# A full-envelope based approach to estimate absorption and scattering attenuation from seismic ambient noise

Malcon Celorio, Laurent Stehly, Emmanuel Chaljub

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Malcon Celorio, Laurent Stehly, Emmanuel Chaljub. A full-envelope based approach to estimate absorption and scattering attenuation from seismic ambient noise. Geophysical Journal International, 2025, 242 (2), 10.1093/gji/ggaf209. hal-05173599

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# A full-envelope based approach to estimate absorption and scattering attenuation from seismic ambient noise

Malcon Celorio<sup>1,2</sup> and Laurent Stehly<sup>3</sup> and Emmanuel Chaljub<sup>3</sup>

Accepted 2025 June 5. Received 2025 May 17; in original form 2025 February 24

## SUMMARY

We present a novel technique for the characterization of small-scale absorption and scattering properties from cross-correlation functions (CCFs) of seismic ambient noise. We use a continuous data set recorded over four years at the Piton de la Fournaise volcano. Attenuation properties are estimated in the frequency range from 0.5 to 4 Hz, by comparing energy envelopes from CCFs with those from the radiative transfer theory (RTT) and the diffusion approximation. Our technique exploits the different propagation regimes observed at long and short propagation distances, which allows us to quantify attenuation properties in two stages: first, we measure absorption from short propagation distances including auto-correlation functions (source-receiver collocated case) to profit from the long coda durations. This set of estimates also allows to observe spatial variation of absorption either from RTT or the diffusion approximation. Once absorption is estimated, we proceed to characterize scattering from long propagation distances where scattering effects dominate absorption. Our inversion strategy to characterize scattering is called the ‘ball-diff’ ratio because we propose to use the ratio of the integrated energies contained in the ballistic and early diffuse regimes. This technique can considerably reduce the effect of the uneven distribution of noise sources. Finally, in order to validate our method, the scattering and absorption properties estimated from CCFs of seismic noise are compared with those from earthquake data, for which we used magnitudes between 1.5 and 2.5. Good agreement was found between the estimates of these two approaches.

**Key words:** Seismic attenuation; Scattering; Absorption; Seismic noise.

## 1 INTRODUCTION

Understanding the attenuation of seismic waves is essential for accurately assessing seismic hazard and investigating small-scale structures in the Earth. Seismic attenuation is controlled by scattering, which is the interaction of seismic waves with small-scale heterogeneities, and absorption, caused by the anelasticity of geological materials or by the presence fluids. Distinguishing these two mechanisms has been possible thanks to the use of phenomenological models such as the radiative transfer theory (RTT; Wu 1985; Margerin et al. 1998) which predicts the transport of energy in media with random inhomogeneities. Reliable estimates of attenuation properties have been performed by comparing observed energy envelopes from seismic data with those predicted by RTT (Fehler et al. 1992; Hoshiba 1993; Carcole & Sato 2010; Ugalde et al. 2010; Eulenfeld & Wegler 2016; Calvet et al. 2023). One of the most widely used inversion technique is the multiple lapse time window analysis (MLTWA; Fehler et al. 1992) which analyses the

space evolution of seismic energy in three time windows representing different propagation regimes from ballistic to multiply scattered energy. Despite its extensive application, MLTWA assumes spatial homogeneity in scattering and absorption, thus, making difficult to interpret the uncertainties of attenuation estimates. An alternative approach is the use of full envelope based methods (e.g. Q<sub>open</sub>) (Sens-Schönfelder & Wegler 2006; Eulenfeld & Wegler 2017; Raniean & Konstantinou 2020; van Laaten et al. 2021), which take into account the whole shape of the envelope for each source-receiver path and makes possible to invert different physical parameters such as source characteristics and site effects. In addition, inversion tools like MuRAT (De Siena et al. 2014, 2016) have been proposed to separately constrain scattering and absorption attenuation at multiple frequencies. MuRAT estimates scattering mechanism from the peak delay of body waves and derives absorption attenuation from the late coda decay. It has been successfully applied in both tectonic and volcanic settings and provides spatially resolved attenuation models.

<footnote n="1"><sup>1</sup>BRGM: French Geological Survey, 45100 Orléans, France. E-mail: malcon0522@hotmail.com</footnote>

<footnote n="2"><sup>2</sup>Grupo Quimbaya, CEIFI, Observatorio Sismológico Univ. Quindio, 630004 Armenia, Colombia</footnote>

<footnote n="3"><sup>3</sup>Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, Univ. Gustave Eiffel, ISTerre, 38000 Grenoble, France</footnote>
Generally, seismic attenuation studies rely on earthquake or active shot data, which are limited to regions with high seismicity and good instrumentation. Furthermore, in the context of small-scale areas (e.g. volcanoes and sedimentary basins), mapping attenuation properties becomes difficult as the sensitivity of earthquake coda might exceed the typical size of the region of interest.

In the last decades, the use of seismic noise has shown to be a powerful approach for studying the earth's interior (Shapiro & Campillo 2004; Stehly et al. 2008). In the context of attenuation studies, while diverse works have focused on the ballistic waves to constrain 'anelastic properties' (Prieto et al. 2009; Zhang & Yang 2013; Liu et al. 2015), some other have exploited the late coda part (Soergel et al. 2019; Ranjan & Stehly 2024). In these studies, the total attenuation is measured without distinguishing the contribution of absorption and scattering. In their pioneering work, Hirose et al. (2019) first utilized noise cross-correlation functions (CCFs) to separate absorption and scattering at Sakurajima volcano. These authors proposed a technique based on MLTWA, which was then applied and extended in subsequent works (van Dinther et al. 2020; Hirose et al. 2022). For example van Dinther et al. (2020) used the MLTWA technique to study scattering properties in the North Anatolian Fault zone, while Hirose et al. (2022) used the technique proposed by Hirose et al. (2019) to constrain attenuation at 18 volcanoes in Japan. Although these studies provide relatively precise attenuation estimates, the fact of using MLTWA-based techniques does not allow to interpret unambiguously the uncertainties of scattering and absorption measurements. In particular, the potential influence of the uneven distribution of noise sources can not be accounted for. This last point is critical because previous theoretical works have emphasized the difficulties to retrieve seismic attenuation from seismic ambient noise, due to the non-uniform distribution of noise sources (Tsai 2011; Hanasoge 2012).

