In this section, we discuss the ground motions (aftershocks) of gorkha earthquake 2015 using wavelet analysis (both CWT and DWT). Wavelet based analysis obviously ameliorate our understanding of this events deeply. It relates energy bursts to seismic wave arrivals. The analysis performed here is mainly focused on CWT and DWT. CWT and DWT based techniques are two possible approaches to improve the clarity of wavelet analysis which is addressed below. Gurley and Kareem (1999) presented numerous ex- amples citing the wavelet analysis usefulness in the identification and characterization of transient random processes involving in ocean engineering, wind and earthquakes. It involves spectral and co-spectral analysis for transient events identification, non-stationary signals simulation, and noisy signals filtration.

### Ground motion analysis

In order to develop the seismic hazard assessment and its mitigation, it is necessary to improve our understanding over ground motion produced by earthquake. In Figs. 4, 5, 6 and 7, the analysis of the strong ground motion characteristics of devastating earthquake after the aftershock events observed on the site TVU, THM, KTP, and PTN respectively are presented. The data used in this paper include the ground acceleration data collected at sampling rate of 100 HZ during an aftershock event occurred on 25 April 2015. The time scale in seconds and ground acceleration in cm/s^{2} are present in horizontal and vertical axis respectively Strong ground motion can be observed in Kathmandu valley. The peak ground acceleration (PGA) > 250 cm/s^{2} was recorded on the EW component at the site TVU and KTP for the after- shock occurred on 06:35 UTC (25 April 2015). At this site the PGA of NS component and vertical component was 100 cm/s^{2}. Due to such high ground motion within Kirtipur(location of TVU and KTP station) area building are severely damaged/partially collapsed. Damage to these buildings is mainly due to the decline of material strength, lack of repair, maintenance and strengthening works. Soil amplification occurs typical to laquestrine and deep clay deposits like in Kathmandu valley. The PGA was around 150 cm/s2 recorded at the site THM on EW and whereas the vertical component was around 120 cm/s2 peak amplitude of the NS component. At PTN site the PGA recorded on EW, NS and vertical component was around 120 cm/s^{2}. The large value of PGA was by reason of isolated shock waves from the origin time. From Figs. 5, 6, 7 and 8, it can be observed that horizontal long duration oscillation with large amplitude at each site. These observation is shore up with the result found by Takai et al. (2016) for Nepal earthquake. Kumar et al. (2016) studied ground motion of Nepal earthquake occurred on 25 April 2015 using the data from sites (one rock site and three sedimentary sites). They found that the largest PGA was 241 cm/s^{2} recorded on the EW component at the rock site and that on horizontal component was 250 cm/s2. They also use USGS data to calculate the ground acceleration and found the value around 0.02 to 0.74 g. The Boorea et al. (2008) equations predict PGA of 0.49 g for the Mw7.8. Goda et al. (2015) inspect time-history data recorded at KATNP. They mentioned that the PGA for Mw 7.8 main shock and the Mw 7.6 aftershock event is about 150–170 and 70–80 cm/s^{2} respectively. According to United States Geological Survey USGS (2015) the fault size of the Gorkha earthquake was around 200 km long and 150 km wide. According to ICIMOD research report, 2016/1 the fracture transmitted 150 km to the east and 60 km to the south of Barpak VDC. In case of large earthquake, the epicentral distance alone is not an effective parameter for ground motion determination because large affected area lies directly within the rupture zone. S-waves arrives at a given point not just come from the epicenter but arrives as a whole from rupture zone. Previously, researcher used multifactor approach for time-frequency analysis of non-stationary ground motion (Kameda, 1975; Scherer, 1994). Researchers Hui Cao and Ming Lai (200, in their paper proved that the LPS (local power spectra) of the process estimated by wavelet transform describe its non-stationary characteristics accurately and the LPS transfer has enough precision to reflect the time-frequency characteristics of earthquake ground motions. They showed that results obtained by non-stationary methods are bigger than those by stationary methods which are in accordant with the conclusions of Wu (1989).

