Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2016) 204, GJI Seismology doi: /gji/ggv414 Estimation of splitting functions from Earth s normal mode spectra using the neighbourhood algorithm Surya Pachhai, 1 Hrvoje Tkalčić 1 and Guy Masters 2 1 Research School of Earth Sciences, The Australian National University, Canberra, Australia. surya.pachhai@anu.edu.au 2 Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA, USA Accepted 2015 September 25. Received 2015 September 18; in original form 2015 February 4 SUMMARY The inverse problem for Earth structure from normal mode data is strongly non-linear and can be inherently non-unique. Traditionally, the inversion is linearized by taking partial derivatives of the complex spectra with respect to the model parameters (i.e. structure coefficients), and solved in an iterative fashion. This method requires that the earthquake source model is known. However, the release of energy in large earthquakes used for the analysis of Earth s normal modes is not simple. A point source approximation is often inadequate, and a more complete account of energy release at the source is required. In addition, many earthquakes are required for the solution to be insensitive to the initial constraints and regularization. In contrast to an iterative approach, the autoregressive linear inversion technique conveniently avoids the need for earthquake source parameters, but it also requires a number of events to achieve full convergence when a single event does not excite all singlets well. To build on previous improvements, we develop a technique to estimate structure coefficients (and consequently, the splitting functions) using a derivative-free parameter search, known as neighbourhood algorithm (NA). We implement an efficient forward method derived using the autoregresssion of receiver strips, and this allows us to search over a multiplicity of structure coefficients in a relatively short time. After demonstrating feasibility of the use of NA in synthetic cases, we apply it to observations of the inner core sensitive mode 13 S 2. The splitting function of this mode is dominated by spherical harmonic degree 2 axisymmetric structure and is consistent with the results obtained from the autoregressive linear inversion. The sensitivity analysis of multiple events confirms the importance of the Bolivia, 1994 earthquake. When this event is used in the analysis, as little as two events are sufficient to constrain the splitting functions of 13S 2 mode. Apart from not requiring the knowledge of earthquake source, the newly developed technique provides an approximate uncertainty measure of the structure coefficients and allows us to control the type of structure solved for, for example to establish if elastic structure is sufficient. Key words: Core, outer core and inner core; Surface waves and free oscillations; Seismic anisotropy; Seismic attenuation; Computational seismology. 1 INTRODUCTION Two classes of seismological observations traveltimes of body waves and frequencies of free oscillations of the Earth (normal modes), have been extensively used to study structure and dynamics of the Earth s deep interior. In particular, traveltime residuals of waves passing through the inner core with respect to predictions from a spherically symmetric Earth model, are mostly used to infer the structure/dynamics of the inner core (e.g. Morelli et al. 1986; Su & Dziewonski 1995; Tanaka & Hamaguchi 1997; Li& Cormier 2002; Niu&Wen2002; Ishii & Dziewonski 2003; Cao& Romanowicz 2004; Tkalčić 2010). The body wave traveltimes can be utilized to produce high-resolution images of the Earth s interior. However, sampling is always limited by an uneven distribution of the source and receiver locations. In contrast, the vibrational patterns of the Earth in response to large earthquakes (i.e. normal modes or free oscillations of the Earth) sample the large-scale structure of the Earth independently of the distribution of sources and receivers. The frequencies and decay rates of Earth s normal modes are used to constrain both velocity structure and dynamics of Earth s interior (e.g. Gilbert & Dziewonski 1975;Bulandet al. 1979; Geller & Stein 1979; Masters & Gilbert 1981; Masters et al. 1982; Giardini et al. 1987; Ritzwoller et al. 1988; Roult et al. 1990; Widmer et al. 1992; Laske & Masters 1999; Resovsky & Ritzwoller 1999a). C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society. 111

2 112 S. Pachhai, H. Tkalčić and G. Masters In addition, the normal mode data are sensitive to lateral density variation and have been used in the past to infer such variation in the mantle (Ishii & Tromp 1999; Resovsky & Trampert 2003). Therefore, Earth s normal modes are complementary data to constrain the Earth s long wavelength 3-D structure, and are often used in studies of the lowermost mantle and the core. In the presence of 3-D structure in the Earth s interior (e.g. lateral variations of elastic parameters, such as density and velocity of shear and compressional waves), the complex spectra (i.e. amplitude and phase) of a multiplet with the angular degree l is split into 2l + 1 resonances (singlets), each centred at a different frequency. Masters & Gilbert(1981) first observed the anomalous splitting of a mode sensitive to the inner core ( 10 S 2 ), and later confirmed the same phenomenon for a number of other modes sensitive to core structure (e.g. Ritzwoller et al. 1986; Woodhouse et al. 1986; Giardini et al. 1987, 1988). After these observations, normal mode splitting has been extensively used to estimate the 3-D structure of Earth s deep interior (e.g. Masters et al. 1982; Ritzwoller et al. 1986; Woodhouse et al. 1986; Giardini et al. 1988; Smith & Masters 1989; Widmer et al. 1992; Resovsky & Ritzwoller 1995, 1998; He & Tromp 1996; Durek & Romanowicz 1999; Masterset al. 2000a; Deuss et al. 2013). In the studies of normal modes of the Earth, the observed time series are transformed to complex spectra in the frequency domain, which serve as primary information to