Filtering of a Ricker wavelet induced by anelastic seismic wave propagation and reflection

A varying Q factor with depth induces modifications of seismic wave features due to anelastic propagation but also reflections at the discontinuities. Standard signal analysis methods often neglect the reflection contribution when assessing Q values from seismic data. We have developed an analytical quantification of the cumulative effects of both the propagation and reflection contributions by considering Kjartansson’s model and a seismic plane wave at normal incidence on a step-like discontinuity. We show that the cumulative effects are equivalent to a frequency filter characterised by a bandform and phase that both depend on the ratio between the elastic and anelastic contrasts. When considering this filter applied to a Ricker wavelet, we establish an analytical expression of the peak-frequency attribute as a function of propagation and reflection properties. We demonstrate that this seismic attribute depends on the anelastic contrast, which cannot be neglected when assessing Q factors: the error in Q estimate is not linearly dependent on the anelastic contrast and we establish an analytical expression for the case where this contrast is weak. An unexpected phenomenon for a step-like interface is an increase in the peak frequency that is observed when the anelastic and elastic contrasts have opposite signs, with a constraint on the anelastic propagation properties. This behaviour allows for assessment of the elastic and anelastic parameters.


Introduction
In the framework of exploration seismology, wave propagation in attenuating media has been extensively studied to understand amplitude loss, frequency content reduction and phase distortion induced by anelastic processes (Kolsky 1956;Futterman 1962;Toksöz & Johnston 1981). Over the last decades, intensive research efforts have been dedicated to the quantification of seismic attenuation through the estimate of the quality factor Q (Dasgupta & Clark 1998;Reine et al. 2009;Tary et al. 2017), the attenuation compensation of seismic data through Q-inverse filtering methods (Wang 2002(Wang , 2008, Q compensation in migration (Wang 2008a;Dutta & Schuster 2014;Zhu et al. 2014;Li et al. 2016) and seismic inversion (Causse et al. 1999;Innanen & Lira 2010;Brossier 2011;Innanen 2011). Most of these studies focused on the attenuation effects on a seismic wave during its propagation in an anelastic medium, which can be characterised by a Q factor varying with depth. However, it has been shown that Q-contrast effects may also affect the seismic wave during its reflection by an anelastic reflector (White 1965;Bourbié & Nur 1984;Lines et al. 2014). This contribution is commonly neglected in analytical modelling. Developing quantitative analyses that takes into account the effect of anelastic propagation and reflection contributions on seismic waves remains of first-order importance in the understanding and exploitation of reflected seismic data. Currently, frequency-dependent seismic anomalies observed on seismic data have been attributed to strong absorptive reflection coefficients associated with contrasts of both P-wave velocities and Q factor properties between two layers Odebeatu et al. 2006;Ker et al. 2014;Wu et al. 2015). In particular, the P-and Q-contrasts are negative/negative or positive/negative for seismic reflectors associated with carbonate reservoirs (Adam et al. 2009;Takam Takougang & Bouzidi 2018), rocks saturated with heavy oil (Gurevich et al. 2008) and hydrate bearing sediments (Guerin & Goldberg 2005;Marin-Moreno et al. 2017). To explain such absorptive reflection coefficients (Castagna et al. 2003), different mechanisms can be considered in the framework of anelastic, i.e. viscoelastic (Lam et al. 2004;Borcherdt 2009;Zhao et al. 2018) or poroelastic Quintal et al. 2009) wave propagation.
In the case of anelastic interfaces, White (1965) introduced the concept of absorptive reflection and Bourbié & Nur (1984) experimentally identified pure anelastic reflection respectively associated with a Q-contrast between two anelastic layers. Lines et al. (2014) confirmed these observations by developing innovative laboratory experiments. The motivation in studying anelastic reflections, related to the imaginary part of the complex seismic impedance controlled by the quality factor (White 1965;Lines et al. 2008;Morozov 2011), is better interpreting reflected wave amplitude (Bourbié & Nur 1984) and phase rotation (Lines et al. 2014). But these studies do not deal with an anelastic upper layer, i.e. anelastic propagation is not considered before the reflection and the impact of the Q-contrast magnitude on the reflected wave is still not quantified. A full understanding of the impact of both anelastic propagation and reflection on an incoming seismic wave has not been provided yet and constitutes the aim of this work.
We develop an analytical analysis to quantify the cumulative effects of both the propagation and reflection contributions by considering Kjartansson's model and a seismic plane wave at normal incidence on a step-like discontinuity. In section 2, we review the anelastic seismic wave propagation and reflection associated with the constant-Q model (Kjartansson 1979) that is widely used in seismic modelling, inversion and analysis (Morgan et al. 2012;Chen et al. 2018;Ker et al. 2019) and we revisit these wave phenomena in terms of frequency filtering. We develop a comprehensive analysis by introducing new constitutive parameters, related to both propagation and reflection. In section 3, we present the filter related to the cumulative effects of both anelastic propagation and reflection and describe its frequency behaviour. In section 4, we investigate the effect of this cumulative filter on a seismic source signal approximated by a Ricker wavelet and develop analytical descriptions of peak-frequency parameters to quantify the wavelet shape distortions induced by the cumulative filter. This investigation provides some Step-like reflector defined by two homogeneous anelastic layers: an anelastic propagation of a normal plane incoming wave in the upper layer and an anelastic reflection of the wave at the discontinuity.
insights and pitfalls related to the use of the peak-frequency attribute when estimating Q values or interpreting seismic data.