Here, we introduce a novel technique for separating absorption and scattering properties from noise CCFs. Our study is carried out at the Piton de la Fournaise, one of the most active and well-instrumented volcanoes in the world (Duputel et al. 2009; Clarke et al. 2013; Obermann et al. 2013; Rivet et al. 2014; Sens-Schönfelder et al. 2014; Mordret et al. 2015). We focus on three frequency bands ([0.5–1], [1–2] and [2–4] Hz) and distances up to 21 km. Absorption measurements are performed at short distances and we exploit long coda durations, whereas scattering is estimated at large distances and short lapse times. We introduce a ratio of ballistic to early diffusive energies to reduce the effect of the uneven distribution of noise sources, enhancing the accuracy of scattering estimates. Our technique is validated by comparing noise-based attenuation estimates with those from earthquake data, ensuring robustness and reliability.

The organization of this article is as follows: in Section 2 we present the cross-correlation functions and corresponding processing for the obtention of the observed energy envelopes. In Section 3, we show the synthetic energy envelopes and their sensitivity to scattering and absorption. Section 4 introduces the inversion techniques for attenuation parameters. In Section 5, we validate our method through comparisons with earthquake data. Finally, some conclusions are presented in Section 6.

# 2 DATA AND PROCESSING

## 2.1 Study area and data set

We use 4 yr of continuous vertical noise records at 17 stations that are part of the Piton de la Fournaise (PF) network (OVPF & IPGP

2008). This seismic array has been continuously monitoring the activity of the Piton de la Fournaise volcano at la Réunion Island since 1980 (Fig. 1a). In this study we use data from January 2018 to December 2021 and the maximum aperture is about 21 km. Note that these stations have different data availability where the minimum number of recorded days is 1000. In Fig. 1(b), we show the location of the seismic stations (blue triangles). Labelled stations are used to illustrate estimation and validation procedures of absorption and scattering properties in the following sections.

Seismic data are downloaded, pre-processed, and the correlations are computed using pycorr, a suite of python codes to process and analyse seismic noise data (Boué & Stehly 2022). Before computing the correlations, the noise records are pre-processed in the following way: data are down-sampled to 20 Hz and corrected for the instrument response. Then, transient amplitudes are removed by applying the “comb-filter” technique (Soergel et al. 2019). To this end, the noise signal is first filtered into four period bands: [0.125–0.25], [0.25–0.5], [0.5–1] and [1–3] s. Each of the filtered signals is normalized by its corresponding envelope to down-weight transient events. Finally, the four filtered and normalized signals are stacked, resulting in a broad-band signal in which transient amplitudes are suppressed while the ambient noise is preserved. Note that different pre-processing techniques such as one-bit normalization is avoided in this study because its nonlinear nature do not allow to retain relative amplitudes between stations, which is essential for our analysis.

## 2.2 Noise correlation functions and observed energy envelopes

We cross-correlate vertical-to-vertical noise records with a sliding window of four hours (4h). Each 4h correlations are summed to obtain the final correlation averaged over a total number of days between 1000 and 1461, depending on the pair of stations. The CCFs are filtered in three frequency ranges ([0.5–1], [1–2] and [2–4] Hz) with a zero-phase 4 poles Butterworth filter. Lower frequencies are excluded due to their large wavelengths relative to the interstation distances while frequencies above 4 Hz are avoided because coherent CCFs are harder to retrieve at higher frequencies. In Fig. 2(a), we show the ensemble of CCFs in the [0.5–1] Hz frequency band, for interstation distances up to 16 km.

Our further analysis rely on the energy envelopes of the CCFs which we compute as:

$$E(t) = \frac{1}{2} [\mathcal{H}^2\{u(t)\} + u(t)^2] \quad (1)$$

where $\mathcal{H}$ is the Hilbert transform, and $u$ is the filtered cross-correlation function. In Fig. 2, we show an example of CCF (b) and its energy envelope (c) in the [0.5–1] Hz frequency band for an interstation distance of 4 km.

In order to quantify the exploitable duration of coda waves, in Fig. 3, we represent the ensemble of energy envelopes as density plots of the signal-to-noise ratio (SNR) for the three frequency bands (a-[0.5–1], b-[1–2] and c-[2–4] Hz). The vertical axis corresponds to the logarithm of the SNR which was computed by dividing the whole envelope by the average energy in the last 250 s of the record. The causal (resp. acausal) part of the CCF is defined as the side with the highest (resp. lowest) average SNR in the first 30 s.

In this work, the exploitable duration of the coda waves is determined by the noise onset, this is the 10 s before the energy envelope reaches the noise level (SNR = 1). In Fig. 4(a), we represent the distance dependence of the noise onset for different receiver couples in
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Study area: (a) Map of La Réunion Island. The red rectangle indicates the location of Piton de la Fournaise volcano, as well as the investigation area; (b) Topographic map of the study area and the PF seismic array. Contour lines are shown every 300 m. Blue triangles represent the seismic stations, with interstation distances ranging from 1 to 21 km.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** (a) Cross-correlation functions in the frequency band [0.5–1] Hz; example of cross-correlation function (b) reconstructed from the receiver couple FRE-FJS (see Fig. 1b), which interstation distance is 4 km, and its corresponding energy envelope (c).</figcaption></figure>

the [0.5–1] Hz band and their corresponding statistical distribution (b). Note that the noise onset decreases with propagation distance. Energy envelopes are subsequently smoothed in time by using a moving average window of duration $2T_0$, where $T_0$ is the central period of the analysed frequency band.

Scattering measurement requires an identification of the ballistic and early coda parts of the CCFs. As illustrated on Fig. 4(c), the arrival time of ballistic waves is found by peaking the maximum amplitude in a time window corresponding to Rayleigh wave arrivals with velocities between 2.5 and $0.5 \text{ km s}^{-1}$ (grey rectangles).

Fig. 4(c) illustrates the steps above mentioned: the blue rectangles comprise the noise onset and the 10 s right before it; black and grey lines correspond to the smoothed and non-smoothed energy

envelopes; and the grey rectangles indicate the range of possible arrival times for the Rayleigh wave. Finally, to improve the quality of our estimates we exclude energy envelopes having SNR in the expected ballistic regime smaller than 4 and coda durations shorter than 30 s. This reduces the number of CCFs from 136 to 135 (resp. 129 and 125) in the [0.5–1] Hz band (resp. [1–2] and [2–4] Hz).