constrain the 3-D structure of the Earth s interior. From the inversion of complex spectra, lateral velocity and density variations can be obtained in two steps. In the first step, the splitting coefficients (also known as structure coefficients) that represent Earth s structure are recovered and can be visualized through a function known as the splitting function (Woodhouse & Giardini 1985). Mathematically, a splitting function σ (θ,φ) is the spherical harmonic expansion of structure coefficients cs t and represent the radial average of structure seen by a multiplet, σ (θ,φ) = 2l s s=0,even t= s c t s Y t s (θ,φ), (1) where Ys t (θ,φ) are fully normalized spherical harmonics of degree s and order t at a given geographical latitude θ and longitude φ. Here, we have assumed that the mode is isolated thus according to selection rules, the mode is sensitive only to structure of even degree. In the second step, 3-D lateral variations of compressional-wave velocity, shear wave velocity, and density are inverted for using linear relationships between the splitting coefficients and lateral structure (Li et al. 1991). In addition, other phenomena such as cylindrical anisotropy of the inner core can be inferred from the degree 2 and degree 4 zonal structure coefficients (i.e. c2 0 and c0 4 ) (Tromp 1995; Dahlen & Tromp 1998). In past studies different methods have been used to estimate the splitting functions based on the inversion of normal-mode data. Ritzwoller et al. (1986) and Widmer et al. (1992) applied a method called singlet stripping (Buland et al. 1979) to compute the splitting functions. According to singlet stripping, all 2l + 1 singlets within a multiplet are assumed to be sensitive only to axisymmetric structure. In such a case, inversion can constrain purely zonal lateral heterogeneity because of its mathematical requirement that the splitting matrix remains diagonal. Therefore, singlet stripping is a method strictly limited to the multiplets that are dominantly sensitive to the axisymmetric structure (i.e. only zero order). Later, an iterative spectral fitting of complex spectra was developed and widely used in the normal mode community (e.g. Ritzwoller et al. 1986, 1988; Giardini et al. 1987, 1988; He & Tromp 1996; Resovsky & Ritzwoller1998, Deusset al. 2013). This method does not assume axisymmetric structure of the Earth, but requires a model of the earthquake source. The assumption made in the global centroid moment tensor calculations is that the earthquake energy is emanated from a single point (Ekström et al. 2012). This may be a good assumption for small and medium size earthquakes. However, the point-source approximation can fail for the very large shallow earthquakes whose waveforms are most useful for harvesting information about the Earth from normal modes. Furthermore, the results of structure coefficient retrieval depend on initial constraints and regularization used to stabilize the inversion. The initial constraints as well as the convergence of the inversion algorithms are challenging to deal with because of the non-linear nature of the problem. With the increased volume of high-quality data from growing numbers of global seismic stations and large earthquakes, a two-step linear inversion approach has been developed (Masters et al. 2000a,b) and successfully applied to study both structure and dynamics of the inner core (Laske & Masters 1999; Masters et al. 2000b). In the two steps linear inversion, the source is eliminated through the means of the property of time series known as autoregression. Despite the increase in the quality of seismic data and the development of new techniques, there are still disagreements between the results of different studies, particularly in the splitting functions of core-sensitive modes (e.g. Li et al. 1991; He & Tromp 1996; Durek & Romanowicz 1999; Resovsky & Ritzwoller 1999b; Masters et al. 2000a). The uncertainties in the splitting functions have led to some controversy over the ability of modes to determine the 3-D density structure of the Earth s mantle (e.g. Ishii & Tromp 1999; Resovsky & Ritzwoller 1999b; Romanowicz 2001; Kuo & Romanowicz 2002). Similar arguments hold for the amplitude and radial variation of anisotropic structure in the inner core (e.g. Woodhouse et al. 1986; Tromp 1993; Durek & Romanowicz 1999; Ishii et al. 2002). This could be due to a non-uniqueness that is omnipresent in the geophysical inverse problems, propagation of data errors to the estimated Earth structure (Snieder 1998), limitations of the methods, and the different regularizations used to stabilize the inversion. For example, Durek & Romanowicz (1999) pointed out that the splitting functions obtained from the inversion are sensitive to the initial model used in the inversion because of the non-uniqueness of the problem. Similarly, Resovsky & Ritzwoller (1999b) also observe that the density model obtained from the inversion of normal mode data depends on prior constraints. Beghein & Trampert (2003) demonstrated that the controversy in inner-core anisotropy is due to regularization, which is used to stabilize the inversion. To remove the effect of regularization and dependence on the single solution, an ensemble of solutions can be collected from the sampling of parameters using a non-linear inversion technique called neighbourhood algorithm (NA; Sambridge 1999a). The example of its application in this field includes an inversion for lateral variation of velocity (Beghein et al. 2002), density in the mantle (Resovsky & Trampert 2003) and anisotropy in the inner core (Beghein & Trampert 2003). A common denominator in these studies is that the previously published structure coefficients were used. However, the robustness of the structure coefficients measured from the inversion of normal mode spectra using traditional methods is still questionable. The quality and uniqueness of the structure coefficients can have a significant effect on the robust measurements of lateral density and velocity variation in the Earth s interior.

