## 5 Thermoelastic Topography of GESA Coating: Topographic Mapping  
In this section, the results of the topographic mapping using the XY-scanning algorithm explained in Section 3 are discussed. As this measurement is very time consuming (\(\sim\)24 h per wavelength for a grid of 21 \(\times\) 40 points), a limited set of four, well-chosen, grating spacings (15, 25, 45 and 200 \(\upmu\)m) was used to sample the dispersion curve in the 2 mm \(\times\) 2 mm surface area. Figure 9 illustrates that the wavelengths that would be needed for SAWs to propagate with a velocity equal to the Rayleigh velocity of the steel substrate are of the order of 1 mm, which is too large for the GILBD experimental setup. Hence, for each position \(x_{k},y_{k},\)_a priori_ information of the Rayleigh velocity of the substrate was used by adding a point \(c_{R}=3024\) m/s in the long wavelength limit (\(\lambda=1\) mm) of the experimental dispersion curve. Figure 10 depicts the wavelength dependent phase velocities resulting from the interpolation of the phase velocities on optimized locations resulting from the optimization algorithm, to a regular rectangular grid consisting of 21 \(\times\) 40 points. The color scale in all images in Figure 10 is the same. Comparing them allows observation of a general trend of increasing phase velocity with decreasing wavelength. At the wavelength of 200 \(\upmu\)m, the spatially averaged phase velocity reaches about 3087 m/s. The velocity increases to 3229 m/s for the 45 \(\upmu\)m grating spacing. As discussed earlier, the latter wavelength corresponds to a SAW penetration depth where the SAW velocity lies in between the values of the coating and the substrate. When the SAW wavelength is decreased towards 25 and 15 \(\upmu\)m, corresponding with wavelengths at the coating asymptote, the average phase velocity increases further to respectively 3230 m/s and 3335 m/s. Due to an insufficient number of data points in the neighborhood of \(x=0.6\) mm and \(y=2\) mm, as a poor signal quality due to high roughness induced scattering of the reflected probe beam light in that region, the values at these points were obtained by effectively extrapolating the known phase velocities, and only serve for display reasons. A dispersion curve consisting of phase velocities evaluated at five wavelengths, including the hypothesized wavelength of 1 mm, can be attributed to each \((x,y)\) coordinate located on the rectangular grid. As in the previous section, three parameters were fitted for each dispersion curve \(v_{\text{SAW}}(f,x_{k},y_{k})\), i.e., the coating thickness, density, and Rayleigh velocity, while the substrate's density and bulk velocities were kept fixed. A least-squares curve fitting algorithm (1 sqcurvefit in Matlab(r)) was used to fit each local dispersion curve, using a SAW propagation dispersion curve function based on references [33,35]. In Figure 11a-c the resulting topographic maps of the best fitting parameters are shown. To ease interpretation, a contour plot with lines of isovalues was added to each surface plot. For further interpretation, the average values and average deviations from average of the 3 fitted parameters, i.e., the Rayleigh velocity, density and thickness of the coating are listed in Table 3. Comparing the values with the ones in Table 2 learns that the deviations are smaller than the respective fitting uncertainties, inferring that the values represent an upper limit for the effective topographic variations. Using the earlier motivated assumption of a value of 0.3 for Poisson's ratio, the Rayleigh  
Figure 11: Surface plots of the best fitting parameters ((**a**) coating thickness, (**b**) coating Rayleigh velocity and (**c**) coating density) obtained by fitting each position dependent acoustic dispersion curve to the coating-substrate model. For the sake of clarity, contours of constant parameter values were added to each surface plot. Around \(x=0.6\) mm and \(y=2\) mm no results are shown due to poor data quality at these positions.