To illustrate the potential directivity effects associated to the uneven distribution of noise sources, in Fig. 5 we show the azimuthal dependence of SNR in the ballistic regime. Ballistic SNR are computed as the mean SNR in the search window of the Rayleigh wave velocity ($0.5–2.5 \text{ km s}^{-1}$, grey rectangles in Fig. 4c). Note that the highest SNR correspond to receiver couples having azimuths near $270^\circ$, which points out in the west direction. This signature is in
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Ensemble of energy envelopes for the three frequency bands: (a) [0.5–1] Hz, (b) [1–2] Hz and (c) [2–4] Hz. Amplitudes are expressed as the logarithm of the signal-to-noise ratio (SNR). The white horizontal line represents the noise level; the colour bar indicates the density of data at given amplitude and time.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Exploitable duration of energy envelopes ([0.5–1] Hz). (a) Ensemble of ‘noise onsets’ as a function of distance and (b) its statistical distribution; (c) example of smoothed energy envelope for the receiver couple **FRE-FJS** (see Fig. 1b). The grey and blue rectangles denote the search window of the Rayleigh wave velocity and the reference window for the noise onset, respectively. Black and grey lines correspond to the smoothed and non-smoothed energy envelopes.</figcaption></figure>

agreement with the works of Brenguier *et al.* (2015) and Nakata *et al.* (2016) who mention that in the Piton de la Fournaise area, the noise sources propagates predominantly westward. It is also worth noting that the azimuthal dependence of SNR is less significant at higher frequencies.

### 3 MODELLED ENERGY ENVELOPES

To measure seismic wave attenuation, we compare observed and theoretical energy envelopes. The latter are computed using the RTT and diffusion approximation. The central quantities of these models are the scattering mean free path ($\ell$), the transport mean
<figure><img src="image_6.png" /><figcaption>Figure 5. Azimuthal distribution of SNR in the ballistic regime for the three frequency bands: (a) [0.5–1], (b) [1–2] and (c) [2–4] Hz. The radius of the rose plots are given by the log10 of the SNR.</figcaption></figure>

free path ($\ell^*$) and the absorption parameter $b$. $\ell$ is defined as the average distance between two scattering events. This parameter also controls the attenuation of the coherent wavefield, which is the part of energy that survives the ensemble averaging process (Celorio et al. 2022). On the other hand $\ell^*$ is defined as the typical distance after which the multiple diffusion regime emerges. This means, that at propagation distances longer than $\ell^*$, the angular dependence of the energy density tends to be uniformized.

The role and the relevance of the scattering length scales $\ell$ and $\ell^*$ has previously been discussed in the works of Gaebler et al. (2015) and Celorio et al. (2022). By comparing estimates from isotropic and anisotropic scattering assumptions, Gaebler et al. (2015) suggest that $\ell^*$ is so far the only scattering length scale that can be obtained with some confidence from seismological observations. The role of $\ell^*$ in the attenuation phenomenon is also discussed by Celorio et al. (2022), who showed, from numerical simulations of SH waves in random media, that the spatial decay of the peak energy is controlled by $\ell^*$. They justify this behaviour from the fact that the so-called ballistic peak is not completely coherent but several multiply scattered components can contribute to it. Regarding the role of the scattering mean free path $\ell$, Celorio et al. (2022) performed some statistical analyses over several realizations of the random media. Their results showed that besides controlling the coherent wave attenuation, $\ell$ defines the transition in distance from coherent to non-coherent behaviour of the peak energy.

The use of non-isotropic scattering considers that scattered waves might take a preferred direction and it makes a distinction between $\ell$ and $\ell^*$. This is the case for some applications at the crustal (e.g. Sens-Schönfelder et al. 2009; Calvet et al. 2023), or at lithospheric scales (Heller et al. 2024), which have been useful to explain lateral anomalies of scattering properties. The distinction between scattering length scales is ignored when assuming isotropic scattering. Due to its simplicity, the isotropic approximation is the most widely used solution to characterize attenuation properties at different geological contexts (Carcole & Sato 2010; Eulenfeld & Wegler 2016; van Dinther et al. 2020). This solution has been proved to be an effective approach to describe the energy transport in the anisotropic case as long as the estimated scattering length scale is interpreted as the transport mean free path (Gaebler et al. 2015).

In extremely heterogeneous media such as volcanoes, multiple scattering tends to randomize the propagation direction of seismic waves at short propagation distances (Dainty et al. 1974; Wegler & Lühr 2001; Prudencio et al. 2013, 2015). Therefore, at sufficiently

long lapse times, energy density is expected to deviate only slightly from the isotropic solution. In such case, a diffusion equation can be retrieved from RTT. This is called as diffusion approximation (Kourganoff 1969; Weaver 1990). In the present work we use both, RTT model and diffusion approximation to estimate absorption parameter. See eq. (S.1) and Fig. S1 in the Supporting Information for more detail about the diffusion approximation.

## 3.1 The radiative transfer theory

For the calculation of synthetic energy envelopes, we use an analytical solution of the 2-D radiative transfer equation that represents the energy transport in isotropic scattering media (Paasschens 1997):

$$E(r, t) = \frac{e^{-ct\ell^{-1}-bt}}{2\pi c r} \delta\left(t - \frac{r}{c}\right) + \frac{e^{\ell^{-1}(\sqrt{c^2 t^2 - r^2}-ct)-bt}}{2\pi \ell \sqrt{c^2 t^2 - r^2}} H\left(t - \frac{r}{c}\right) \quad (2)$$

where $E(r, t)$ is the energy density at a source-receiver distance $r$ and lapse time $t$ (relative to the origin of the signal). $\ell$ and $b$ are the mean free path and absorption parameter. $c$ is the velocity at which the energy is being transported. $\delta$ and $H$ are the Dirac delta and Heaviside functions, respectively. The first term on the right side of the equation describes the contribution of the direct wavefield, which decay with distance is determined by $\ell$. The second term of eq. (2) refers to the backscattered energy contained in the coda part, in which amplitudes are inversely proportional to $\ell$ and its decay with time is controlled by the absorption parameter $b$. Note that the frequency dependence of the energy envelope in eq. (2) is implicitly included in parameters $\ell$ and $b$.

