Harmonic–percussive separation refers to the problem of decomposing a signal into its harmonic and percussive components. This topic has received much attention in recent years (Rafii et al., 2018) and has numerous applications in the field of MIR and musical signal processing.

Within a general context harmonic signals show an overtone structure in the spectral domain. We call overtones one or more clear narrowband frequency peaks that are integer multiples of the fundamental frequency (the first frequency peak in the spectrum). Harmonic signals have relatively stable behavior over time and can be identified in a short-time Fourier transform (STFT) spectrogram by horizontal structures referring to constant frequencies along the time axis.

In contrast, percussive signals form vertical structures in an STFT spectrogram that contain energy in a wide range of frequencies. Therefore, it is a straightforward strategy in most HPS algorithms to try to separate the horizontal structure from the vertical structure in the spectrogram corresponding to harmonic and percussive components, respectively. The horizontal lines in the spectrogram could correspond to either harmonic signals or monochromatic signals.

One of the simplest and fastest HPS approaches is median filtering (FitzGerald, 2010). For simplification we name this algorithm MED in this study. Median filters are usually used to remove noise from an image or a signal. Using a median filter a sample will be replaced by the median of neighboring samples within a window of a specific length (the specific length is the kernel size of the median filter). The entire signal is processed using a sliding window analysis. Within the HPS, two median filters are applied to the amplitude of the STFT spectrogram of a signal. One median filter is performed along the time axis of the spectrogram to suppress percussive events and enhance harmonic components. Another median filter is applied along the frequency axis in order to enhance percussive events and suppress harmonic components. The two resulting spectrograms are then subsequently used to create two masks, which are applied to the original signal spectrogram separately to generate two spectrograms of harmonic and percussive components, respectively. For creating the harmonic and percussive signals in the time domain the phase of the original signal is added to each spectrogram, and the time domain signals are reconstructed using the inverse STFT.

Another powerful approach in HPS proposed by Rafii and Pardo (2012) is based on calculating a similarity matrix. We name this algorithm SIM here. This approach is a repetition-based separation, which identifies repeating elements in the spectrogram by looking for similarities by means of a similarity matrix. Within the SIM algorithm, first similar time frames in the spectrogram are identified through a similarity matrix. Then a median filter is applied only to the frames identified as similar to constitute the repeating spectrogram model that corresponds to harmonic components. The non-repeating spectrogram that corresponds to the percussive component of the data is obtained by subtracting the repeating spectrogram from the original spectrogram. For creating repeating and non-repeating signals in the time domain, the phase of the original signal is added to each spectrogram and the time domain signals are reconstructed using the inverse STFT. Details of this approach are discussed in the following section.

The motivation for using HPS for noise reduction of OBS data stems from the different characteristics of earthquake and OBS noise signals as described in Sect. 2. Earthquakes are broadband transient signals, while the signals of OBS noises are more narrowband compared to earthquakes. We combine two modified HPS algorithms to separate those signals in a two-step procedure. We divide the frequency content of the signal into two frequency ranges: the MED frequency range covers the frequency range between 0.1 and 1 Hz, whereas the SIM frequency range contains the complementary frequency range, i.e., all frequencies except the band between 0.1 and 1 Hz. Then two different algorithms are applied to these ranges. In the first step, we use the SIM algorithm and separate only harmonic or monochromatic signals from the original records in the SIM frequency range. The reason is related to the frequency content of the noise and earthquake signals and how the SIM algorithm separates them. For a better understanding, we first explain how the algorithm works and then present more detail about this selection. In the second step, to reduce noise from MED frequency range we apply MED. There we target harmonic (or monochromatic) as well as narrowband signals with gliding frequencies named current-induced harmonic tremor signal in Sect. 2. The overall schematic diagram of our HPS noise reduction algorithm along with an example is shown in Fig. 2.