Principles of the approach
We assume a 1D plane-wave seismic propagation and a steplike reflector defined between two anelastic layers M 1 and M 2 (figure 1). As a consequence, a reflected seismic wave ′ is the result of two contributions induced on the incoming wave , i.e. (i) a propagation in the upper anelastic layer and (ii) a reflection at the anelastic interface. To develop a quantitative analysis of these effects, our proposed method consists of defining frequency filters (bandform and phase) associated with these two contributions in terms of both modulus and phase expressed in the frequency domain.

Modulus and phase associated with the anelastic propagation contribution
Seismic propagation occurs in the upper anelastic layer M 1 characterised by a density 1 , a reference velocity c 1 and a quality factor Q 1 . The attenuation quantified by Q 1 induces a complex P-wave velocity v * 1 ( ), which depends on the angular frequency of the seismic wave. Based on Kjartansson's model (Kjartansson 1979), v * 1 ( ) can be expressed as a function of Q 1 and c 1 according to: 2 where the reference angular frequency h ensures the compliance with the Kramers-Kröning dispersion relation and corresponds to the highest frequency of the seismic source frequency content (Wang & Guo 2004;Wang 2008b). In the following, we consider the approximate form of the complex velocity: which induces an error lower than 1.5% for a quality factor larger than 5. According to this approximation, the phase velocity and attenuation associated with the real and imaginary parts of the complex wavenumber k * 1 = ∕v * 1 ( ), respectively, can be used to highlight that the anelastic propagation acts as a filter defined by the modulus M p ( ) = exp(− By introducing the ratio: which has the dimension of a frequency, and assuming Q 1 >> 1, we can finally write: which highlights that the anelastic propagation in the upper layer acts as a filter that depends on the frequency parameter Γonly. This means that the filter related to the anelastic propagation is unchanged when both the travel time and the attenuation proportionally increase or decrease. The frequency response of the filter associated with wave propagation in the upper anelastic layer is described by the partial derivatives of both the modulus and phase with respect to the angular frequency . This is given as: and highlights that the modulus is a monotonous decreasing function of , with a limit value of Additionally, the phase reaches a minimum value: this occurs at the angular frequency h ∕e where e = exp(1) ( Table 1). This means that the anelastic propagation acts as a low-pass filter associated with a frequency-dependent phase rotation.