In contrast with the observed energy envelopes, the ballistic regime of RTT does not take into account the effect of the band-pass filter, instead direct waves are simply represented by a Dirac type pulse. In order to improve the comparisons between RTT and the observed energy envelopes, the effects of the band-pass filters are incorporated in the synthetics by convolving RTT envelopes with the energy envelope of the impulse response of the filter. (see Fig. S2 in the Supporting Information). A similar strategy was adopted by Celorio et al. (2022).

In eq. (2) it is assumed that surface waves are dominant in the direct wavefield and also in the late coda, even though it is known that some conversion from surface to body waves is possible to occur in the coda (Margerin et al. 2019).
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Effect of scattering (a, b and c) and absorption (d, e and f) at different propagation distances (2, 12 and 22 km). RTT envelopes were convolved with the energy envelope of the [0.5–1] Hz filter. Colourbar at the top represents different values of the transport mean free path (in km), and that at the bottom corresponds to variations of absorption parameter (in s⁻¹).</figcaption></figure>

## 3.2 Example of forward modelled envelope

In order to evaluate the effects of scattering and absorption parameters on the synthetic energy envelopes, in Fig. 6 we show envelopes predicted by RTT at three different propagation distances *r* (2, 12 and 22 km). Scattering effect (Figs 6a–c) is evaluated by defining diverse values of *ℓ* ranging from 0.5 to 10 km and absorption parameter is fixed as *b* = 0.05 s⁻¹. At short distances (*r* = 2 km) effects of scattering are almost negligible except for the highest levels of scattering, where *ℓ* is even shorter than the minimum propagation distance (2 km). At larger distances (*r* = 12 and 22 km) energy envelopes become more sensitive to scattering, especially around the ballistic peak.

Absorption effect (Figs 6d–f) is evaluated by using different values for *b*, from 0 to 0.2 s⁻¹ and setting *ℓ* as 5 km. These calculations suggest that absorption controls the decay of energy regardless of the propagation distance.

## 4 INVERTING FOR ABSORPTION AND SCATTERING ATTENUATION

The aforementioned synthetic envelopes are now used to estimate attenuation parameters (scattering and absorption) through the comparisons with observed data. Our inversion technique is based on a 2-D grid search for the *ℓ* and *b* parameters. The comparisons between observed and modelled envelopes are performed for values of *ℓ* (resp. *b*) ranging from 0.1 to 40 km with an increment of 0.2 km (resp. from 0 to 0.2 s⁻¹ with an increment of 0.01 s⁻¹). As shown in Fig. 6, scattering and absorption can generate different signatures on the appearance of modelled energy envelopes, which sensitivity is strongly dependent on the propagation distance and the lapse time. Thus, we estimate absorption and scattering in two stages: (i) *b* is measured by using both CCF at short distances and

auto-correlations functions (zero-offset case). In this way, we determine the average absorption as a function of frequency. (ii) This average absorption is further fixed and the grid search is performed only in terms of the scattering parameter (*ℓ*) for longer propagation distances.

### 4.1 Absorption estimates

We perform two sets of absorption estimates: first by using CCFs with short propagation distances that provide longer coda and secondly, we use auto correlation functions (ACFs).

#### 4.1.1 Results for short propagation distances

We consider CCFs with propagation distances up to 5 km in the [0.5–1] Hz band. To facilitate the comparisons between observations and predictions, energy envelopes are normalized with respect to a 10-s window right before the earliest noise onset (80 s) observed among all CCFs in this distance range. We use both models: RTT and diffusion approximation as the latter can also model the distribution of energy at long lapse-times (see Fig. S1 in the Supporting Information). Observations and predictions are compared by using all the points of the record, from the arrival time to 80 s. For a given path (receiver couple), the misfit between observed and modelled energy envelopes is given by the sum of the square of the residuals (SSR):

$$SSR = \sum_{t=t_0}^{t_f} \left[ \log \left( \frac{E_{\text{RTT}}(t)}{E_{\text{obs}}(t)} \right) \right]^2 \quad (3)$$

Where $t_0$ is the arrival time of the direct Rayleigh wave and $t_f$ corresponds to 80 s. By choosing the best-fit parameters we find the corresponding *b* and *ℓ* parameters for a specific path. Note that, at this point, the estimation of *ℓ* is not relevant as our distance range is not sensitive to such parameter. In Figs 7(a) and (b), we show an
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Example of absorption measurement at station pair **FRE-FJS**. The 2-D grid search with the normalized sum of squared of residuals from the acausal (a) and causal (b) parts of the CCF; (c) Comparison of observed energy envelope and the best fit-synthetic envelope when using RTT (red) and diffusion approximation (blue). Thin vertical grey lines bound the exploited portion of the energy envelope. Propagation distance of 4 km. Case [0.5–1] Hz.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) Observed ACF from station SNE and its synthetic counterpart from the best-fit parameters of RTT and diffusion approximation. (b) Misfit function for the obtention of *b*. Case [0.5–1] Hz.</figcaption></figure>

example of absorption estimate for the receiver couple **FRE-FJS** illustrated in Fig. 2 (*r* ~ 4 km). This inversion was performed by using both, RTT and diffusion models, from which we obtained similar estimates. Note that for both sides of the CCF, absorption parameter is well resolved, with a central value of 0.05 s⁻¹ and error bars ranging from 0.04 to 0.06 s⁻¹. In contrast, the mean free path is poorly constrained, with central values of 10 and 1 km but a wide uncertainty ranges extending from 1 to 20 km. These error bars are obtained as the range of parameters *b* for which the misfits are at most 10 per cent larger than the minimum. Fig. 7(c) shows the observed and the best-fit synthetic envelopes. The thin grey lines indicate the region in which the fitting procedure was performed. Note that right after the ballistic regime (grey area), the agreement between RTT (red) and diffusion approximation (blue) is perfect.

### 4.1.2 The use of ACF: the source–receiver collocated case

Since short propagation distances exhibit a longer coda duration, we now explore the ACFs, which represent the source–receiver collocated case. Even though the peak amplitude of ACFs just represents the recorded noise level at the station, the long lapse time has shown to be helpful to study the energy decay of coda waves (Wegler &

Sens-Schönfelder 2007; Soergel et al. 2019; Ranjan & Stehly 2024). We measure absorption attenuation from ACFs having coda durations of minimum 100 s (resp. 85 and 70 s) in the [0.5–1] Hz band (resp. [1–2] and [2–4] Hz bands). In Fig. 8, we show an example of ACF (a) in the [0.5–1] Hz band, and the corresponding misfit function (b) for the station SNE. We used both, RTT and diffusion approximation. For both models we found the same *b* values 0.06 s⁻¹ with error bars of the order of 10 per cent.

The global best-fit absorption parameter is found by minimizing the misfit function given in eq. (3) over the ensemble of ACFs for the 17 seismic stations. The results for the three frequency bands are summarized in Table 1. Note that these results are given in terms of absolute values of the absorption parameter (in bold) and the corresponding error bars.

Since our technique allows us to measure absorption attenuation at each energy envelope, either from CCFs or ACFs, it is convenient to visualize lateral variations of this parameter from ACF's estimates.

Overall, absorption increases with frequency. Fig. 9 shows that at low frequency ([0.5–1] Hz), *b* lateral variations seem to exhibit two main signatures. In the first one, absorption decreases as one gets farther from the volcanic summit. This pattern is certainly consistent
<table><caption>**Table 1.** Global best-fit estimates of absorption when using ACFs ($r = 0$ km). Best-fit parameters were obtained from RTT and diffusion models at the three frequency bands.</caption><thead><tr><th rowspan="2">Model</th><th colspan="3">b (s<sup>-1</sup>)</th></tr><tr><th>[0.5–1] Hz</th><th>[1–2] Hz</th><th>[2–4] Hz</th></tr></thead><tbody><tr><td>RTT</td><td><strong>0.05</strong> (0.046–0.061)</td><td><strong>0.08</strong> (0.072–0.091)</td><td><strong>0.11</strong> (0.099–0.118)</td></tr><tr><td>Diffusion approximation</td><td><strong>0.05</strong> (0.046–0.061)</td><td><strong>0.08</strong> (0.072–0.091)</td><td><strong>0.11</strong> (0.099–0.118)</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Lateral variations of absorption parameter from individual measurements when using ACFs. Results are shown for the three frequency bands: (a) [0.5–1] Hz, (b) [1–2] Hz and (c) [2–4] Hz. The colour map stands for different values of *b*. The number of stations gets reduced with frequency as coda duration decreases.</figcaption></figure>

with the phase velocity maps obtained by Mordret *et al.* (2015) at the Piton de la Fournaise volcano. By using continuous data over four years, these authors found lower values of Rayleigh wave velocity at regions near the volcanic summit. The second signature of our absorption estimates at the [0.5–1] Hz denotes certain degree of heterogeneity at stations right above the volcanic summit. This behaviour might be consistent with the works of Prôno *et al.* (2009) and Lénat *et al.* (2012) who reported high spatial variability of geophysical properties in the shallow layers. Such heterogeneity near the volcanic summit is related to diverse processes such as the presence of lava flow, the high fracturation and even the presence of cooled magma bodies.

In contrast to the previous observations, the estimates of *b* in the [1–2] and [2–4] Hz bands exhibit spatial variations that appear to be less correlated with the surface geology of the volcano. Using beamforming techniques, Nakata *et al.* (2016) found that above 3 Hz, noise sources in the Piton de la Fournaise volcano are increasingly dominated by body waves. Furthermore, Mayor *et al.* (2016) suggest that coda waves may become more sensitive to deeper structures when the proportion of body waves increases in the equipartition process. Based on this hypothesis, the observed lateral variations of absorption at higher frequencies may reflect the influence of deeper geological bodies. It is also worth noting that the dispersion of *b* values is smaller at higher frequencies (see the ensemble of absorption measurements and associated error bars in Fig. S3 in the Supporting Information).

## 4.2 Estimating scattering using the ballistic and early diffuse regimes

The second step of our full-envelope based approach consists in estimating scattering attenuation from receiver couples with sufficiently long propagation distances. Absorption is now a fixed parameter, which values are shown in Table 1. The range of propagation distances for scattering estimation is chosen as 8–15 km in the [0.5–1] Hz frequency band, and those for the [1–2] and [2–4] Hz bands are chosen as 4–12 and 4–10 km, respectively. Our strategy is based on the $Q_{open}$ method proposed by Sens-Schönfelder & Wegler (2006),

and largely applied to constrain attenuation, source and site amplification parameters (Eulenfeld & Wegler 2016, 2017; Ranjan & Konstantinou 2020; van Laaten et al. 2021). As shown in the analyses of modelled energy envelopes (see Fig. 6), for a wide range of mean free paths, early lapse time components are more sensitive to scattering than the late coda. As mentioned by Sens-Schönfelder & Wegler (2006), isotropic scattering can not take into account several effects occurring at the ballistic pulse, such as peak broadening and wandering effects. These discrepancies can complicate the comparisons between the modelled and observed energy envelopes. To circumvent these difficulties they proposed to average the energy contained in a short window around the ballistic pulse. In our case, we use the average energy contained in a first window with duration $2T_0$ ($T_0$ is the central period at given frequency band) starting at the peak amplitude. In contrast to Sens-Schönfelder & Wegler (2006), Eulenfeld & Wegler (2016), Eulenfeld & Wegler (2017), Ranjan & Konstantinou (2020) and van Laaten et al. (2021), we use an additional 8-s window right after the first one. The reason to use a second window is the high sensitivity of the early coda to scattering.

### 4.2.1 Inversion strategy 1: average energy criterion

In a first attempt to estimate $\ell$, we search to fit modeled to observed envelopes by comparing average energies in the two windows. For a given receiver couple, the misfit function is given by:

$$ \text{SSR} = \sum_{w=1}^{2} \left[ \log \left( \frac{E_{\text{RTT}}(t_w)}{E_{\text{obs}}(t_w)} \right) \right]^2 \quad (4) $$

Where *w* is a given window in which energy is averaged. For the example in Fig. 10(a), the synthetic envelopes from the best-fit parameters are also shown with the previously fixed absorption $b = 0.05(s^{-1})$ in the [0.5–1] Hz band. The ensemble of individual measurements are shown in Figs 10(b), (d) and (f) for all the azimuths and frequency bands. We find values of $\ell$ ranging from 3 to 40 km in the [0.5–1] Hz frequency range, and up to 20 km for the rest of frequency bands.