3 Estimation of splitting functions using NA 113 Figure 1. (a) Sensitivity of P-wave velocity (solid) and S-wave velocity (dashed) for 13 S 2 mode used in this study. Comparison between the observed (solid lines) and predicted (dashed lines) phase spectra (top) and amplitude spectra (bottom) of the 1994 Bolivia earthquake for inner core sensitive mode 13 S 2 recorded at (b) KEV (c) ATD and (d) COL seismic stations shown on the map. The spectra are predicted using the receiver strips obtained from linear inversion. Vertical dash lines represent the degenerate frequency of 13 S 2 mode. (e) Station locations (triangles) for the 1994 Bolivia earthquake are also shown. In order to address these challenges, here we present another application of a derivative-free, direct search NA in the measurement process itself, namely to estimate the structure coefficients and their uncertainties. The algorithm takes advantage of all the information in the previous iteration and focuses search on the regions of the parameter space (i.e. structure coefficients) with minimum misfit. To narrow down the search space during the course of iteration, NA utilizes a geometrical concept based on Voronoi cells (Voronoi 1908) and requires two tuning parameters: the number of new models generated in each iteration, and the number of best models with the lower misfit selected in each iteration. After demonstrating the applicability of NA to estimate the splitting functions through a synthetic test, we present its application to 13 S 2, a relatively welldocumented normal mode that is sensitive to the inner core of the Earth (Fig. 1a). The major advantage of the NA seems to lie in its ability to find an optimal solution for fewer structure coefficients in a short time without the need for regularization and an earthquake source model in the inversion. In addition to finding the optimal solutions, the NA can explore a complete model space. Consequently, the ensemble of solutions can be extracted to calculate an approximate uncertainty of the parameters (Sambridge 1999b). 2 THEORY AND METHOD In order to implement NA, several ingredients are required: a forward model, a misfit function, and data. The misfit function is used to rank the models based on the misfit between the predictions and the observed data, the forward model is used to compute the model prediction from a given set of physical parameters (structure coefficients of a multiplet). As we work in the frequency domain, the complex spectra of the Earth s normal modes are the primary information to estimate the 3-D structure of the Earth s interior (i.e. structure coefficients). However, the complex spectra of the multiplet computed from seismic recordings around the globe cannot be straightforwardly used as data in our inversion. This is because the spectra of a multiplet at any seismic station do not only depend on the Earth s internal structure seen by that multiplet but also on the source and the station location. Figs 1(b) (d) show examples of amplitude and phase spectra that depend on the location of the stations around the globe (Fig. 1e). Figs 1(b) (d) also show the predicted spectra computed from the receiver strips using our forward model (eq. 2), which is described in the Section 2.1. The earthquake source model is unknown and can be complicated for large earthquakes. The structure coefficients (ultimately Earth s structure) are therefore estimated in a number of steps starting from these observed complex spectra. In the first step, we remove the station location information from the complex spectra using a method called receiver stripping (Masters et al. 2000a,b). According to this method, spectra of a multiplet with angular degree l from all available stations around the globe are reduced to a set of 2l + 1 spectra known as receiver strips (Fig. 3). These receiver strips are termed data in our inversion. We then use the autoregressive property of the receiver strips to remove the source information [for more details on receiver strips and autoregression, we refer readers to the original work of Masters et al. (2000a,b)]. According to this property, data at current time can be predicted by the data at the previous time lagged by δt. In the following section, we briefly summarize the method to formulate the forward model and

4 114 S. Pachhai, H. Tkalčić and G. Masters prepare the data (i.e. receiver strips) in order to estimate the structure coefficients. 2.1 Forward problem According to the isolated mode approximation (Woodhouse & Girnius 1982), the displacement seismogram u(t) for a specific mode of vibration at any particular station j can be expressed as: u j (t) = l R jm exp [i (H mm + Iω 0 ) t]a m (0), m= l u(t) = R exp[i (H + Iω 0 ) t] a(0) = R d(ω,t), (2) where d(ω,t) = exp[i(h + Iω 0 )t], R is the receiver vector matrix. Each row of R is a vector of 2l + 1 elements and represents a product of the displacement eigenfunction at the Earth s surface and spherical harmonics that describes the motion of a particular mode of vibration at an individual station (Dahlen & Tromp 1998). For the angular degree l, the azimuthal orders m and m span values from l to l (i.e. 2l + 1intotal).ω 0 is the degenerate frequency of a multiplet. I is an identity matrix and a(0) is the source excitation vector consisting of 2l + 1 elements. H is the splitting matrix of dimension (2l + 1) (2l + 1) for an isolated multiplet, and includes information about the rotation, ellipticity and 3-D structure. It is expressed as: H mm = ω 0 ( a + bm + cm 2 ) δ mm + ω 0 s=0 seven s t= s or γ mm t ls c t s. (3) The first term on the right-hand side includes the effect of Earth s rotation and ellipticity represented by a, b and c (Dahlen 1968), while the second term incorporates the effect of 3-D structure. If the splitting matrix is diagonal then the Earth structure is axisymmetric (e.g. rotation/ellipticity). As the rotation and ellipticity of the Earth are well determined, the splitting matrix can be readily corrected for these effects. γ mm t ls is the Gaunt integral and it is represented by the integral over three fully normalized spherical harmonics (e.g. Edmonds 1960; Woodhouse & Dahlen 1978). Using the convention of Edmonds (1960), the Gaunt integral can also be expressed in terms of the Wigner 3- j symbols and are readily computed. cs t are the structure coefficients that depend linearly upon the Earth s internal heterogeneity of the harmonic degree s and order t (e.g. Masters et al. 1982; Giardini et al. 1987). Since we consider only isolated modes in our inversion, the mode is sensitive to only even-order structure up to a harmonic degree s = 2l (eq. 3). Using the complex spectra obtained from the displacement seismograms at seismic stations around the globe, the receiver strips d(ω,t) can be derived by multiplying eq. (2) with the inverse of receiver vector matrix and expressed as; d (ω,t) = R 1 u(ω,t). (4) This equation reduces all the observed spectra for an event to 2l + 1 spectra known as receiver strips. This approach has two major advantages: selection of events to study the Earth interior and identification of problematic stations (Masters et al. 2000a). An event with poor excitation of receiver strips can be discarded in the analysis. Additionally, the complex spectra for all stations u(t) are predicted by substituting receiver strips obtained from eq. 4 into eq. (2). The fit between the predicted and measured amplitude/phase spectra can be used to remove noisy and problematic station spectra. Examples of fit between observed and predicted data using receiver strips are shown in Figs 1(b) (d) and supplementary material (Figs S1a and b). The spectra shown in Fig. 1 are considered to compute the receiver strips as their fit are good. In contrast, an example of a poor fit is shown in Fig. S1 resulting in noisy receiver strips, particularly for m = 1 (Fig. S1c). When such spectra are discarded in the inversion, the receiver strips become stable and signal-to-noise ratio is quite high. The same procedure was applied for all other events. The examples of receiver strips of 13 S 2 mode for four earthquakes (1994 Bolivia, 1994 Fiji, 1995 Chile and 1996 Flores) are shown in Fig. 2. In these examples, each receiver strip contains a single resonant frequency peak, which suggests that the structure sensed by this mode is axisymmetric. This means that the splitting matrix is dominated by the zonal structure coefficients (i.e. t = 0). This was demonstrated by Ritzwoller et al. (1986) through synthetic experiments. These receiver strips still contain information about the earthquake source. To eliminate the earthquake source information from the receiver strips, a small lag time δt s is added in eq. (4) and the equation is then solved using an autoregressive property of the receiver strips (Masters et al. 2000a): d (ω,t + δt) = P(δt) d(ω,t), (5) where d(ω,t + δt) are the receiver strips for the time lag δt. The dimension of d(ω,t + δt) is(2l + 1) number of frequency points. Similarly, P(δt) is the propagator matrix of dimension (2l + 1) (2l + 1) and is given by: P (δt) = exp[i (H + Iω 0 ) δt]. (6) Figure 2. Receiver strips for 13 S 2 mode computed from (a) the 1994 Bolivia, (b) the 1994 Fiji Islands, (c) the 1995 Chile and (d) the 1996 Flores events. The amplitudes of individual singlet are normalized by its own maximum amplitude and arranged from m = 2 to 2 (from back to front).