Modulus and phase associated with the anelastic reflection contribution
The seismic reflection that occurs at the anelastic interface is modelled by a step-like discontinuity between the upper anelastic layer M 1 where the seismic wave propagates, and a lower anelastic layer M 2 characterised by a density ] and the reflection coefficient can thus be expressed as: where we introduce the parameter that quantifies the anelastic contrast between the two layers.
To understand the filtering effects induced by the anelastic reflection coefficient on a normal incident plane wave, both the modulus and phase of R * have to be expressed analytically. Such developments require expressing the anelastic reflection coefficient as the sum of its real and imagery parts. To do so, we use the decomposition R * ( ) = R E + R A ( ) established by Bourbié & Nur (1984) where represents the elastic impedance contrast that depends on the velocity and density parameters, and represents the anelastic contribution that is complex and depends on (equation (9)) and frequency. By introducing Table 2. Frequency behaviour of the filter associated with anelastic reflection (modulus M r and phase P r ).
the parameter: this allows the reflection coefficient R * to be expressed as: which tends to R E as approaches 0 as expected for an elastic reflector. As a result, it is now straightforward to establish analytical formulations of the modulus M r = These equations highlight that an anelastic reflection acts as a complex frequency filter that depends on both the anelastic contrast and the ratio R E ∕ between the elastic and anelastic contrasts. The frequency response of the filter associated with the reflection on an anelastic interface is described by the partial derivatives of both the modulus and phase with respect to the angular frequency . We can show that where the modulus increases or decreases with depending on R E ∕ and the phase is a monotonous decreasing function of . The asymptotic values of the partial derivatives when tends to 0 are +∞ and 0, respectively, and when tends to h (Table 2). This means that the frequency filter associated with an anelastic reflection is asymmetric with respect to R E ∕ whereby: -if R E ∕ ≤ 0, the modulus and phase are monotonous decreasing functions of : the anelastic reflection acts as a low-pass filter, the phase response is nonlinear and ranges between an opposite and quadrature phase; -if R E ∕ > 0, the modulus is characterised by an extremum, i.e. M r m = | |∕4, that is located at the critical angular frequency and is associated with a quadrature filter. At lower and higher frequencies, the modulus decreases and increases with , respectively, i.e. the anelastic reflection is not a simple lowpass filter and acts as a complex attenuating filter, ranging between an opposite and in-phase response.

Modulus and phase associated with the anelastic propagation contribution
The frequency filter that takes into account both a seismic propagation in an upper anelastic layer and a reflection by an anelastic interface is the cumulative frequency filter. It is defined by the following modulus M and phase P: The global filter is characterised by a frequency-dependent phase rotation because P p ( ) has a minimum located at the angular frequency h ∕e and P r ( ) is a monotonous decreasing function of : additive contributions of the anelastic propagation and reflection effects occur when < h ∕e and the opposite occurs when > h ∕e. The frequency response of the phase can thus be assessed analytically based on equations (5b) and (15b). Table 3. Frequency behaviour of the filter associated with anelastic reflection (modulus M and phase P).
Similarly, the cumulative filter is characterised by a frequency-dependent modulus but the frequency response defined as does provide straightforward analytical solution. The frequency response depends on R E ∕ and its behaviour can be summarised as follows (Table 3): -if R E ∕ ≤ 0, M p and M r are both monotonous decreasing functions of the angular frequency , i.e. the cumulative filter is equivalent to a low-pass-filter with additive contributions of the anelastic propagation and reflection effects; -if R E ∕ > 0, M p is still a monotonous decreasing function of but M r is characterised by a minimum value M r m located at the critical frequency c . The number of roots for equation (19) depends on the physical parameters. Consequently, a numerical solution is required if further quantitative analysis is to be performed, allowing the descriptions reported in Table 3.
A numerical application with realistic values of the physical parameters is now performed to quantitatively assess the contributions of both the anelastic propagation and reflection in the frequency domain for different elastic R E and anelastic contrasts.