The azimuthal distribution of $\ell$ shows that the variability is higher for receiver couples with azimuths near the predominant direction of the noise sources (270°) and having high SNR. On the other hand,
<figure><img src="image_11.png" /><figcaption>**Figure 10.** Scattering estimates from the fitting of average energies at early lapse times for long propagation distances. (a) Observed energy envelope for station pair SNE-HDL and corresponding best-fit synthetic envelope in the [0.5–1] Hz frequency range; (b), (d) and (f) ensemble of individual measurements at the three frequency bands and their azimuthal dependence: circles indicate the estimates for each individual energy envelope (including causal and acausal), and colours represent the corresponding SNR; (c), (e) and (g) global fit of $ℓ$ for causal (solid line) and acausal (dashed line) parts of the CCFs.</figcaption></figure>

measurements from energy envelopes with lower SNR exhibit a lower variability of $ℓ$. A global estimation of scattering (Figs 10b-g) can be performed by fitting the ensemble of energy envelopes through the sum of eq. (4) over all the paths. We show global estimates for the causal (solid line) and acausal (dashed line) parts of the CCFs in the three frequency bands with significant discrepancies between them. The lower resolution in the causal part can be explained by the huge variability of individual measurements of $ℓ$ near the dominant direction of the noise sources. It is worth noting that the resolution of the global scattering estimates increases

with frequency. The ensemble of individual measurements indicate that the azimuthal dependence of the SNR (which is also frequency dependent as shown in Fig. 5) can strongly affect the resolution of scattering estimates.

### 4.2.2 Inversion strategy II: the ballistic-to-diffusive energy ratio

The results shown in Fig. 10 reveal a significant influence of the distribution of noise sources on scattering estimates. This observation aligns with previous theoretical studies (Tsai 2011; Hanasoge
2012), which highlighted the challenges of constraining direct wave attenuation when noise sources are nonuniformly distributed.

Despite this, existing studies on scattering measurements from noise CCFs (Hirose et al. 2019; van Dinther et al. 2020; Hirose et al. 2022) have not explicitly addressed this issue, either because their methods only provide average values or because stacking over different paths is performed. For instance, Hirose et al. (2019, 2022) estimated scattering attenuation at different volcanoes in Japan using the space evolution of seismic energy in short-time windows. In their approach, they averaged the causal and acausal parts of the energy envelopes, which masked potential directivity effects on scattering estimates. On the other hand van Dinther et al. (2020), used MLTWA to investigate lateral variations of scattering properties at the North Anatolian Fault Zone by dividing the area into four subregions. Although their results were globally consistent with the geology, the influence of the noise sources distribution on scattering estimates was not assessed since all the horizontal components of the CCFs tensor were averaged.

In this section, we develop a technique to reduce the effects of noise directivity on $\ell$ estimates (observed in Fig. 10). To that end, we propose to fit the ratio of the energy of the two windows corresponding to the ballistic regime and the early coda (see Section 4.2.1). We rely on the idea that the ratio of observed average energies in the early regimes is less sensitive to the influence of noise directivity, as the energy of both windows is expected to depend on the distribution of the noise sources. Furthermore, the ballistic-to-diffusive ratio is a crucial parameter in the scattering phenomenon as it is an indicator of the energy redistribution. We refer to this inversion technique for scattering as ball-diff ratio (ballistic-to-diffusive ratio), which misfit function is given by the square of the residuals (SR) as:

$$SR = \left[ \log \left( \frac{C_{RTT}}{C_{obs}} \right) \right]^2 \qquad (5)$$

Where $C_{RTT} = E_{1RTT}/E_{2RTT}$ and $C_{obs} = E_{1obs}/E_{2obs}$, with $E_1$ and $E_2$ being the average energy contained in the ballistic and early diffuse windows, respectively. Thus, eq. (5) could be rewritten as:

$$SR = \left[ \log \left( \frac{E_{1RTT} E_{2obs}}{E_{2RTT} E_{1obs}} \right) \right]^2 \qquad (6)$$

In Fig. 11(a), we present the same case of energy envelope as in Fig. 10 for the [0.5–1] Hz frequency band (station pair SNE-HDL). Here, the envelope is normalized with respect to the average energy contained in the early diffused field window ($TW_2$). In the causal part of the correlation, we find $\ell$ to be 6.6 km with the ball-diff ratio, compared to 9.6 km obtained using the previous approach (Figs 11 and 10). Moreover, the ball-diff ratio provides better agreement in scattering estimates between the causal and acausal parts of the CCFs.

The ensemble of individual measurements for the whole set of receiver couples are shown in Fig. 11(b) ([0.5–1] Hz). It can be seen that the variations of $\ell$ are significantly smaller than those in Fig. 10(b), specially at azimuths near 270°, which corresponds to the predominant direction of the noise sources.

Similarly, the global misfit functions 11(c) evidence that scattering measurements of the causal and acausal parts are much closer, from 13.4 and 7.8 km when using the average energy criterion, to 8.6 and 7.8 km for the ball-diff ratio. The same type of improvement is observed at higher frequencies (see Figs 11d–g).

Table 2 summarizes the results of absorption and scattering attenuation for the three frequency bands and their corresponding error

bars. Note that absorption parameter is presented in terms of absorption length in order to show which attenuation mechanism is more dominant. b parameter is converted into absorption length $\ell_a$ from the expression $\ell_a = c/b$. The ensemble of these results suggests first, that both, absorption and scattering, increase with frequency. And secondly, that scattering attenuation dominates over absorption in the whole frequency range, which is generally expected for volcanic environments as reported in previous works (e.g. Prudencio et al. 2013; Hirose et al. 2019, 2022).