5 Estimation of splitting functions using NA 115 We take the transpose of eq. (5) so that the number of frequency data points becomes equivalent to the number of rows in eq. (5): d T (ω,t + δt) = d T (ω,t) P T (δt), This leads to D n+1 = D n P T (δt), (7) where D n and D n+1 represent the receiver strips with n frequency points in each column of 2l + 1 singlets for the zero and δt time lag, respectively. The phase differences between the n and n + 1 lags in eq. (7) are mainly due to the degenerate frequency of the mode which is much larger in amplitude than the elements of H. Therefore, eq. (7) is multiplied by e iω d δt on both sides and the degenerate frequency in eq. (6) is replaced by ω 0 ω d.where,ω d is an estimate of degenerate frequency. Masters et al. (2000a,b) solve eq. (7) to compute the propagator matrix using the receiver strips for different time lags. The splitting matrix H is then obtained by computing the eigenvalue decomposition of the propagator matrix P(δt). However, at this step we apply the derivative free search approach (i.e. NA) to estimate the structure coefficients. In order to take advantage of NA, we compute the propagator matrix (eq. 6) using the splitting matrix computed from eq. (3) for given model parameters (i.e. splitting coefficients). The receiver strips for a time lag δt are predicted from the combination of receiver strips at zero time lag and the propagator matrix (i.e. right-hand side of eq. 7). In our sensitivity tests, the splitting function coefficients are taken from published material; synthetic waveforms are constructed from eq. (3) and eq. (2) (with and without added noise), and it is then tested how well the splitting coefficients are recovered by NA. For the real waveform data, the splitting coefficients are directly inverted. 2.2 Misfit function The NA does not require assumptions about the data noise and the nature of the misfit function (Sambridge 1999a). Instead, relative values of the misfit between the model predictions and observed data are used to rank the order of models. Therefore, NA allows different choices of misfit functions to control the influence of data in the search process. In the case of an isolated mode and N events, the dataset includes 2l + 1 complex spectra with N n frequency points in a given frequency window. Therefore, we define the L2- norm misfit function used in our inversion to be E = 1 1 (2l + 1) N N j 2l+1 ω 2 i=1 ω 1 D obs D pred 2 dω. (8) Where ω 1 and ω 2 are the bounds of the frequency window in which the data are considered in the inversion. We call D obs as observed data and represent the receiver strips at the time δt multiplied by e iω d δt (i.e. e iω d δt D n+1 ). Similarly, D pred is the model prediction and represents the product of the receiver strips at zero time lag D n and the predicted propagator matrix P(δt) (i.e.d n P T (δt)). Here, the observed data are computed by moving the beginning and ending time of the time window through δt, which is 40 s. We performed the inversion for differently defined misfit functions (i.e. L1 and L2 norms of real and imaginary parts of phase and amplitude, as well as different possible combinations of these). We find that the absolute L2-norm of the complex spectra provides comparatively efficient sampling and faster convergence than other misfit functions. 2.3 Model parametrization As our formulation for the receiver strips assumes that the normal mode multiplets are isolated, there are (l + 1) (2l + 1) complex elastic splitting coefficients cs t (i.e. s = 0,2,2l and t = stos). Thus the inner core sensitive mode 13 S 2, with angular degree l = 2, has 15 complex elastic splitting coefficients. Because the elastic splitting matrix is Hermitian, the coefficients for negative order t can be obtained from the coefficients of positive order t (i.e. cs t = ( 1) t (cs t ),where represents the complex conjugate and t is the absolute value of t). Therefore, the 13 S 2 mode is parametrized with nine complex coefficients (i.e. 18 independent parameters including the real and imaginary parts). The coefficient for s = 0andt = 0 represents the perturbation to the degenerate frequency. The algorithm generates randomly distributed N models of elastic structure coefficients within the parameter bounds. Our parameter bounds span a wide frequency range namely from 37.0 to 37.0 μhz. For the imaginary part of cs 0, we fix the bounds to 0.0 without sampling, to satisfy their theoretical definitions. Forward modelling is performed for all randomly generated models to compute the propagator matrix and receiver strips. Subsequently, the misfit is computed between the observed and the predicted receiver strips and the best M models of structure coefficients with the lowest data misfit are determined. Subsequent iterations are repeated in the same way with the best M models of structure coefficients, and their Voronoi cells are updated in each iteration. The selection of tuning parameters M and N is important for efficient sampling and faster convergence, and is discussed in Section DATA PREPARATION In this study we consider isolated spheroidal modes of oscillation sensitive to the Earth s inner core. Only vertical components of long period seismograms are analysed since these have the best signalto-noise ratio. For each event, we examine time series consisting of up to 15 days of continuous recording obtained from global seismic networks following the origin time. Glitches, aftershocks, other small events and clipped segments are removed from the time series in the first instance. Instrument responses are deconvolved from all the recordings. For each time series, the first hr after the event origin time are discarded to remove the strongly attenuated and low Q-value modes sensitive to mantle structure, and the seismograms are then cut at different lengths depending on the mode of interest (we use approximately Q-cycles of the mode being analysed). The recordings are discarded if more than 50 per cent of the signal is found to be unusable. In normal mode analysis, the spectral resolution achieved depends on the choice of the taper. For example, a boxcar window gives the best resolution, but it leaks a significant amount of energy from the central lobe to side lobes. Here, we apply the Hanning taper, which broadens the central lobe of the spectral window but suppresses side lobes. The application of the Hanning taper is a good compromise between a visual separation of singlets and signal-to-noise ratio degradation in the frequency domain (Dahlen 1982) and leads to the optimal record length of Q cycles. The amplitude spectra of individual recordings for the selected mode are visualized in the frequency domain and all recordings with low signal-to-noise ratio are discarded. Examples of receiver strips