Numerical investigation
The aim of the numerical evaluation, performed over a large range of physical parameters, is to quantify the frequency filtering effects that include both the anelastic propagation and reflection contributions.
For the anelastic propagation contribution, we consider Q 1 = 100 and = 100 ms, which corresponds to Γ = 1273 Hz. The attenuation (20 log M p ) and the phase P p are represented as functions of the normalised frequency f n = ∕ h in the frequency bandwidth 0-150 Hz (figures 2a and 3a). For the anelastic reflection contribution, which depends on R E ∕ , we determine the attenuation (20 log M r ) and phase P r for three elastic reflection coefficients R E = −0.025, 0 and 0.025 and ranging from 0 to 0.2 with Q 1 = 100 (figures 2b1-3 and 3b1-3). Note that changing Q 1 to a lower value while keeping constant only affects the propagation contribution, both in amplitude and phase according to equations (4a) and (4b), but does not affect the reflection contribution. The case R E = 0 corresponds to a pure anelastic reflection induced by the constant-Q contrast only: for instance, when ≃0.1, the attenuation is about 30 dB. Note that this is similar to the attenuation of 32 dB associated with the pure elastic case = 0 with R E = ± 0.025. The cumulative effects of both the anelastic propagation and reflection are solved numerically to display the attenuation (20 log M) and P (figures 2c1-3 and 3c1-3) of the cumulative filter.
As expected by the previous analytical developments, (i) the filter associated with the anelastic propagation is equivalent to a low-pass filter with a frequency-dependent phase rotation, (ii) the filter associated with the anelastic reflection is equivalent to a low-pass filter with a frequency-dependent phase rotation when R E ≤ 0 and characterised by a critical frequency when R E > 0, and (iii) the cumulative filter is equivalent to a low-pass filter with a frequency-dependent phase rotation when R E ≤ 0. The numerical investigation highlights three main effects induced by the interaction of the propagation and reflection contributions when R E > 0: (i) it removes the critical frequency c when > c where c is a critical value of above which the modulus of the cumulative filter becomes a monotonous decreasing function of the frequency ( c ≃0.11 in the present case), (ii) modifies . We show that these critical curves ( c ′ , c ′ ) and ( c ′′ , c ′′ ) depend on (figure 4) and as a result, the cumulative filter modulus can be characterised by zero, one (case → 0) or two critical frequencies, i.e. the filter can be a low-pass, a band-pass or a high-pass filter depending on and when R E > 0 (figure 2c3).
This result means that for some particular cases, high frequencies of a seismic signal may be less attenuated than low frequencies when anelastic reflection is taken into account: this unexpected behaviour is controlled by the sign of the parameter D described in equation (12). These quantitative analyses are of first importance to better understand 5 Figure 2. (a) Modulus spectrum associated with the anelastic propagation contribution (normalised frequency f n = ∕ h ). Modulus spectra associated with the anelastic reflection contribution for a negative (b1), null (b2) and positive (b3) R E . (c1, c2 and c3) Modulus spectra associated with the cumulative effects of both the anelastic propagation and reflection for the three R E values. The dashed curves follow the critical frequency c (equation (17)) associated with the anelastic reflection contribution. The critical value c is indicated with a black star. The solid and dot-dashed curves follow the minimum and maximum amplitude values, respectively (numerical solution). the effects of both the anelastic propagation and reflection phenomena on seismic signals.

