2.2.3. *Time Lag Method*

Independent of the method described in Section 2.2.2, the propagation direction of NLIWs was estimated using the distance between two locations of the NLIW front observed at different times, with the assumption that the NLIWs propagate across the two measurement locations with an angle orthogonal to the constant phase lines at a constant speed (Figure 2a,c). The observed propagation speed was estimated by dividing the distance between the two locations *Dobs* by the arrival time lag *Tobs*, for example, *cobs* = *Dobs Tobs* . Then, *cKdV*.*iw* was calculated from *cobs* and angular difference *θtl* = |*φobs* − *φtl*| between *φobs* (direction from the first measurement location to the second measurement location) and the propagation direction of NLIWs *φtl* as

$$c\_{\rm KdV.iw} = c\_{\rm obs} \cos(\theta\_{l\rm l}) = \frac{D\_{\rm obs}}{T\_{\rm obs}} \cos(|\phi\_{\rm obs} - \phi\_{\rm ll}|). \tag{16}$$

Finally, the *φtl* was obtained from Equation (16) as

$$\phi\_{tl} = \phi\_{obs} \pm \cos^{-1}\left(\frac{c\_{KdV,iw}T\_{obs}}{D\_{obs}}\right). \tag{17}$$

The propagation direction estimated using the method described above (*time lag* method) also has an angular ambiguity caused by the sign of the arccosine part. Thus, a physically reasonable direction is selected. The *φtl* and *φobs* are angles in degrees measured counter-clockwise from the east.
