.102345. URL https://www.sciencedirect.com/science/article/pii/S0167473023000322.

Yoshimi Watanabe, Hisashi Sato, and Yasuyoshi Fukui. Wear properties of intermetallic compound reinforced functionally graded materials fabricated by centrifugal solid-particle and <i>in-situ</i> methods. *Journal of Solid Mechanics and Materials Engineering*, 2(7):842-853, 2008. doi: 10.1299/jmmp.2.842.

M. T. Nair. On morozov's discrepancy principle for nonlinear ill-posed equations. *Bulletin of the Australian Mathematical Society*, 79(2):337-342, 2009. doi: 10.1017/S0004972708001342.

Per Christian Hansen. Analysis of discrete ill-posed problems by means of the l-curve. *SIAM Review*, 34(4):561-580, 1992. doi: 10.1137/1034115. URL https://doi.org/10.1137/1034115.

Finn Lindgren, Håvard Rue, and Johan Lindström. An explicit link between gaussian fields and gaussian markov random fields: the stochastic partial differential equation approach. *Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 73(4):423-498, 2011. doi: https://doi.org/10.1111/j.1467-9868.2011.00777.x. URL https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2011.00777.x.

Michael Sutton, Jean-José Orteu, and Hubert Schreier. *Image Correlation for Shape, Motion and Deformation Measurements. Basic Concepts, Theory and Applications*. 01 2009. ISBN 978-0-387-78746-6. doi: 10.1007/978-0-387-78747-3.

Andre P. Ruybalid, Johan P. M. Hoefnagels, Olaf van der Sluis, and Marc G. D. Geers. Comparison of the identification performance of conventional fem updating and integrated dic. *International Journal for Numerical Methods in Engineering*, 106(4):298-320.

D. Thomas Seidl, Assad A. Oberai, and Paul E. Barbone. The coupled adjoint-state equation in forward and inverse linear elasticity: Incompressible plane stress. *Computer Methods in Applied Mechanics and Engineering*, 357:112588, 2019. ISSN 0045-7825. doi: https://doi.org/10.1016/j.cma.2019.112588. URL https://www.sciencedirect.com/science/article/pii/S0045782519304591.
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Lu Hai and Meng-Ze Lyu. Modeling tensile failure of concrete considering multivariate correlated random fields of material parameters. *Probabilistic Engineering Mechanics*, 74:103529, 2023. ISSN 0266-8920. doi: https://doi.org/10.1016/j(probengmech.2023.103529. URL https://www.sciencedirect.com/science/article/pii/S0266892023001182.

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