The SIM algorithm is explained in the following: from the original OBS record SO (SO represents the original restituted OBS signal) we derive the STFT named X, which is a complex-valued spectrogram.

The complex-valued spectrogram X is separated into its amplitude and phase components using Eq. (1): 1X=V⋅exp⁡1j⋅φ, where φ is the phase of X, V=|X| is the amplitude of X, and j is the imaginary unit.

All of the spectrogram modifications will be applied to the amplitude spectrogram V. The cosine similarity (the similarity between two vectors of an inner product space) between the STFT time frames is calculated through the multiplication of the transposed V by V with the normalization of the V. This is shown in Eq. (2): 2Ska,kb=∑i=1nVi,kaVi,kb∑i=1nVi,ka2∑i=1nVi,kb2, where S is the similarity matrix. Each point (ka,kb) in S is the cosine similarity between time frame ka and kb of V,∀ka,b∈1,m, where m is the number of time frames and n is the number of frequency channels for each time frame. Once the similarity matrix is calculated we use it to determine the time frames most similar to each single time frame. For time frame ka we compare all the values in Ska,ki for i∈[1,m]. We identify similar frames for time frame ka by choosing the upper 2 % of the all time frames with the highest similarities.

Finally, all time frames similar to any frame k in V are stored in a temporary array K. Those similar time frames are used to create a repeating spectrogram model W. The corresponding frame in W is obtained by taking the median of K for each frequency at each time frame k. Those time–frequency bins, which are similar with little deviation between repeating frames, are captured by the median and constitute the repeating spectrogram model. This spectrogram contains only similar and repeating patterns. The time–frequency bins with large deviations between repeating frames would constitute non-repeating transient patterns and would be suppressed by median filtering.

The nonnegative spectrogram V is the sum of two nonnegative spectrograms of repeating and non-repeating patterns; hence, W (the repeating spectrogram model) should always have smaller values or at most be equal compared to V. To ensure this, a repeating spectrogram model W̃ is defined by taking the minimum between W and V. The non-repeating spectrogram model is derived by subtracting W̃ from V.

We use these two (the repeating and the non-repeating) spectrogram models to create two time–frequency masks for repeating and non-repeating patterns, respectively. Instead of the binary mask, which is used in Rafii and Pardo (2012), we use a soft mask via Wiener filtering (Vaseghi, 1996), which is more flexible and usually leads to a better result. The calculation of the soft masks is shown in the following equations: 3M1=W̃2W̃2+V-W̃2,4M2=V-W̃2V-W̃2+W̃2, in which M1 and M2 are repeating and non-repeating masks, respectively. We multiply the masks with the input amplitude spectrogram V to separate the repeating and non-repeating components. The element-wise multiplication of the masks by the input amplitude spectrogram V is shown in the following equations: 5R=M1⊗V,6NR=M2⊗V, in which R and NR denote repeating and non-repeating amplitude spectrograms, respectively.

The resulting R and NR spectrograms are shown in Fig. 2a for a specific and/or typical example of an OBS recording. As can be observed in the R spectrogram, the low-frequency harmonic or monochromatic signals below 0.1 Hz in particular are well captured. We applied the SIM algorithm only to the frequency band below 0.1 Hz and above 1 Hz. In the frequency band from 0.1 to 1 Hz the signals remain unchanged by this procedure. This is the first constraint we consider for the SIM algorithm. In the field of noise reduction using signal processing techniques, a very important point is to not modify the signals of interest for analysis. P and S waveforms in teleseismic earthquake signals often have frequency content in the range of 0.1 to 1 Hz with a dominant frequency around 0.3 Hz. Oceanic microseism noise, which is usually present in OBS data, has a dominant frequency around 0.1 to 0.3 Hz. As P and S phases have a similar dominant frequency as the microseism noise wave field, superposition of both wave fields could happen in this frequency range. They could interfere constructively or destructively, so the resulting amplitude could be higher or lower compared to the original P- or S-phase amplitudes. Considering these interferences, using the SIM algorithm, may result in creating fake higher amplitude for these phases or losing part of their amplitude in the noise-reduced signal. But this could be problematic only when the amplitude of the noise changes over time. For a noise signal with almost constant amplitude, the SIM algorithm can extract the true amplitude of the noise even in the interference moments. However, the microseism noise has slightly varying amplitude over time.