#### CWT analysis

Figs. 8, 9, 10 and 11 display the result of CWT. This analysis helps to understand the patterns relative to seismic events occurred on 25 April 2015 Nepal. These figures show a time-frequency sketch of ground motion noted by TVU, THM, KTP and PTN stations, respectively. In these figures, the horizontal axis denotes the time scale and the vertical axis denotes the period. In these figures, it is possible to represent the energy distributions of signal in both frequency and time, and also, to analyze some specific aspects. The first aspect is to equate the amplitude amplification by discontinuities that exist in the signal and another is to observe sudden changes. The after-shock event starts at 06:11:25.95 UTC (25 April 2015). The occurrence of energy bursts at different times can be noted in the time history easily. These figures demonstrates the existence of multi scale feature of the signal. And also shows how the energy transference is taking place. These characteristics give the impression on the scalogram through the scattering of frequencies. The color scale present on the right side of the panel has same unit of real data. In the scalogram, stronger wavelet power areas are displayed as pink and the lower wavelet power areas are displayed as red. The scalogram of NS, EW and vertical motion shows different highest intensity power areas displayed as pink color and blue color. By comparing Figs. 8, 9, 10, and 11 it is observed that there is a clear power area for 30 s i.e., there is clear ground motion during the earthquake within this time. Similarly, within this time the less intense power areas are seen at many places covered by green and blue color. In Fig. 8, we can observe long duration horizontal motion with large frequency during the earthquake. There is a pink, blue and green band in between the period range of 64–32 (0.015–0.031 Hz) seconds for NS and EW ground motion while those bands observed in the period range of 64–8 (0.015–0.125 Hz) second for vertical motion. In other words, this range of period is the most remarkable for all constructions which have the same frequency as the foremost frequency of earthquake. The nature and power areas in Fig. 10 are similar to Fig. 9. In Fig. 11, there is pink, blue and green band of power area around period 64–16 (0.015–0.0625 Hz) seconds for NS and EW ground motion while those bands observed in the period range of 64–8 (0.015–0.125 Hz) second for vertical motion. In Fig. 11 the nature of scalogram is similar to that of Fig. 10. By comparing Figs. 8, 9, 10, and 11 it is also observed that the effect of motion starts from around 06:11:31UTC to 06:11:51UTC. Similarly, in Fig. 10, the energy burst of the highest peak amplitude with period approximately 32–16 (0.031–0.0625 Hz) around the time 06:11:35UTC. The maximum wavelet power area has intensity around 23 unit as shown in Fig. 11 for EW scalogram. By comparing Figs. 8, 9, 10, and 11 it is also observed that the highest frequency spectrum of ground motion record in horizontal scalogram.

The less intense areas are seen at the comparative scales for every interval of each component. This suggests that the periodicity is constantly communicating throughout the earthquake. Also we see that the zone with less periodicity occur frequently and the zone with the high periodicity occur less frequently. This is exactly in accordance with the frequency periodicity inverse relation i.e. the high frequency components with less periodicity and high intensity are found to lie within period range 64–32 s (0.015–0.031 Hz). And hence we can infer that any construction with natural frequencies lying within this frequency band or close enough to this have high chances to suffer the resonance and get destroyed. Thus we found CWT to be good tool to study the nature of seismic waves.

#### DWT analysis

Figures 12, 13, 14 and 15 illustrates the results obtained from DWT analysis. Daubechies, (1992) define this wavelet transform as a mathematical tool which helps to filter signals into different frequency components. So that we can analyze each component with its resolution accorded to scale. It is used for dynamic analysis of structures tempted by earthquake loading (Salajegheh and Heidari, 2002; Salajegheh and Heidari, 2005; Heidari and Salajegheh, 2006) and optimize the structural analysis (Salajegheh and Heidari, 2004a; Salajegheh and Heidari, 2004b; Salajegheh et al., 2005). Gurley and Kareem (1999) used this method for modeling, analysis, and simulation of non-stationary processes by decomposing random processes into localized orthogonal basis functions. In this paper, we used second order Daubechies orthogonal wavelet transform of level j = 1and 2. The optimize description of second order Daubechies orthogonal wavelet transform can be found in (Domingues et al., 2005; Gonz’alez et al., 2014; Klausner et al., 2014b, Adhikari and Chapagain, 2016, Adhikari et al., 2017). Actually wavelet transformation determines how a chain of wavelet functions characterize the signal. The wavelet transformation results consist different order coefficients allied with two independent variables, dilation and translation. The scale is a process of analyzing the frequency contents, while translation typically represents time. The main steps involve are to calculate and analyze the wavelet coefficients of one decomposition levels, and to select the wavelet coefficient thresholds that allows to detect singularity present in the earthquake ground motion signal. In those figures the time interval of data sampling is 0.005 s but the recorded data length varies for each event. For this study, record length of 81.92 s (16,384 = 214 samples) is picked. The discrete wavelet transform require samples of power 2 for calculation.

Figures 12, 13, 14 and 15 show the behavior of Daubechies wavelet coefficients dj (for j = 1and 2) of ground motion signal during aftershock events that occurred on 25 April 2015. The wavelet coefficients versus the corresponding wavelet time with detail sub bands (d1and d2) are plotted in those figures. The frequency bands for each sub band are specified for ideal filters. To avoid Gibbs effect, the actual filters decay slowly across the ideal bands which results into some partial overlap between these intervals. Therefore, the wavelet map indicates the position (both in time and frequency) of the wavelets used for approximation. By the Perceval equality, the square of those wavelet coefficients would give the analogous term in the series of signal energy. Such a map indicates energy distribution of the signal on time-frequency plane. For visual comparison, the original signals is shown at the top of each figures. The principle behind the use of amplitude of wavelet coefficients is to symbolize the local regularity present in the signal (Mallat, 1989; Domingues et al., 2005; Ojeda et al., 2011, Gonz’alez et al., 2014, Adhikari and Chapagain, 2016, Adhikari et al., 2017). The peak amplitudes of wavelet coefficients point out significant fluctuation of signal. The resulting singularity patterns is equivalent with the sudden energy release in the Earth’s crust.