Peak-frequency attributes associated with a Ricker source wavelet
In this section, we investigate the effect of the cumulative filter associated with both anelastic propagation and reflection on a seismic signal by approximating the seismic incoming signal (t) with a Ricker wavelet. This wavelet, widely used to model a seismic source in geophysical prospecting (Wang 2015a), is a zero-phase signal expressed in the frequency domain by Wang (2015a): where p = 2 F P , with F P the peak frequency defined as the frequency associated with the maximum amplitude of Ψ( ). The peak frequency can be analytically described in the framework of propagation effects of anelasticity (Zhang & Ulrych 2002;Tary et al. 2017) for evaluating the Q factor. The present work extends these developments by including the reflection contribution. This cannot be undertaken when considering the central frequency (Wang 2015a), which can only be solved numerically. The use of the peak frequency constitutes a quantitative support for seismic analysis to better understand the physical phenomena discussed in the present study. The signal ′ (t) measured after a seismic propagation of the Ricker wavelet through the anelastic upper layer and a reflection at the anelastic interface is defined by the modulus M ′ ( ) = Ψ( )M( ) and the phase P( ). The peak frequency F P is changed into the peak-frequency attribute F ′ p defined as the solution ′ p = 2 F ′ p of M ′ = 0 (Ker et al. 2012;Wang 2015b). In the following, the amplitude A ′ p and phase ′ p of ′ (t) at F ′ p are the peak-frequency attributes used as complementary attributes in the present quantitative analysis of the wavelet distortion induced by the anelastic processes. The shape changing of the seismic wavelet, during propagation in the upper anelastic layer and reflection at the interface with the lower layer, is implicitly considered in the present analytical solution. The peak-frequency attribute is influenced 6 Figure 3. (a) Phase spectrum associated with the anelastic propagation contribution (normalised frequency f n = ∕ h ). Phase spectra associated with the anelastic reflection contribution for a negative (b1), null (b2) and positive (b3)R E . (c1, c2 and c3) Phase spectra associated with the cumulative effects of both the anelastic propagation and reflection for the three R E values. The dashed curves follow the critical frequency c (equation (17)) associated with the quadrature filter induced by the anelastic reflection contribution. The solid curve follows the quadrature filter induced by both the anelastic propagation and reflection (numerical solution). by the increasing asymmetry of the amplitude spectrum of the reflected wavelet generated by both propagation and reflection filters. In addition, the phase attribute ′ p is influenced by the effect of the wavelet shape changing induced by the cumulative filter on the phase spectrum of the reflected wavelet.
Based on the analytical developments presented in the previous section, we show that the peak-frequency attribute of the reflected signal associated with a Ricker seismic source wavelet is given by where the parameter quantifies the contribution of the anelastic reflection coefficient. We can show that B ranges between ±1∕2 . Equation (21) can be solved numerically but to develop an analytical expression, we consider that B( ) is dominated by the contribution of the peak-frequency attribute F p p related to the anelastic propagation, i.e. B( ) ≃ B( p p ) where p p = 2 F P p is the solution of (ΨM p ) = 0. We show that The present work focuses on the impact of the anelastic contrast on the peak-frequency modification and thus requires analysing F ′ p by fixing Q 1 in the upper layer, i.e.
we investigate the behaviour of F ′ p with changes of Q 2 only. The limit F ′ p∞ when Q 2 tends to 0, which corresponds to tending to infinity, is outside the validity range of the present work but a rough estimate is given in the Appendix. When Q 2 tends to Q 1 , which corresponds to tending to 0, we show that the limit is F ′ p0 = F p p . This analysis is limited to a Q 2 value between 0 and Q 1 as results associated with negative values are identical to those obtained by changing the sign of R E (i.e. the consequence of the dependency on the sign of R E ∕ as described in Section 3). We demonstrate that the . When tends to 0, the curve c ′ tends to the critical frequency c (equation (17)) associated with the anelastic reflection contribution (dashed curve). partial derivative of F ′ p in relation to Q 2 is expressed as (24) When R E ≤ 0, the peak frequency is monotonous and F ′ p ≤ F p p , with an increasing trend with Q 2 meaning that F ′ p decreases with . For the case R E = 0, F ′ p is a constant similar to F ′ p∞ .
When R E > 0, F ′ p is not monotonous but characterised by two extrema located at two particular Q 2 values. Only one corresponds to Q 2 < Q 1 and is expressed by It does not depend on R E and is located at and then decreases to F p p . Additionally, we note that The existence of the unexpected phenomenon F ′ p ≥ F p associated with a step-like interface is based on the condition < Q 1 2 F p , which deals with the anelastic propagation in the upper layer only.
For the peak-frequency attributes A ′ p and ′ p , we develop analytical expressions in relation to Q 2 , i.e. with when Q 1 is fixed based on approximations detailed in the Appendix and used here. The analysis predicts that A ′ p is a monotonous increasing function of if R E ≤ 0, but is characterised by an which differs from e . The analysis also predicts that ′ p decreases or increases monotonically with from ′ p∞ to p p (see Appendix) if R E < 0 or R E > 0, respectively, and is similar to ′ p∞ if R E = 0.