Before moving to the second step we introduce a second constraint parameter, which we use in the SIM algorithm. Surface waves of teleseismic events usually show a dispersed narrowband signal and correspond to mainly horizontal patterns of short duration (on a daily scale) in the spectrogram. Given the way the HPS separates harmonic from transient signals, the surface wave train may be erroneously recognized as the harmonic component and thus be separated as noise signal. In order to prevent this and preserve the whole frequency content of the earthquake, we define a so-called waiting factor for the similarity calculation, introducing a minimum time distance between two consecutive similar frames. For the problem of retaining teleseismic surface waves we found that a waiting factor of at least 2 h prevents the algorithm from pruning surface waves from the transient signal part. The rationale is that the duration of a teleseismic event is usually less than 2 h, whereas the noise components have a longer duration. Using this waiting factor prevents separating any harmonic component of the earthquake signal as noise component. As a side effect this constraint causes short-duration harmonic–monochromatic noise signals to also not be well captured. However, these types of signals are not common in OBS data (see Sect. 2).

In the second step of our algorithm, to target noise signals in the frequency range of 0.1 to 1 Hz, we use MED as it is described in Sect. 3.2. We apply this second part of the noise removal procedure only to a restricted frequency band of 0.1 to 1 Hz. The dominant noise in the MED frequency range is the current-induced harmonic tremor signal (see Sect. 2).

First we create the X′ spectrogram, which is equal to X in the MED frequency range and is equal to zero outside this band. Then we apply a horizontal median filter to X′ in order to separate harmonic components. This results in the harmonic spectrogram H, which contains nearly horizontal patterns captured by the median filter.

Now we have two separated spectrograms for noise signals: R, which is derived from the first step, and H, which is derived from the second step. Summing these two spectrograms will build the noise spectrogram N. Subtracting N from the input amplitude spectrogram V will construct the transient spectrogram T.

As can be seen in Fig. 2a in step 2, the dominant energy of the narrowband signals with gliding frequencies in the range of 0.1 to 1 Hz (the current-induced harmonic tremor noise as introduced in the Sect. 2) is captured in the noise spectrogram N, but part of it remains in the transient spectrogram T. Signals with changing frequency which do not form complete horizontal lines in the spectrogram are difficult to capture by our HPS algorithm, so some of their energy remains in the final spectrogram.

In order to reconstruct the noise-removed signal in the time domain we must add phase information to the spectrogram. We separated the complex-valued spectrogram X into its amplitude V and its phase component using Eq. (1), and all the further modifications have been applied to the amplitude spectrogram V. The phase of input signal SO is mostly affected by the phase of noise signals as they have the dominant energy in the signal. Therefore, we use the phase information of SO in order to reconstruct the noise signal. We add this phase to the noise spectrogram N using the following equation: 7N′=N⋅exp⁡1j⋅φ, where N′ is the complex-valued noise spectrogram. We reconstruct the noise signal NS from the complex spectrogram N′ using the inverse STFT. Finally, the OBS denoised signal HPS (HPS here represents the SO signal after the HPS processing) is obtained by subtracting the noise signal from the input OBS signal SO using the following equation: 8HPS=SO-NS.

Many typical noise signals observed in OBS recordings are harmonic, monochromatic, or narrowband signals with gliding frequencies (see Sect. 2). In order to extract the expected narrowband noise signals from the STFT we require a high-frequency resolution in the spectral domain, therefore making it necessary to use sufficiently long time windows for the spectral analysis. Here we use a fast Fourier transform (FFT) window length of 163.84 s with an overlap of 75 %, corresponding to an FFT size of 16 384 at a sampling frequency of 100 Hz, which corresponds to a frequency resolution of 0.006 Hz.

