The data obtained from experiments and analytical simulations are now analysed by three proposed metrics, thereby delivering a systematic methodology for diffuse field prediction on any structure. In most of the ultrasonic applications, the fundamental frequency is examined hence the wave propagation at the fundamental frequency in diffuse state is critical to these techniques. As a consequence, the metrics should assess the diffusivity of wave components particularly at the fundamental frequencies, which can be accomplished by introducing a gaussian windowed filter in post-processing.

Specifically, the experimental FMC signals, *f*_{n,m}(*t*), the decayed white noise, \(\hat{f}_{n,m}^{W} \left( t \right)\) and the white noise, \(f_{n,m}^{W} \left( t \right)\) were selectively processed by a gaussian windowed filter and then examined by three different metrics (*η*_{area}, *η*_{max} and *η*_{phase}), in order to predict a diffuse field (as illustrated in Figure 4). Note that *f*_{n,m}(*t*), \(\hat{f}_{n,m}^{W} \left( t \right)\) and \(f_{n,m}^{W} \left( t \right)\) are input to the identical filter independently. The metrics were plotted against the gate start time, *t*_{r}, from 0.005 ms to 0.85 ms (this range was indicated by limits of hardware reception gain). In addition, the window length, *T*, was initially selected as 0.12 ms as a consequence of the standard parameter used in NUI. Therefore, the outcome of this process will help determine the earliest approach of the diffuse field as well as understand the underlying physics behind the formation of diffuse field in real structures.

First, the relationship between metric, *η*_{area}, and gate start time, *t*_{r}, on the aluminium sample is displayed in Figure 5. This figure indicates the variations in windowed energy, which were post-processed from experimental FMC signals and white noise. A gaussian windowed filter from \(2\omega_{0} /3\) to \(4\omega_{0} /3\) was selectively implemented to those data prior to evaluations through proposed metrics. The results from simulated white noise in Figure 5 indicate the benchmarks for an idealised diffuse field. Most importantly, the effect of exponential decay extracted from experimental data and the gaussian windowed filter has measurable contributions to the metric, *η*_{area}. Their corresponding influence can be observed in Figure 6, which demonstrates that the incoherent signals within a narrow band tend to have much higher value in *η*_{area} (i.e., less diffusivity), and the metric on decayed signals is likely to inherit the characteristics of their decay rate.

The metric, *η*_{area}, on experimental FMC signals in Figure 5 suggest a good agreement with previously observed variation of the amplitude envelope measurements about 15% [3]. Specifically, the experimental results using the metric *η*_{area} suggest that a diffuse wave field occurs at 0.1 ms by observing the converged start point and the filtered data provide the higher value in *η*_{area} within predicted diffuse field*,* which is consistent to the observation from white noise. Most importantly, this predicted gate start time for a diffuse field is same as the empirical one used in NUI experiments [8] that delivers good NUI performance on crack detection.

The metric (*η*_{max}) indicating the variation in maximum energy is plotted in Figure 7 with respect to gate start time, *t*_{r}. Its results are very similar to those in Figure 5, but the overall variations increase by 20 percent and small fluctuations with increasing *t*_{r} due to too small number of samples in a statistical problem (i.e., when calculating the standard deviations of overall energy in a wave field). This fact will be confirmed later by the window size study (a converged relationship between *η*_{area} and window length *T* as presented in Figure 9) Therefore, the metric (*η*_{area}) is favoured over the metric (*η*_{max}) as a consequence of the most representative measurement in overall wave field energy.