Numerical applications to quantify the anelastic Ricker wavelet distortions
The analytical expressions developed previously allow general considerations of the Ricker wavelet distortion induced by the anelastic processes. To illustrate the results, we consider the following parameters: the peak frequency of the Ricker source wavelet is F P = 50 Hz, R E = −0.025, 0 and 0.025, Q 1 = 100 and = 100 ms. We determine the reflected seismic signal as a function of in the range [0; 0.2], which corresponds to a Q 2 range of [5; 100], both in the time and frequency domains (figures 5 and 6). Note that because the Ricker frequency content is theoretically unlimited, the reference upper frequency F h has to be defined in order to be able to perform the analyses. In practice, this may be controlled by a threshold on the signal-to-noise ratio: an arbitrary threshold of 0.3 % corresponds to F h = 150 Hz. When tends to 0, the waveform associated with R E < 0 (figure 5a1) is in the opposite phase with the one associated with R E > 0 (figure 5a3), which is similar to the shape of the Ricker source wavelet, and for R E = 0 (figure 5a2), the amplitude of the waveform tends to 0. When increases to 0.2, reflected waveforms for the three cases look similar but strongly differ for intermediate values. This behaviour is quantified by the associated modulus and phase spectra displayed with respect to (figure 6), in accordance with the analytical results developed in the previous section. In particular, the peak-frequency attribute F ′ p (figure 6, blue curves) decreases monotonously with when R E < 0, is constant when R E = 0 and is characterised by an extremum when R E > 0 where F ′ p can be larger than F p (figure 6, dashedblack lines). It is interesting to note that (i) measuring   (21)) by the anelastic cumulative filter for a negative (left), null (middle) and positive (right) R E . The solid and dot-dashed-black curves follow the minimum and maximum amplitudes of the cumulative filter, respectively (numerical solution). (b1-3) Phase spectra of the signals displayed in figure 5: a mask defined on the modulus spectrum by a threshold of −24 dB has been applied to focus on the main representative frequency content. < Q 1 2 F p being satisfied, and (ii) whatever the value of R E , the peak frequency measured on the reflected signal is limited to the range [44; 52 Hz] (figure 7a1-3). This highlights relative changes of the peak-frequency attribute limited to [−8; 4%]. Importantly, can be assessed from F ′ p when R E < 0 but uncertainties increase when F ′ p tends to 44 Hz. The two peak-frequency attributes associated with F ′ p are also used to quantify the Ricker wavelet distortions. First, the amplitude A ′ p (figure 7b1-3, blue curves) increases with when R E ≤ 0, and is characterised by an extremum when R E > 0. Amplitude variations are about 10 dB when R E ≠ 0 and about to 20 dB when R E = 0. Second, the phase ′ p (figure 7c1-3, blue curves) strongly depends on both and the sign of R E : it highlights a phase rotation that decreases with in a limited range [2.5; 3 rad] when R E < 0, is constant to about 2 rad if R E = 0, and varies from in-phase to opposite phase when R E > 0 in the range [−0.1; 1.7 rad]. As a result, the phase of the cumulative filter significantly contributes to the variations in the reflected wavelet observed in the time domain (figure 5).
It is important to remember that the phase spectrum of the reflected signal associated with a zero-phase Ricker source wavelet is given by the phase of the cumulative filter. In order to focus on the phase rotation around the main frequency content of the reflected signal, we apply a mask defined by an amplitude threshold, fixed here to −24 dB on the modulus (figure 6b1-3): the difference between the minimum and maximum values of the phase monotonously increase with when R E < 0 but is characterised by an extremum when R E > 0 where most of the phase variations occurs between c ′ and c ′′ (figure 7c1 and 3, dashed lines).

Discussions on the peak-frequency attribute: pitfalls and insights
As quantified by the numerical example described above, the second order contribution of the anelastic propagation can be neglected in most realistic cases, i.e. F 2 p Γ 2 << 1. This allows us to neglect the case where a seismic wave vanishes in the upper layer and considering for sake of simplicity. Recall that Γ = 4Q 1 ∕ .
The peak-frequency attribute F ′ p is commonly used to assess the quality factor Q 1 of the upper layer. The approach is based on the assumption that = 0, i.e. the anelastic contribution of the reflection is negligible. In this case, a first approximation consists in F ′ p ≃ F p p which allows relating Q 1 and F ′ p according to . But this relation induces errors when anelastic reflection is not negligible. Based on the present study, we are able to quantify the modification of F ′ p when is taken into account, and as a result, the impact on the Q 1 assessment. The modification of F ′ p with relatively to F p p is quantified by ΔF = (F ′ p − F p p )∕F p p , which highlights a dependency with both , R E and (figure 8a1 and 2), and induces a relative modification of Q 1 quantified by ΔQ 1 (figure 8b1 and 2). When R E < 0, ΔF is negative and its amplitude increases with both and by up to 10% (figure 8a1): assessing Q 1 from F ′ p , whose relationship is nonlinear, results in an underestimation of 20% that increases to 75% at large To go beyond these particular numerical examples, it is interesting to observe the approximated linear dependency of ΔF with when is low. We can demonstrate that at large R E ∕ values, i.e. roughly larger than unity, ) and the modification of the peakfrequency attribute can be approximated as This result analytically predicts a significant modification of F ′ p even at low values. In addition, it also demonstrates a weak dependency with Q 1 and , quantified by F p ∕Γ, implying a low contribution of the effects of the anelastic propagation in the relative modification ΔF of the peak-frequency attribute. Finally, we show that the error in assessing Q 1 when is low can be expressed by the approximation which is nonlinear in and depends on Q 1 ∕Q 2 . Note that an underestimation of Q 1 is expected if R E < 0 and that according to equation (29), the error in Q 1 increases when F P decreases. For example, the error is almost doubled when F P decreases from 50 to 30 Hz.

