Abstract

Wave equation tomography attempts to improve on traveltime tomography, by better adhering to the requirements of our finite-frequency data. Conventional wave equation tomography, based on the first-order Born approximation followed by cross-correlation traveltime lag measurement, or on the Rytov approximation for the phase, yields the popular hollow banana sensitivity kernel indicating that the measured traveltime at a point is insensitive to perturbations along the ray theoretical path at certain finite frequencies. Using the instantaneous traveltime, which is able to unwrap the phase of the signal, instead of the cross-correlation lag, we derive new finite-frequency traveltime sensitivity kernels. The kernel reflects more the model-data dependency, we typically encounter in full waveform inversion. This result confirms that the hollow banana shape is borne of the cross-correlation lag measurement, which exposes the Born approximations weakness in representing transmitted waves. The instantaneous traveltime can thus mitigate the additional component of nonlinearity introduced by the hollow banana sensitivity kernels in finite-frequency traveltime tomography. The instantaneous traveltime simply represents the unwrapped phase of Rytov approximation, and thus is a good alternative to Born and Rytov to compute the misfit function for wave equation tomography. We show the limitations of the cross-correlation associated with Born approximation for traveltime lag measurement when the source signatures of the measured and modelled data are different. The instantaneous traveltime is proven to be less sensitive to the distortions in the data signature. The unwrapped phase full banana shape of the sensitivity kernels shows smoother update compared to the banana–doughnut kernels. The measurement of the traveltime delay caused by a small spherical anomaly, embedded into a 3-D homogeneous model, supports the full banana sensitivity assertion for the unwrapped phase.

INTRODUCTION

In seismic inversion, the goal is to find an accurate subsurface velocity model that produces, with the proper physical simulation, data similar to the observed ones. For this task, many inversion schemes exist spanning the whole wavefield theoretical frequency spectrum from traveltime tomography to the full waveform inversion (FWI). For traveltime tomography, a high-frequency assumption is typically employed and the objective function is based on the misfit in arrival traveltimes (Dziewonski 1984; Van Der Hilst et al.1997). The model is thus updated by the backprojection of the traveltime residuals along rays or using the adjoint state. Nevertheless, ray-tracing-based methods fail to model the wave behaviour in complex media and more attention, recently, has been paid to finite-frequency-based analysis and wavefield modelling (Woodward 1992; Marquering et al.1998; Dahlen et al.2000; Montelli et al.2004; Tromp et al.2004) in order to overcome these shortcomings.

In FWI (Tarantola 1984; Virieux & Operto 2009), the misfit represents the difference between the modelled and observed data. It takes advantage of the whole broad-band data, which give better resolution compared to traveltime tomography. However, the biggest limitation of FWI is the highly nonlinear nature of the misfit function (Bunks et al.1995; Virieux & Operto 2009).

Another approach, referred to usually as the wave equation tomography (WET; Devaney 1984; Pratt & Goulty 1991; Luo & Schuster 1991; Woodward 1992) had been proposed in the past to tackle the same problem. Here, the misfit function for WET is given in traveltimes, but the modelling and model update is based on solving the wave equation and specifically on the wavefields. After generating synthetic data using a predicted model, the traveltime difference is measured generally by identifying the maximum of the cross-correlation between the two waveforms (observed and modelled). Though this misfit function, like the conventional high-frequency asymptotic-based traveltime tomography is quasi-convex, computing the maximum of the cross-correlation is only valid under certain circumstances (Hörmann et al.2002; De Hoop & Van Der Hilst 2005; