## 3 A Post-Processing Algorithm to Enforce Traction Continuity along the Interface
### Inverting for the Polynomial Coefficients
\right]A_{2,[n]^{\prime}}^{+},\,x_{2}=0.\]  
Similarly, the condition for continuity of shear stresses is enforced by Different methods can be used to solve the system of equations in Equation (12) together with the constraints in Equations (14) and (15) for the polynomial coefficients; here we use Matlab (The MathWorks, Natick, MA, USA) function _lsqlin_. Once the coefficients are obtained, the displacements at the pixels of the central column (\(n_{x2}\) pixels above and \(n_{x2}\) pixels below the interface) are replaced by those predicted by the corresponding polynomials, and the procedure is repeated for all columns and images. Note that the system of equations to be solved is small because the local polynomials should represent the local variations in displacements and stresses well. Thus, the computation time was small, and the modification of all pixel columns within an image was done within 2-3 s for the images analyzed here (400 pixels along the interface). Figure 4 shows the interface-parallel (a) and normal (b) displacements at the center of the field of view (\(j=190\)) for the rupture of Figure 1. Both the displacements obtained by Vic-2D (with the \"fill boundary\" algorithm) and displacements modified by TC enforcement are shown. The maximum difference in displacements between the two solutions was smaller than 0.001 pixels, which is equivalent to 0.5% for \(u_{1}\) and 1.5% for \(u_{2}\), but it was enough to enforce the continuity of traction. Figure 3: (**a**) A speckled image with two subsets of \(41\times 41\) pixels above and below the interface (yellow squares). The post-processing algorithm is performed on the regions inside the subsets marked by the red rectangles. (**b**) The geometry of the pixels over which the local polynomials approximating the displacement fields are defined. The polynomials approximating \(u_{2}^{+}\), \(u_{1}^{+}\) extend over a rectangle above the interface, while those approximating \(u_{2}^{-}\), \(u_{1}^{-}\) extend over a rectangle below the interface; both rectangles include \(n_{x1}\times n_{x2}\) pixels. The pixels within each rectangle are divided into a group of \(n_{x1}\times n_{x2,inter}\) pixels that are closer to the interface, and a group of \(n_{x1}\times n_{x2,cont}\) pixels that are closer to the center of the subset. The observed displacements in the latter group are used together with continuity of stresses constraints on the interface to construct the polynomials. (**c**,**d**) Two different combinations of \(n_{x2,inter}\) and \(n_{x2,cont}\) and their relationship with Vic-2D extrapolation. In the first combination, the group of pixels \(n_{x1}\times n_{x2,cont}\), which are used as data in the inversion, include pixels where the displacement values were obtained by the “fill boundary” algorithm of Vic-2D. When the gradients in displacements in the direction parallel to the interface are large, as in the case of the rupture analyzed in this study, direct calculation of strains and stresses using the modified displacement fields may not result in completely continuous tractions at the interface. The calculation of strains and stresses for pixels in column \(j\) involves the displacements in columns \(j\)-1 and \(j+1\), as it requires displacement gradients with respect to \(x_{1}\) (Equations (1)-(4)). When enforcing TC, these gradients are computed using the polynomial centered at column \(j\). Once the displacement fields have been updated using the procedure described above, strains can be computed from the updated displacement fields, using finite difference schemes, as discussed in Section 2.3. However, the finite differences would be using the displacements of columns \(j\)-1 and \(j+1\), which are obtained from other polynomials than that used to enforce TC, those centered in columns \(j\)-1 and \(j+1\). As such, the spatial derivatives of the modified displacement field with respect to \(x_{1}\) would be slightly different than those obtained during the enforcement of TC, using the polynomial centered at column \(j\).