### Data characteristics

The broadband seismograms are obtained from the Broadband Array in Taiwan for Seismology (BATS; Kao and Jian 2001)—an island-wide seismic network deployed by the Institute of Earth Sciences (IES) of the Academia Sinica and the Central Weather Bureau (CWB; Shin et al. 2013) of Taiwan. The infrasound records are obtained from the three sensors (Model 21 Chaparral Physics^{Footnote 1}), deployed in a campaign mode by IES during the relevant time. Figure 1 shows the location of the used seismic and infrasound stations; only those of high signal-to-noise ratio (SNR) are adopted in this study.

The broadband seismometers, equipped with a symmetric triaxial force feedback sensor designed with a flat sensitivity to ground velocity from 0.02 to 100 Hz, can efficiently detect ground motions. The infrasound instrument, with the nominal sensitivity of 0.4–2.0 volts/Pa at 1 Hz (90–18 Pa full-scale range) and frequency response between 0.1 and 200 Hz, records signals that span the low audio/infrasound boundary ranging from explosions to avalanches.

The shockwave records, retrieved from 12 seismic and 3 infrasound stations in northernmost Taiwan (Fig. 1), have the sufficiently high signal-to-noise ratio (SNR > 4) as judged from the records within the windows of 4-s length after and before the first emergence. In other parts of the island, seismic stations of the array have relatively low shockwave signal strengths and hence were not used.

### Data processing and event recognition

We produce the time–frequency wavelet spectrograms (using the real-valued Morlet wavelet; Morlet et al. 1982; Chao et al. 2014) of all our seismograms for the frequency range of 0.1–25 Hz. The wavelet spectrogram is effective and robust for detecting and characterizing moderate to high-SNR signals in a time series and capable of revealing non-stationary periodicities (Chao et al. 2014). We filter the data within a band-pass frequency range of 5–25 Hz to reduce the acausal peak signals along the time series. Figure 2a, c shows the record (filtered as well as unfiltered) of co-located seismic and infrasound station SXI1, and Fig. 2b, d plots the corresponding wavelet spectrogram.

The filtered time series waveforms at all the stations are plotted in Fig. 3 lined up with the distances from the intercept of the protracted trajectory with the ground surface (hereafter called termination point; calculated below using the inversion). The shockwave signal manifests the characteristic “N” shape in infrasound records and reversed “N” shape in seismic records (Ishihara et al. 2003; Langston 2004; Pujol et al. 2005), which we identify in dilated waveform record at each station in Fig. 4.

### Inversion for trajectory parameters

We make two assumptions, following Ishihara et al. (2003), to invert for the trajectory parameters: the meteor penetrates linearly through the atmosphere with a constant velocity, and the shockwave speed of 320 m/s is uniform in the atmosphere. The linear trajectory assumption is reasonable because the meteor’s high propagation speed implies negligible effects of gravity (Tatum 1999) and the nearly constant velocity in the upper atmosphere prior to its termination (Halliday et al. 1996; Brown et al. 2002).

We define two rectangular coordinate systems (Fig. 5), describing the geographical coordinates (*x, y, z*) and the meteoric trajectory (*X*, *Y*, *Z*); its respective origins are taken at—(25.16 N, 121.44 E, 0), the location of Tamsui town center, and the intercept point of the meteoric trajectory with the x–y plane (termination point). The x–y plane is the tangential plane to the Earth’s surface at the origin. We constrain the meteoric trajectory by six pertinent parameters, namely the speed (\( v \); constant), azimuth (\( \gamma \)), elevation angle (\( \delta \)), the termination point (\( x_{0} ,y_{0} \)), and the expected time at the termination point (\( t_{0} \)). The two coordinate systems and the trajectory parameters are related by the following set of non-linear equations (Eq. 1; Ishihara et al. 2003):

$$ { \sin }\beta = \frac{c}{v} $$

(1a)

$$ \left( {\begin{array}{*{20}c} X \\ Y \\ Z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\cos \gamma \sin \beta } & {\sin \gamma \sin \delta } & { - \cos \delta } \\ { - \sin \gamma } & {\cos \gamma } & 0 \\ {\cos \gamma \cos \delta } & {\sin \gamma \cos \delta } & {\sin \delta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {x - x_{0} } \\ {y - y_{0} } \\ z \\ \end{array} } \right) $$

(1b)

$$ \frac{{\sqrt {X^{2} + Y^{2} } }}{\tan \beta } - Z = v\left( {t_{\text{pre}} - t_{0} } \right) $$

(1c)

$$ {\text{F}}\left( {v, \gamma , \delta , x_{0} ,y_{0} , t_{0} } \right) = \sqrt {1/\left( {N - 6} \right)\mathop \sum \nolimits \left( {t_{\text{obs}} - t_{\text{pre}} } \right)^{2} } $$

(2)

We can solve using the least-squares method for the best set of the six parameters by minimizing the objective function in Eq. 2 (where *N* is the number of stations). However, minimizing the objective function—a non-linear problem with a large number of local minima in 6-parameter space—by any gradient-based method runs a likely risk of the solution to get trapped in a local minimum. To circumvent this risk and to seek a trustworthy solution, we conduct a non-linear iterative direct search of the model space using the global optimization technique of the Genetic Algorithm or GA (Sen and Stoffa 1995; Yamanaka and Ishida 1996). Computationally less demanding than the grid search method, the GA inversion is capable of efficiently search a very large model space, where the model parameters can be freely defined with a few assumptions and restrictions.

The GA works, on a user-defined set of values including the search space and the size of the population, to generate stochastic population in the successive generation such that the objective function value decreases progressively. The initial population of model parameters is randomly chosen within the given search range. The simple GA undergoes a set of operations on the model population to produce the next generation: selection, coding, crossover, and mutation. In the selection scheme, the model parameters exhibiting the higher fitness value (lower objective function value relative to others) are selected and replicated with the given probability such that the total population size remains constant; then the population members are randomly paired among themselves. In the coding scheme, the decimal values associated with each population is converted to binary system forming a long bit string (analogous to a chromosome). In the crossover scheme, some part of the long bit string of binary model parameters is exchanged with their corresponding pair to produce a new population. In the mutation scheme, some randomly selected sites (with given probability) of the new set of the binary model population are switched. These sets of operations will continue until some pre-defined termination criteria for the technique are satisfied.

Specifically, the total population size for GA is chosen to be 100, guided by the number of parameters in our model. We decide the selection and mutation probability (0.7 each) by experimentation. In each generation, the new population is created with two elite members (individuals giving better fitness value), which replaces the worst two members of the next generation. The termination criterion of the GA process was taken to be the tolerance value (difference of objective function value between successive generations), which was 1E − 6. We set up the GA to leap all the greedy traps of a local minimum in the path toward the optimum value and then we opt for the gradient-based Quasi-Newton method to converge quickly for the best parameters possible.

We tested the reliability of our 6-parameter inversion procedure through 10,000 Monte Carlo simulations. The statistics of the residuals tells a Gaussian distribution with zero mean (not shown), as expected of an effective estimation method.