We use a kernel size of 80 for the median filter in the MED algorithm. The larger the kernel size, the more noise signal would be captured. However, using very large sizes could introduce waveform distortions. As discussed in Driedger et al. (2014), the kernel size is not critical as far as not using extreme values. Our tests show that a kernel size of 80 is the largest size which leads to a safe separation without capturing any energy of the teleseismic signal.

The values of parameters, which we used for this study, are presented in Table 1. These are our recommendations for noise reduction of teleseismic earthquakes. One can tune the parameters based on other specific applications such as denoising local earthquakes or extracting specific signals like microseism signal.

In this section we aim to demonstrate the reliability of our HPS noise reduction algorithm and evaluate the improvement of the OBS data. We applied the method to synthetic and real teleseismic earthquake data recorded by the OBS station D10 of the DOCTAR array (e.g., Hannemann et al., 2016 2017). The synthetics were calculated for a source–receiver epicentral distance of 40∘ (focal depth: 45 km, focal mechanism: double couple, source duration: 4 s) by using the full wave field software qseis (Wang, 1999) and a modified average ak135 velocity model including a water layer (Kennett et al., 1995). The crustal structure of the velocity model is adapted to the oceanic crust (crustal thickness is 6.6 km) in that area, and the water depth is fixed to 4.9 km. Real oceanic noise of the vertical, radial, transverse (ZRT) components recorded by the station D10 is added to the corresponding components of the synthetic teleseismic signal. We created synthetics for three different noise situations at the beginning (N1), during (N2), and after (N3) tidal currents (Fig. 3), each with a theoretical SNR of 1–10 between noise and P onset on pure synthetic Z. Throughout the whole paper the SNR is defined as the root mean square (rms) of the signal divided by the rms of the noise. For further details of synthetic data creation see Fig. S2. For the comparison with real data, we selected 46 teleseismic events in total with magnitudes Mw>5.6 and epicentral distances of 30–160∘ (see Fig. S1). Here only events for which a P onset could be visually identified were used. The pre-selection of the events is taken from Hannemann et al. (2017) and expanded by some events with low magnitudes (see Table S1 in the Supplement). In the following, we will discuss the improvement of the records by comparing the seismograms and spectrograms of synthetic and real data. We also illustrate the improvement for two seismological applications (teleseismic surface wave group velocity analysis and receiver function analysis). For some observations, e.g., checking the phase arrival of the teleseismic body waves, we rotated the arbitrarily oriented horizontal components of the real data into the ZRT system. The orientation angles are taken from the previous study on the DOCTAR array (Hannemann et al., 2016).

Comparing the spectrograms and waveforms of the synthetic example we see a significant improvement of SNR in the HPS processed data set on all components (e.g., Figs. 3 and S3–5 for the real data). The continuous spectral lines of the assumed electronic noise are removed from the data, as are most of the spectral lines related to tremor episodes of head buoy strumming. During the tides, we observe a reduction of the spectral amplitudes for the tilt noise, as well as for the general background noise (Figs. 3 and S3–5) on the horizontal components. The results from the spectrograms are confirmed by the spectra (Fig. 2b), which show the removal of the spectral peaks of the electronic noise (0.05, 0.1, 0.15 Hz) and the tremor episodes (0.5–1 Hz). The amplitude and phase information of the synthetic earthquake is preserved in the HPS signal (see Fig. 3).

To quantify the improvements obtained when using our method, we calculated the cross-correlation of the teleseismic waveform, the SNR of the teleseismic body-wave phases, and the rms of the teleseismic waveform before and after denoising. Because most of the oceanic noise occurs at frequencies below 1 Hz, which is also the most interesting frequency range for the OBS analysis, a 1 Hz low-pass filter is applied to the signals before all result calculations.