This generates seismic waves on the fault slip surface. By comparing Figs. 12, 13, 14 and 15, it is observed that there is no effect of ground motion up to 18 s. The effect of ground motion starts from 20 s to 80 s with higher and lower amplitude discontinuously. In Figs. 12, 13, 14 and 15 we can observe several peaks of different amplitude in lower order frequency but in higher order frequency we can observe significant fluctuation. Here in Fig. 12, square wavelet coefficients show several high amplitude peaks corresponds to vertical ground motion while peaks corresponds to NS and EW ground motion are of lower amplitude. In Fig. 13, several sharp peaks for vertical ground motion at the time of aftershock. Here some sharp peaks are also seen for NS and EW ground motion but the amplitude is lower than that of Vertical ground motion. In Fig. 14, singularity peaks of high amplitude can be found in all direction ground motion signal but number of such peak is high for vertical ground motion signal.

In Fig. 15 also, number of singularity peaks of high amplitude found on vertical ground motion signal. We can say that NS, EW and vertical motion as observed by four different stations where singularity present in signals observed during peak of aftershock. Moreover, it can be noticed in all figures that the highest wavelet coefficient amplitudes seems to be associated with the Rayleigh wave arrival. The rupture directivity and fling step effects might be two crucial factors for the strong ground motion observed in Kathmandu valley. The rupture directivity is also a key factor causing the large ground motion during the Gorkha earthquake (Koketsu et al., 2016). Comparing NS, EW and Vertical motion, most of the high amplitude signals are associated with vertical motion, because in that direction energy release is high. These sudden variations accounts for the high destruction caused due to the Earthquake. Similarly from all panels we see that the singularity follows the pattern of transference of energy. The small amplitudes observed in the wavelet coefficients for the Earthquake means the energy transfer process is smooth; while the larger amplitudes indicate that there are impulsive energy injections superimposed to smooth background process. This transference of the energy even supports for the cause of destruction.

Soon after the main event seismic energy damped but the period of damping is different. This can be seen on DWT curve where square wavelet coefficient is high at the time of aftershock and the coefficient decrease with time. The damping period depends upon the types of site i.e. sedimentary or rock site. At rock site seismic energy highly damped. For each case, it is observed that in the higher-frequency sub bands signal has less energy. This is important to most of the coefficients in these sub bands which is being eliminated by the thresholding. In those plots the low-amplitude high frequency pulses are curved in low approximation levels, while the largest amplitude pulses are still signified quite well. This is the characteristic of data compression by thresholding. By this process the high-frequency components are sieved where they are small and are well-preserved, and where they are significant. From CWT analysis, the high frequency components with less periodicity and high intensity power areas are found to lie between time 20–80 s. It is interesting to found singularity-at the same time-on the signal from DWT graph.

From these figures we can see the two level decomposition are sufficient to notice the singularity patterns. Taking the fact that the highest amplitude of the wavelet coefficients indicate singularities, in all cases singularity patterns were identified associated with the intense earthquake effect. When the ground motion is under less intense condition i.e. times where motion was comparatively low the accelerations value recorded in the seismograms can be represented by the smooth functions, and the wavelet coefficients show low amplitude accordingly. As the Earthquake had the moment magnitude around 8.3 we certainly expected to see highest amplitudes at most places and indeed we can see it at level d1. In other words we can say that DWT or those coefficients are able to identify sudden variations in ground motion that occurred during Earthquake. The wavelet coefficients of two levels i.e. first and second decomposition level of the wavelet transform show indeed a better time localization and then are locally associated with the higher frequencies during the Earthquake. Thus in our case we concluded that the decomposition levels has proven to be sufficient to isolate the singularity patterns. In each components and for all the decomposition levels we can see the transient variations which are almost similar; the only difference is maximum value of amplitude. This value is slightly different in each case and the reason behind may also be the orientation or the direction the seismic wave that is being recorded in the seismogram. Thus using DWT in earthquake analysis can provide us with different advantages. This can give us the good view of the sudden variations occurring during the Earthquake. The singularity patterns can show us the positions of occurrence of higher frequency components which in our case were almost entirely distributed for the considered interval. This can also show how the energy transference takes place during the phenomenon.