6 116 S. Pachhai, H. Tkalčić and G. Masters Table 1. Event and receiver information for earthquakes used in this study. Event ID Event name Event origin time Depth (km) Moment (dyne-cm) No. of stations M060994A Bolivia June M030994E Fiji March M073095A Chile July M061796A Flores June for four events are shown in Fig. 2, and event information is given in Table 1. This table also presents the number of stations considered in our analysis. We started with the analysis of the Bolivia 1994 earthquake, as this event was recorded by a number of high quality digital stations operating at that time. Next, we considered three more events that happened in consecutive years. We are particularly interested in the inner core sensitive mode 13 S 2 since it is well excited and isolated in the frequency spectrum, and all of the events excited this mode well. Among these events, the receiver strips obtained from the 1994 Bolivia earthquake are of the highest quality, amplitudes of the receiver strips 10 times higher than those for the Fiji event. Additionally, the receiver strips with m =±1 are strongly excited, while the strip with m = 0 is weakly excited. The receiver strips with m =±2 are most excited while the m = 0 strip is weakly excited for the Fiji event. The receiver strips for the Flores 1994 earthquake are comparatively weakly excited. We note that the receiver strips for both observed and synthetic waveforms are prepared in the same way. 4 RESULTS AND DISCUSSION 4.1 Inversion results for synthetic data To illustrate the feasibility of the NA technique, we present the results of inversion using synthetic waveforms. In our synthetic test, we consider an isolated inner-core sensitive mode 13 S 2. To compute the synthetic data, we first transform the previously published raw splitting coefficients A st and B st (He & Tromp 1996) to the splitting coefficients (cs t ) using the formulation described in Giardini et al. (1988): c t s = ( 1)t (2π) 1/2 (A st ib st ) for t > 0 c t s = (4π)1/2 A st for t = 0 (9) c t s = ( ) (2π)1/2 A s t + ib s t for t < 0. Where t is the absolute value of order t. We compute the splitting matrix(h) for the splitting coefficients computed in eq. (9). The splitting matrix is a part of eq. (2), and is needed to compute time series of up to 90 hours for 100 stations distributed around the globe. In this case, earthquake information from the Bolivia 1994 earthquake was used. The complex spectra for individual stations are computed after a Hanning window is applied for the same record length that was used for the real data. The receiver strips are computed from these synthetic complex spectra. Before we perform the inversion, we test the effect of noise on the receiver strips (both amplitude and phase). To examine the effect of noise on the receiver strips, Gaussian random noise with a standard deviation σ = 10 per cent of the maximum amplitude of the signal (i.e. noisy signal = signal + σ n,wheren is the Gaussian random noise) is added to the synthetic waveforms. The amplitude spectra (Fig. 3a) are fairly stable over the frequency window of the multiplet while the phase spectra (Fig. 3b) are highly affected by Figure 3. The effect of noise on (a) the amplitude and (b) the phase spectra of the receiver strips for 13 S 2 mode. The receiver strips are obtained from the synthetic data without noise (thick grey), with observed noise (red dashed line), and with 10 per cent uncorrelated Gaussian random noise (thin blue line). Note that synthetic data were computed for the Bolivia 1994 earthquake. noise, especially farther away from the degenerate frequency. Note that the same level of noise is added in all the time series. In contrast to the theoretical Gaussian random noise, realistic noise can be correlated due to several factors such as: frequency response of recording instrument, microseisms, atmospheric effects, tidal effects, etc. The assumption of uncorrelated noise in the observed data can be impractical and can significantly influence the parameter estimates. We therefore estimate noise from the observation with an appropriate frequency spectrum recorded by the seismic instrument. We call this type of noise the observed noise. Furthermore, observed noise is considered in the inversions of synthetic data presented in this paper. To estimate the observed noise, we request 100 hours of recording before an earthquake. Small earthquakes, glitches, instrument responses, and tidal effects are first removed from recordings. The observed noise is then added to the synthetic seismograms in the time domain and frequency spectra are computed for all stations distributed around the globe. Then the complex spectra are computed from the time series with and without noise. The linear inversion of the complex spectra is performed to estimate the receiver strips. Note that we use the real and imaginary parts of the complex spectra as our data in the inversion. The effect of noise in the real part of the receiver strips is shown in Fig. S2(a), but is barely visible in these spectra as the signal amplitudes are high. Therefore, residual errors between the strips with and without noise are computed. Fig. S2(b) shows the residual errors in the real part of the receiver strips when the Gaussian random noise is considered. Similarly, the residual errors in the real part of the receiver strips due to observed noise are shown

7 in Fig. S2(c). One of the obvious differences in Figs S2(b) and S2(c) is that the period of noise is greater and amplitudes smaller for the residuals obtained from the inversion of synthetic data combined with the observed noise in comparison to those containing Gaussian random noise. This is due to the correlation between time points in the case of observed noise. The imaginary part of receiver strips contains similar effects, and is not shown here. Fig. S2 further illustrates the correlation of the residual errors and suggests a strong correlation of residuals in the case of data with the observed noise (Fig. S2d). The effect of different level of noise in the receiver strips is analysed and summarized in Table S1. With the increased level of noise, the errors in the receiver strips increases. The receiver strips are computed with and without noise for two time lags. The phase spectra of the receiver strips with observed noise are shown in Fig. 4(a). Because of the time lag and degenerate frequency of the mode, the phase spectra computed after 40 s (i.e. by one sample) are shifted to the right. For the inversion, we define the frequency window ( mhz, Fig. 4a) where the phases remain stable after the noise is added (Fig. 3b). This is in agreement with the choice of the frequency range used in Masters et al. (2000a). The observed and predicted phase spectra for the best model (the model having the lowest misfit) are shown in Fig. 4(b). Amplitude spectra are not shown, as the amplitude and position of the peak do not change with the time lag. Green circles in Figs 4(c) and (d) represent the final recovered values for the real and imaginary parts of the structure coefficient. The recovered structure coefficients are also compared with the true values with uncertainties in Figs 4(e) (f). This figure illustrates that all structure coefficients are recovered well even after adding noise on the synthetic data. The effect of noise is only visible in degree-two zonal structure coefficient (C 20 ) and is overestimated by an insignificantly small value. An inversion was also performed without adding noise, which successfully recovered all the parameters and is omitted here. Furthermore, the splitting matrices are computed for the recovered splitting coefficients with and without noise and we show these results in Fig. 5. We find that the noise degrades the splitting matrix (i.e. central point with m and m = 0 slightly overestimated) but most of the features of the splitting matrix are well recovered. We further compute the difference between the true and the recovered splitting functions obtained from the noisy data (Fig. 5d). The inversion is performed for several time lags from 40 to 280 s. However, the level of misfit between the observed and the predicted data increases linearly with an increasing time lag (Fig. S2), which suggests that the inversion is insensitive to the lag time. Masters et al. (2000a) also derives the similar conclusion in the two-step inversion. Finally, the splitting functions are computed to visualize Earth structure (Fig. 6). The splitting function for the true value of the structure coefficients is shown in Fig. 6(b). The splitting functions for the best models of the structure coefficients recovered from the inversion for the synthetic data with and without added noise are shown in Figs 6(c) and (d). As expected, the addition of noise slightly degrades the splitting functions (due to an overestimation of degree-two zonal splitting coefficient), but all the patterns of splitting functions are well recovered. In the case of 13 S 2 mode, there are only nine complex numbers to represent the structure seen by this mode. However, the number of structure coefficients increases significantly with the increase of angular degree. To test the feasibility of our inversion approach for higher angular degrees, we synthesized the data for 8 S 5 mode that has little sensitivity to the inner core (Fig. 7a). In this case, there are 36 complex numbers to represent the structure sensed by this mode, but the available structure coefficients are truncated at degree six. Estimation of splitting functions using NA 117 Therefore, there are 15 complex numbers to represent the lateral variations of the structure. We compute the synthetic seismogram similar to the procedure adopted for 13 S 2 mode from the normal mode catalogue (He & Tromp 1996). The observed noise is added to all the synthetic seismograms. Then the linear inversion of the complex spectra is applied to compute the receiver strips, which isolates the receiver location effects and collapses all the spectra into 11 spectra (Fig. 7b). Receiver strips are also computed for another time lag with 40 s. The NA inversion is applied to this data and splitting coefficients recovered from the inversion are shown in Figs 7(c) (f). To illustrate the evolution of sampling, Figs 7(c) and (d) shows the samples of the splitting coefficients from a single run. These structure coefficients are compared with the true values of the structure coefficients with uncertainties (Figs 7e and f). This shows that the inversion recovers all the complex structure coefficients reasonably within the uncertainty limit provided by He & Tromp (1996). The splitting matrices computed for the true and recovered values of the structure coefficients with and without the addition of noise are shown in Figs S4(a) (c). The difference between the recovered and true values is also shown in this figure. The difference is insignificant, suggesting that the inversion performs well for high angular degree as well. The comparison of the inversion results between 13 S 2 and 8 S 5 modes demonstrates that the mode with higher angular degree requires more intensive sampling and takes a longer time to converge but is still successful. 4.2 Inversion results for observed data After demonstrating the feasibility of NA through synthetic experiments, we perform the inversion for observed inner core sensitive mode 13 S 2. Since this mode attenuates slowly, we start 10 hours after the event origin time to suppress neighbouring low-q modes. A detailed description about data pre-processing is presented in Section 3. The receiver strips for four events are computed and showninfig. 2. Since all the singlets are not excited equally well, we have to use more than one event to achieve well-constrained 3-D structure seen by this mode. Due to the fact that it is well documented, we first used the 1994 Bolivia earthquake to compute the splitting coefficients. For this event, zonal structure coefficients for both degree 2 and 4 are dominant. The 1994 Fiji earthquake is then added to the Bolivia event in a second inversion. In this case, the degree 2 structure coefficients become dominant. Then the Chile 1995 earthquake is also considered in the inversion to verify the stability of the obtained structure. Each time multiple events are considered in the inversion, a form of weighting of events can be applied, however, such weighting can highly influence the inversion results. For example, considering the moment magnitude of the event gives roughly equal weights but can down-weight the receiver strips that contain high signal-to-noise ratio. This is because not all events excite all the receiver strips equally well. If large events do not excite a singlet, a higher weight has to be given to the smaller events with well-excited singlets. Therefore, we do not apply such weighting in our inversion. In contrast, the event with weakly excitation of receiver strips is not considered in the inversion (e.g. the Flores 1996 earthquake). The structure coefficients obtained in three inversions using different events are compared in Fig. S5. This shows that the structure coefficients remain stable and do not change significantly with the addition of another earthquake. An example of sampling of the structure coefficients (cs t ) when two events are used is shown separately in Figs 8(a) and (b). In this case, Bolivia 1994 and Fiji

8 118 S. Pachhai, H. Tkalčić and G. Masters Figure 4. (a) Phase spectra of the synthetic receiver strips of 0 s time lag (red line) and 40 s time lag (blue line) for 13 S 2 mode. The noisy synthetic data are utilized to compute the receiver strips, while noisy synthetic data are prepared by adding observed noise on the synthetic seismogram prepared for the Bolivia earthquake. The frequency window used in the inversion is shaded by grey colour in the background. (b) Phase spectra of the receiver strips used in the inversion (solid line) and recovered from the inversion (dashed line). Amplitude spectra are not shown here, as they do not change with time lag. (c) Sampling for the real and imaginary parts of the structure coefficients. Green circles represent final values used to compute the splitting matrices/functions while black solid circles represent the true values of the structure coefficients. The colour bars represent the density of the model. The light yellow colour towards the end of the axis bounds represent lower density of models while the dark yellow colour towards the centre of the axis represent the higher density of models. The close up of the true (asterisks) and recovered (diamonds) values for the (e) real and (f) imaginary parts of the structure coefficients for 13 S 2 mode.