We calculated the correlation coefficient for synthetic SO and HPS compared with the synthetic earthquake signal for different SNR and noise realizations and plotted it in Fig. 4a. The high correlation coefficients for HPS and synthetic compared with SO and synthetic in all cases demonstrate a significant noise reduction. Furthermore, they indicate that the HPS denoising preserves the earthquake signal and does not introduce significant waveform distortions since the HPS is more similar to the synthetics compared to SO. In the following we show other measures that confirm the preservation of the earthquake signal.

For the SNR calculation we used a signal window of 30 s starting from the theoretical onset (direct P on Z component, direct S on R and T component, and Love wave on the T component) and a noise window of 60 s starting 70 s before the theoretical onset. For the Love wave, the SV phase (R component), and P phase (Z component) the SNR increased significantly (Fig. 4b). We find that the noise type properties influence the perceived SNR improvement. It appears that there is no SNR improvement on the T component for noise situation N1 (Fig. 4b, the second panel). N1 is taken from the tidal current event's beginning, where there is a significant variation in noise frequencies over time. In this instance, the signal and noise have comparable frequency ranges. Despite the SNR showing no increase, a visual check of the matching trace reveals a definite improvement in the waveform for the SH wave on the T component. The results from the cross-correlation (Fig. 4a) confirm the improvement and preservation of the waveform. The SNR should not be utilized alone to assess the improvement by the HPS noise reduction approach since we are concentrating on the preservation of the waveform and the SNR comparison strongly depends on the noise situation. The improvement of the traces by the HPS noise reduction approach is confirmed by the study of the cross-correlations, rms (which is explained in the following paragraph), and the pure waveforms, even though the SNR does not improve in all instances.

The rms amplitudes of noise-free R-component synthetic, SO, and HPS signals are estimated over 8 s windows with 80 % overlap and plotted in Fig. 4c. Comparing the rms amplitude of the synthetic, SO, and HPS we see that the synthetic and HPS have similar amplitude ranges, while SO has a much higher amplitude. This shows a significant noise reduction in HPS along with preserving the energy of the earthquake and all the phase arrivals. As there is some noise remaining after denoising we see some differences in the overall shapes of the rms amplitude of the synthetic and HPS (especially after minute 24, which is almost at the end of the energy of the synthetic signal). However, HPS shows peaks on the arrival times of seismic phases of the synthetic, which means that the energy of seismic phases is preserved after denoising. The minor changes in seismic phase shapes of the synthetic and HPS are also due to the remaining noise. The seismograms and spectrograms related to this example are presented in Fig. 3. Figure 4d shows a comparison of the rms amplitude of the original noise in SO (blue curve), the remaining noise in HPS after denoising (red curve), and the synthetic earthquake (green curve) signals. Besides a high noise reduction in HPS, the plot shows that the remaining noise is independent of the pattern of the synthetic earthquake, which confirms that the denoising process does not affect the earthquake energy in the HPS signal.

By applying our HPS noise reduction algorithm, we aim to improve seismological analyses, especially those involving the analysis of teleseismic body and surface waves. Valuable constraints of the Earth's structure in oceanic regions can be taken from the analysis of the SH wave field like Love waves, which are not influenced by the water column, but often cannot be analyzed due to strong noise on the horizontal components. SV waves are also often masked by noise but are, for instance, important for tomography studies or S and SKS shear wave splitting analysis (e.g., Silver and Chan, 1991). Other techniques using the SV wave field like the Z / R ratio of the teleseismic Rayleigh waves (Tanimoto and Rivera, 2008) or receiver functions (RFs) (Langston, 1979) also rely on clear radial component readings. In the following we will show the improvement which was achieved for the SH arrivals and for the group velocity analysis of teleseismic Rayleigh and Love waves, as well as for the receiver function analysis.