## 5 VALIDATION OF THE METHOD: COMPARISONS WITH EARTHQUAKE DATA

To validate our technique, estimates from noise CCFs are now compared with those from earthquake data. To that end, we measure attenuation by using our full-envelope based approach on seismic records coming from local earthquakes. In their pioneering work, Hirose et al. (2019) compared measurements of absorption and scattering from seismic noise with those from active shots, obtaining similar results. In the present work, we perform comparisons with local earthquakes in the Piton de la Fournaise volcano with magnitudes ranging from 1.5 to 2.5 and reported depths of ~0 km. These volcano-tectonic events occur under the Dolomieu crater, just above the possible location of a shallow magma reservoir (Peltier et al. 2009; Prôno et al. 2009; Duputel et al. 2022). They are associated to a ring-fault system at the summit caldera (Duputel et al. 2022). Given the shallow depth of these events and short epicentral distances, the wavefield is dominated by surface waves.

We used nine earthquake events which are located in the area near the volcanic summit (see Fig. S4 in the Supporting Information). The range of epicentral distances is similar to the interstation distances used to compute CCFs of seismic noise in the previous sections. In Fig. 12, we show the recorded seismograms for an earthquake with magnitude (based on shaking duration) $M_d = 2.0$. We present the waveforms at different distances (a) as well as their corresponding Fourier amplitude spectra (b). Solid and dashed lines indicate the frequency bands of interest ([0.5–1], [1–2] and [2–4] Hz). While low frequency (below 1 Hz) is mostly composed by seismic noise, higher frequencies are dominated by the earthquake energy. However, we shall limit our earthquake-based measurements up to 4 Hz in order to perform the comparisons with noise-based results in a common range of frequencies ([0.5–4] Hz).

Similar to the processing of noise CCFs, earthquake seismograms are filtered in three frequency bands ([0.5–1], [1–2] and [2–4] Hz), and energy envelopes are calculated from eq. (1). SNR were computed from the ratio between the energy envelopes and the average energy in a 30 s time window immediately preceding the event. The same selection criteria are applied: only envelopes with SNR higher than 4 and coda durations longer than 30 s were considered. For the [0.5–1] Hz band, however, coda durations are often shorter than 15 s, making it challenging to perform reliable attenuation measurements below 1 Hz. Consequently, only envelopes in the [1–2] and [2–4] Hz frequency bands are used (see the ensemble of earthquake envelopes in Fig. S5 of the Supporting Information).

Following the procedure described in Section 4, absorption is estimated from envelopes with short epicentral distances ($r < 2$ km), and scattering is estimated for distances larger than 4 km. To illustrate the similarities between reconstructed noise CCFs and
<figure><img src="image_12.png" /><figcaption>**Figure 11.** Same as Fig. 10 for the ball-diff ratio technique.</figcaption></figure>

<table><caption>**Table 2.** Global estimates of absorption ($l_a$) and scattering ($l$) from noise auto-correlation functions and the ball-diff ratio technique, respectively.</caption><thead><tr><th>Frequency band (Hz)</th><th>l (km)</th><th>l<sub>a</sub> (km)</th></tr></thead><tbody><tr><td>0.5-1</td><td><strong>8.60</strong> (6.00-11.00)</td><td><strong>23.00</strong> (19.17-28.75)</td></tr><tr><td>1-2</td><td><strong>3.80</strong> (3.00-5.40)</td><td><strong>14.38</strong> (12.78-16.43)</td></tr><tr><td>2-4</td><td><strong>2.80</strong> (2.40-3.20)</td><td><strong>10.45</strong> (9.58-11.50)</td></tr></tbody></table>

earthquake records, in Fig. 13 we compare the observed energy envelopes from an event with magnitude 2.0 and the CCFs having similar propagation path. We compare the ACF from station SNE with the earthquake record having an epicentral distance of 0.2 km

(Fig. 13b). We can see that both energy envelopes exhibit similar decrease with time. As an example of long propagation distance, the CCF between stations SNE and HDL is compared with the recorded signal at station HDL from the same earthquake (Fig. 13c). In this case, the shape of both envelopes suggest that direct waves have been significantly diffused.

It is worth mentioning that envelopes were normalized by the energy at a 10 s window right before the 60 s lapse time. Next, we measure absorption and scattering from the earthquake records at stations SNE and HDL, respectively.

The results of these individual estimates are shown in Figs 14(a) and (b). We show the comparisons between the observed energy en-
<figure><img src="image_13.png" /><figcaption>**Figure 12.** Earthquake data: (a) example of recorded waveforms for an earthquake with magnitude 2.0 and corresponding (b) Fourier amplitude spectra where the colours denote different epicentral distances.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 13.** Comparing energy envelopes from an earthquake record and noise CCFs having similar paths. (a) Location of the event (yellow circle) and the two seismic stations. (b) Recorded earthquake waveform at a very short propagation distance (0.2 km) in black, and an ACF of noise at the same location in grey. (c) Earthquake waveform recorded at 7.4 km and CCF of a receiver couple with identical path. The black vertical line indicates the arrival time of Rayleigh waves.</figcaption></figure>

velopes (earthquake and CCFs) and the best-fit RTT envelopes. For both ranges of propagation distance, results are in good agreement. The absorption parameter found at the short epicentral distance and the ACF is exactly the same (0.11 s⁻¹). Regarding scattering attenuation, we find a mean free path of 2.0 km using the earthquake records and 2.4 km using the noise correlation.

In order to synthesize the comparisons of attenuation estimates from noise correlations and earthquakes, we estimate the average attenuation as we did in Section 4. The mean attenuation estimates are found by minimizing the ensemble of square of residuals between observations and RTT predictions for all the earthquake records.

Fig. 14(c) shows the global measurements at the [1–2] and [2–4] Hz bands from earthquake data (black), and a comparison with the results from the noise-based approach (blue). These results are summarized in Table 3.

Overall, attenuation estimates from earthquake data are in good agreement with those from seismic ambient noise. The agreement of

absorption parameters is better than that from scattering estimates. Even though, error bars of *ℓ* are relatively small for both cases, the resolution when using earthquake data is higher. This might be due to the difficulties to reconstruct direct waves at high frequency when using CCFs of seismic noise. However, these similarities demonstrate the feasibility to retrieve attenuation properties from seismic ambient noise similar to those observed with earthquake data.