9 Estimation of splitting functions using NA 119 Figure 5. Real (top row) and imaginary (bottom row) parts of the elastic splitting matrix for 13 S 2 mode: (a) used to compute the synthetic data, (b) recovered from the inversion without noise and (c) with observed noise added on the synthetic data. (d) Difference between (c) and (a). Figure 6. (a) Distribution of the stations around the globe at which synthetic seismograms are computed for the Bolivia earthquake. Elastic splitting functions for 13 S 2 mode (b) used to compute the synthetic spectra, recovered from the inversion (c) without adding noise in the synthetic data (d) after adding observed noise in the synthetic data earthquakes are utilized. Green circles represent the final models considered to compute the splitting matrix and splitting functions. The structure coefficients obtained from our inversion are further compared with the previously published structure coefficients of He & Tromp (1996) and Deuss et al. (2013). The real parts of the structure coefficients closely agree with those from Deuss et al. (2013) while the imaginary parts do not agree well with the published results. In this figure, we also present approximate

10 120 S. Pachhai, H. Tkalčić and G. Masters Figure 7. (a) Sensitivity kernel of P-wave (solid line) and S-wave (dashed line) velocities for 8 S 5 mode. (b) The receiver strips for 8 S 5 mode computed from the synthetic time series prepared from the normal mode catalogue of He & Tromp (1996) and adding observed noise. Sampling of the (c) real and (d) imaginary parts of the structure coefficients. The colour bars represent the density of the model. The light yellow colour towards the end of the axis bounds represent lower density of models while the dark yellow colour towards the centre of the axis represent the higher density of models. The green circles represent the recovered values of the structure coefficients. True values are not shown because they correspond to the recovered values. The close up of the true (asterisks) and recovered (diamonds) values (asterisks) of the (e) real and (f) imaginary parts of the structure coefficients for 13 S 2 mode. The structure coefficients up to degree 6 were sampled. The error bars are from He & Tromp (1996).

11 Estimation of splitting functions using NA 121 Figure 8. Parameter sampling for the (a) real and (b) imaginary parts of the structure coefficients for 13 S 2 mode when the observed data from Bolivia 1994 and Fiji 1994 earthquakes are used in the inversion. The green circles are the final models of the structure coefficients considered to compute the splitting matrices/functions. The yellow/dark colour bars represent the density of the model. Comparison of the (c) real and (d) imaginary parts of the structure coefficients shown in (a) and (b) with the previously published coefficients. Diamonds are from He & Tromp (1996), circles are from Deuss et al. (2013), and asterisks are from the inversion of data from Bolivia 1994 and Fiji 1994 events. Vertical lines represent the ±1 standard deviation error bars. Error bars in this study are obtained from the ensemble models selected based on the procedure shown in Fig. S6. (e) Real and (f) imaginary parts of the elastic splitting matrix computed for structure coefficients shown as asterisks in (c) and (d).

12 122 S. Pachhai, H. Tkalčić and G. Masters uncertainties for Cst coefficients. To obtain approximate uncertainties, we use thousands of models obtained in the inversion when misfit does not further decrease as shown in Fig. S6. Note that the first part of the NA (applied in this paper) is an optimization and cannot give the accurate uncertainties. More rigorous parameter values and their uncertainties could be estimated applying the second part of the neighbourhood algorithm (i.e. the appraisal part, Sambridge 1999b). However, an assumption about the distribution of the noise in the complex spectra is required to implement appraisal part. An improper noise assumption can mislead the parameter estimation and uncertainties. As this understanding is beyond the scope of the work presented here, we do not apply the appraisal part in this paper. The real and imaginary parts of the splitting matrix computed from these coefficients are visualized in Figs 8(e) and (f). This shows that the diagonal elements are dominant for real part of the elastic splitting matrix. We apply a sensitivity analysis to examine the stability of structure coefficients to the number of events used in the inversion. From this test, we find that two events can constrain the structure for this specific mode. The splitting functions obtained from the sensitivity test for the combination of single event and pairs of events are shown in Figs S7 and S8. The structure coefficients obtained from this test are shown in Fig. 9 with error bars. Fig. 9 illustrates that the structure coefficients computed from a single event are highly erroneous and are not sufficient to adequately constrain the structure for 13 S 2 mode. However, adding one more event in the analysis provides more satisfactory results. It is clear that the addition of a third event does not have any further significant effect on the estimation of these coefficients. The structure coefficients obtained from this test are visualized in terms of splitting functions in Fig. 10. Fig. 10(a) illustrates that the splitting functions are dominated by both degree-2 and degree-4 structure. With addition of Figure 9. Sensitivity analysis for varying number of events to illustrate the stability of structure coefficients for 13 S 2 mode. Mean value of the real (a) and imaginary (b) parts of the structure coefficients obtained from the inversion using one event, two events and three events. Vertical lines represent the ±1 standard deviation error bars. Error bars are obtained from the sensitivity analysis. Figure 10. Resulting observed splitting functions computed for varying number of events for 13 S 2 mode: (a) one event, (b) two events and (c) three events. The splitting functions in the case of different events and their different combinations are shown in Figs S7 and S8. (d) Resulting splitting functions from Laske & Masters (1999). Note that Laske & Masters (1999) used nine events.