Conclusions
In the present work, we have quantified the effects induced by both anelastic propagation and reflection by performing analytical developments of the equivalent cumulative filter defined in the framework of Kjartansson's model for a step-like reflector. In particular, the cumulative filter is characterised by an asymmetrical frequency behaviour that depends on the ratio R E ∕ between the elastic reflection coefficient of the reflector and the anelastic contrast between the upper and lower layers. In most cases, the cumulative filter acts as a low-pass filter in agreement with the classical understanding that higher seismic frequencies are more attenuated than low frequencies. The band-or high-pass behaviour occurs only for a limited range of source peak frequency, parameter and travel time , and when the signs of and R E are the same: this unexpected behaviour constitutes a key result.
In order to quantify the impact of the cumulative filter on an incoming seismic wave at normal incidence, we have developed analytical expressions for a Ricker source wavelet defined by a peak frequency. Because of the anelasic reflection, the reflected seismic signal is modified: its peak frequency and related amplitude and phase are used as attributes to quantify the distortions of the incoming source signal. We 11 show that the peak frequency depends on the sign of R E ∕ . The effect of a negative ratio is to amplify both the shift of the peak frequency and the phase rotation induced by the anelastic propagation contribution. With a positive ratio, the effect is to decrease the shift of the peak frequency. We also highlight that the peak frequency can be larger than the source peak frequency, an unexpected phenomenon for a step-like interface controlled by the two critical frequencies of the cumulative filter when < Q 1 2 F p . The consequence of the cumulative filter is to introduce fluctuations in the peak frequency that cannot be explained by considering anelastic propagation alone as is commonly assumed when estimating the Q factor from the peak frequency. For a 50 Hz Ricker source wavelet, the modification of the peak-frequency attribute relative to the propagation contribution can be as high as 10%, with a strong impact when assessing the quality factor of the layer above the anelastic reflector. Errors arise as soon as is not negligible and follow a nonlinear relationship with .
The accuracy of the Q value estimation based on the peak frequency of a seismic wave reflected by an anelastic reflector depends on many physical parameters, as analytically demonstrated in this work that considered a Ricker wavelet. Consequently, estimating the quality factor of the upper layer from this seismic attribute alone is not a well-constraint inversion problem. A suggestion would be to consider further quantification in taking advantage of the full set of peak-frequency attributes introduced in this study in a simultaneous inversion scheme to assess both the elastic reflection coefficient and the constant-Q parameters defining the anelastic reflector.

Asymptotic limits when tends to infinity
The limit when is large is equivalent to Q 2 tends toward 0 for Q 1 fixed in the upper layer. This is outside the validity range of the approximations used in the present study but as a first estimate, the asymptotic limits of the peak frequency and phase attributes can be expressed by: . (32b) 1 − 16D 2 ( p p ) 1 + 16D 2 ( p p ) .
12 Behaviour of A ′ p with Q 2 : The behaviour of the parameter D with the frequency is inversely proportional to the frequency and with F ′ p >> 1 similar in magnitude to F p p , a first approximation consists in replacing D( ′ p ) by D( p p ), which gives (35) A minimum amplitude is located at the quality factor

Behaviour of
By considering both Q 1 >> 1 and << 1s, which corresponds to a frequency Γ much larger than the seismic frequency, the behaviour of ′ p with Q 2 can be approximated to: .
(37) Since ranges between −1 and 1, the peak phase attribute increases or decreases monotonically with Q 2 from ′ p∞ (equation (32b)) to p p (equation (30c)) when R E is negative or positive, respectively.