## 6 CONCLUSION

This work aims to estimate the absorption and scattering properties using the full envelope of seismic noise correlations. We tested this approach at the Piton de La Fournaise volcano.

We used continuous data recorded over four years at 17 stations. Our approach takes into account, separately, the sensitivity of the energy envelope of coda waves to the absorption and scattering as a function of the interstation distance. Thus, estimates were
<figure><img src="image_15.png" /><figcaption>**Figure 14.** Comparison of attenuation estimates from earthquake records and cross-correlation functions of seismic noise. Observed and synthetic energy envelopes, first at short propagation distances (a) and then at long propagation distances (b). Subscripts **qk** and **ccf** indicate the approach, either from earthquake data or CCFs. (c) Global estimates from both approaches at the two frequency bands.</figcaption></figure>

<table><caption>**Table 3.** Comparisons of global estimates of absorption and scattering from earthquake data and noise correlated signals. Noise-based estimates were obtained from the full-envelope based approach.</caption><thead><tr><th rowspan="2">Frequency band (Hz)</th><th colspan="2">ℓ(km)</th><th colspan="2">ℓ<sub>a</sub>(km)</th></tr><tr><th>Earthquake</th><th>CCF</th><th>Earthquake</th><th>CCF</th></tr></thead><tbody><tr><td>1–2</td><td><b>3.20</b> (2.80–4.00)</td><td><b>3.80</b> (3.00–5.40)</td><td><b>14.38</b> (12.78–19.17)</td><td><b>14.38</b> (12.78–16.43)</td></tr><tr><td>2–4</td><td><b>2.60</b> (2.60–3.00)</td><td><b>2.80</b> (2.40–3.20)</td><td><b>10.45</b> (9.58–11.50)</td><td><b>10.45</b> (9.58–11.50)</td></tr></tbody></table>

performed in two stages: first, absorption was measured by using short propagation distances and secondly, scattering was estimated at longer distances. In addition to the short propagation distances, we also used ACFs for the estimation of absorption. Obtaining absorption measurements at each station enabled us to observe lateral variations of this parameter. In the [0.5–1] Hz frequency band, we found that absorption measurements exhibit lateral variations, with shorter absorption length within the caldera. At higher frequency, we found no clear lateral variation of absorption, and no obvious correlation with the surface geology. This may indicate that coda waves are sensitive to deeper structures in these frequency bands.

Scattering attenuation was measured from the ballistic and early diffused energies at long propagation distances. While it is known that noise directivity might influence scattering estimations, existing studies from CCFs have not faced this issue for two main reasons: first, because energy envelopes from different paths are averaged, and secondly, because their inversion techniques only provide average values of attenuation properties. Here we proposed to measure scattering attenuation from the ballistic and early diffused regimes at each energy envelope, separately for causal and acausal parts of the CCFs. Initially, we fitted the ballistic and early diffused energies, which revealed a clear bias in $ℓ$ towards the predominant direction of noise sources. To circumvent this issue we proposed to fit the energy ratio between these two regimes (ball-diff ratio technique). Thus, we were able to mitigate the impact of the uneven distribution of noise sources on scattering estimates, as well as to improve the resolution of our measurements.

A potential advantage of this full envelope approach, is that it makes it possible to measure the absorption and scattering lengths for each pair of station, rather than getting an average value over

a dense network like previous noise-based studies. This opens the possibility to derive 2-D maps of absorption and scattering.

Finally, we validated this method by comparing the noise-based estimates with those from earthquake data. We found excellent agreement in the [1–2] and [2–4] Hz frequency bands. It was possible to extend the measurements to the [0.5–1] Hz band using noise, but not with earthquake records. This highlights the potential of noise-based full-envelope analysis as a powerful tool to extract attenuation properties, even in frequency bands and spatial scales that are difficult to access with earthquake data alone.

## AUTHOR CONTRIBUTIONS

Malcon Celorio (Conceptualization, Data curation, Formal analysis, Methodology, Writing—original draft), Laurent Stehly (Conceptualization, Methodology, Writing—review & editing) and Emmanuel Chaljub (Conceptualization, Methodology, Supervision, Writing—review & editing).

## ACKNOWLEDGMENTS

MC acknowledges the financial support of MINCIENCIAS, call 860 (convocatoria de doctorados en el exterior 860) and the French-Colombian collaborative project ECOS-NORD, grant number C21U01 2021-2023. The noise data pre-processing and the calculations of cross-correlation functions were performed at the High Performance Computing Center of the Grenoble University on the Dahu cluster of the GRICAD-CIMENT infrastructure (https://gricad.univ-grenoble-alpes.fr). We are grateful for the fruitful discussions with Ludovic Margerin, Pratul Ranjan, Bérénice Froment, German Prieto and Carlos Vargas.
## SUPPORTING INFORMATION

Supplementary data are available at GJIRAS online.

**Figure S1.** Convergence of radiative transfer envelopes (solid black line) towards diffusion approximation (dashed blue line). Envelopes were band-pass filtered at [0.5–1] Hz and correspond to a medium with $ℓ = 5$ km and $b = 0.05 \text{ s}^{-1}$.

**Figure S2.** Calculation of synthetic energy envelopes. (a) RTT envelope for 2-D medium with $ℓ = 5$ km and $b = 0.05 \text{ s}^{-1}$); (b) energy envelopes of the impulse responses for three band-pass filters ([0.5–1], [1–2] and [2–4] Hz); (c) convolution of band-pass filter envelopes with that from RTT model.

**Figure S3.** Absorption estimations and associated error bars for each station in the (a) [0.5–1], (b) [1–2] and (c) [2–4] Hz bands.

**Figure S4.** (a) Spatial distribution of earthquakes (yellow circles) and seismic stations (blue triangles) in the study area; (b) histogram of earthquake magnitudes.

**Figure S5.** Ensemble of envelopes of the earthquake data, for the three frequency bands (a-[0.5–1], b-[1–2] and c-[2–4] Hz). Amplitudes are expressed as the logarithm of the SNR. The white horizontal line represents the noise level; the colourmap indicates the density of data at given amplitude and time.

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper.

## DATA AVAILABILITY

The seismic ambient noise data used in this study are publicly available and can be accessed from the IRIS Data Management Center at https://ds.iris.edu/mda/PF/.

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