2.2.4. Estimation of Propagation Speed and Direction

Two propagation directions of NLIWs estimated from the two methods were used to estimate the optimal propagation direction and successive propagation speed. First, a consistent direction between the two directions derived from each method was selected to minimize the ambiguity where the two methods yield angular difference |*φds* − *φtl*| typically less than 30◦ . For example, each one (bold green and blue colors in Figure 2b,c) among two *φds* and two *φtl* are selected as more consistent between the two methods and physically reasonable, while inconsistent *φds* and *φtl* (deemed green and blue colors in Figure 2b,c) among the two *φds* and two *φtl* were not selected. To optimize the propagation speed and direction, the difference in consistent propagation directions from the two methods |*φds* − *φtl*| was minimized by iteratively changing *cKdV*.*iw* at intervals of 0.01 m·s −1 instead of using the constant propagation speed derived from the two-layered KdV theory. The updated propagation speed was determined from the iterations, and the final propagation direction of NLIWs *φiw* was determined by averaging the two directions (*φds* + *φtl*)/2 when |*φds* − *φtl*| reached its minimum value for the updated propagation speed. Iterations were performed for a range of 30% deviation from *cKdV*.*iw* (typically requiring 38 iterations to reach the minimum), which is comparable to the range of the interannual variation of propagation speed reported in a previous study [47], as discussed in Section 4.