The change in the metric (*η*_{phase}), by which the extent of phase coherence between signals fired by two neighbouring transmitters is reflected, with increasing *t*_{r} is displayed in Figure 8. As mentioned before, the expected range in *η*_{phase}, indicating the change from coherent field to diffuse field, is from 0.5 to 1. As a consequence of a phase-dependent method, the filter and the attenuation rate barely have influence on white noise, which can be reflected by the overlapped curves (in cyan and magenta colours) in Figure 8. Furthermore, the metric (*η*_{phase}) on white noise (uncorrelated signals) is consistently with a value of 1 as expected. For the experimental data, the filter also has little effect on this metric. The diffuse state might be indicated from *t*_{r} at 0.07 ms due to the corresponding *η*_{phase} value at 1 although the metric then decreases by approximately 0.15 and then converges to the value around 1.02. This small decrease is possibly because the surface waves not only attenuate more slowly than the bulk waves [30], but also become diffuse much later due to high aspect ratio of this structure (6:1). That is to say, the bulk waves initially at higher intensity will reach diffuse state earlier due to significantly more reflections from two horizontal boundaries closer to each other in z-direction (as illustrated in Figure 2). This fact will be further confirmed by examining *η*_{phase} on a specimen with lower aspect ratio (1.04:1), as presented in Figure 12. The metric, *η*_{phase}, evaluating experimental data have the small decrease prior to the previously predicted diffuse field start time (0.1 ms), which might be attributed to the existing surface wave components in high intensity before 0.1 ms in Figures 3 and 7.

The window length, *T*, is studied by plotting the metrics (*η*_{area} and *η*_{phase}) against *t*_{r}, because it is important to select the most representative samples in a predicted diffuse field with the smallest window required. In theory, the smaller number of sample (e.g., the single sample used in the metric *η*_{max}) are more likely to provide the large error when measuring standard deviations of the statistical energy in diffuse wave field. However, there should be an upper limit for the size of sample to be estimated, which delivers a saturated measurement of diffusivity.

Therefore, the size of window length, *T*, was varied from 0.0004 ms to 0.2 ms. As presented in Figure 9, the metric, *η*_{area}, was first plotted against *t*_{r}, in order to explore the effect of window size on variation between windowed energy received at different location. It clearly demonstrates that the relationship between the size of *T* and *η*_{area} is exponentially inverse. In particular, the *η*_{area} converges with increasing size of *T* from the empirical *T* with 0.12 ms (used in NUI imaging and previous studies) by observing the overlapped curves between *T* with 0.12 ms and 0.2 ms in Figure 9. It should be also noted that the small size of *T* (e.g., 0.0004 ms and 0.002 ms) contributed to more fluctuations and larger offset. As a consequence of the results, the size of *T* with 0.12 ms is suggested to be used.

Similarly, the same window length study was performed by examining the metric, *η*_{phase} with increasing gate start time, *t*_{r}. The results in Figure 10 suggest that the larger window size delivers more rapidly converged value in the idealised diffuse state with increasing *t*_{r}. In addition, the metric, *η*_{phase} becomes converged at *T* of 0.12 ms with increasing window size and the three neighbouring pairs was able to provide the same prediction of diffuse field as the entire array transmitters.

Furthermore, the robustness of these two metrics is examined on the same type of steel CT specimen used in experiments [11] for performing NUI technique on crack monitoring, whereby the size of *T* was used as 0.12 ms and the resulting time traces from 0.005 ms to 0.4 ms was combined from nine sets of FMC data were acquired independently with different gains (from 36 dB to 70 dB). Note that the white noise here is also produced from combining simulated random signal and exponential decay rate extracted from the experimental resulting time traces. The results in Figure 11 demonstrate that the metric *η*_{area} on filtered experimental data implies good agreement with the empirical values used in NUI works [11] (*T* of 0.12 ms with the gate start time, *t*_{r}, at 0.1 ms) and that of filtered white noise by observing the converged start point at approximately 0.07 ms (where *η*_{area} is around 15% as the indicated diffuse state). In addition, the metric *η*_{phase} as presented in Figure 12 identifies the convergence to 0.95 (very close to the idealised diffuse state, 1) from around 0.07 ms in good agreement. It should be noted that the surface of this steel sample in contact with the probe has one-third length of the aluminium one, so that the state of surface waves is expected to become diffuse three times faster due to more reflections from boundaries. As a consequence, the small decrease observed after the predicted diffuse start time (0.07 ms) in Figure 8 is probably attributed to the surface waves as discussed before.