13 another event, the splitting functions become dominated by degree- 2 structure (Fig.10b). The splitting functions obtained by using two and three events (Fig. 10b c) do not differ significantly. This resulting image resembles the previously published results (Fig. 10d, Laske & Masters 1999). Note that Laske & Masters (1999) use nine events to obtain these splitting functions. Estimation of splitting functions using NA Discussion This paper demonstrates the feasibility of using the parameter optimization (Neighbourhood Algorithm) through both synthetic and observed data examples. The splitting functions obtained from the inversion closely match with those retrieved through the two-step linear inversion (Laske & Masters 1999; Masterset al. 2000a), but somewhat disagree with that obtained from the linearized inversion (He & Tromp 1996). This could be due to the fact that the first step (the autoregressive part of the analysis) is common between our inversion approach and the two-step linear inversion. Additionally, the source information is eliminated through the autoregression of receiver strips in both approaches. However, the linearized inversion is regularized to stabilize the inversion. Consequently, this can generate inconsistent results among different studies based on the choice of the regularization parameters (e.g. He & Tromp 1996; Deuss et al. 2013). Furthermore, from the sensitivity analysis of our inversion results it appears that only two events are sufficient to constrain the inner core structure sensed by 13 S 2 mode, although this is only true when the Bolivia 1994 earthquake is one of the events in the analysis. According to Masters et al. (2000a), the splitting matrices do not change significantly after adding the third event in the inversion. However, they could not constrain the structure for 13 S 2 mode using a combination of any two events. In the case of linearized regressions, it is well established (e.g. figs 8 10 of Resovsky & Ritzwoller 1998) that improvements in the resolution of deep mantle splitting functions resulting from the addition of data from each of the 2 or 3 best signal-to-noise events one by one can be matched by the addition of other events only if increasingly many events are added each time. In contrast to the linearized iterative spectral fitting (e.g. He & Tromp 1996; Deuss et al. 2013) and the two-step linear inversion (Laske & Masters 1999; Masterset al. 2000a), the method introduced here utilizes a derivative-free parameter search algorithm that has an ability to explore the entire parameter space. The splitting coefficients of He & Tromp (1996) and Deuss et al. (2013) are in many cases very inconsistent with one another given their reported uncertainties. Such inconsistency is characteristic of linearized regression solutions of ill-posed inverse problems for which regularization distorts the model space, creating the illusion of stable solutions in different (depending on the regularization) parts of what really is a broad class of best-fitting models. As this is a search algorithm, there is no need to perform the matrix inversion and stabilization of the inversion. A particular advantage of this method is that the search is guided by the misfit function and a tuning of only two parameters is needed, i.e. the number of models generated in each iteration (N) and the number of best models considered for the next iteration (M). Therefore the search concentrates on those portions of the model space (the splitting coefficients) where the misfit is low. However, the tuning of these two parameters (N and M) is not straightforward and it is problem-specific. A higher ratio of N to M narrows down the search, while a smaller ratio allows easier exploration of the parameter space. We perform the inversion with different combinations of these parameters (from 2 to 30 for both N and M) and find that the NA converges quickly to the optimal model when N = 25 and M = 2. These tests were performed for the inversion of synthetic data and the parameters were chosen based on the faster convergence towards the true value and the computational time. The computational time increases with larger values for both M and N become larger and larger. In our inversion, we start with a randomly generated model within the parameter bounds and a large M so that NA can explore a wide parameter space. The inversion is then restarted from the predetermined models (best models obtained from the previous run) with lower value of M to achieve a faster convergence and increase the computational efficiency. The process is repeated until the misfit does not change. Examples of the effect of different values of M and N on convergence is presented in Fig. S9. Fig. S9 indicates that the inversion takes a long time to find the true parameter value when a large value of M is used. This is because a large value of M explores wider regions of the parameter space. Apart from tuning of the two control parameters, the computational time and the number of samples required to fill the parameter space become crucial for higher dimension of the parameter space (i.e. number of unknowns). According to Sambridge (2001), at least 2 d number of samples are required for d unknowns to fill the complete parameter space. This means that at least samples are required for the case with 18 elastic structure coefficients. If we run the algorithm for 3000 iterations, it takes 30 minutes on a single processor (2.9 GHz Intel Core i7). If we increase the number of iterations to , more than 10 hours of CPU time is required. We attempted an inversion including the anelastic structure coefficients which represent intrinsic attenuation rather than scattering attenuation because normal modes are insensitive to small-scale features. In this case, 4.2 billion samples with 0.1 billion iterations are required to fill the space of 36 structure coefficients (18 for elastic and 18 for anelastic). This demands significantly higher computational cost and limits the simultaneous inversion of elastic and anelastic coefficients. We perform a synthetic experiment in which we treat Earth structure as mathematically elastic+anelastic, but fix all anelastic contribution to zero in the first step of the inversion. In the second step, we centre and narrow down the parameter bounds for the elastic structure coefficients to the values determined in the first step, and sample only the anelastic contribution. In that case, NA correctly identifies all the anelastic structure coefficients as zero. For the observed data, it was found that the misfit decreases after the second step of the inversion because of the increased complexity of the model. We also perform inversions varying the record length of the time series and find that an increase in record length reduces the convergence time since the longer record provides better resolution of the spectra. However, one needs to be cautious about the trade-off between the record length and the noise of the spectra. Additionally, we perform inversions varying the time lag and find that the minimum misfit increases with an increase in the time lag (Fig. S3) but the inversion fully recovers the splitting coefficients. From both the synthetic experiments and the inversion of the observed data, we conclude that the quality of the receiver strips (by quality, we mean high signal-to-noise ratio) is crucial for robust measurement of the splitting coefficients. In our inversion scheme, we consider the data from a narrow frequency window over which the signal is stable as suggested by our synthetic experiment. In this paper, we presented application of NA to only isolated normal modes with different angular degrees, which provide the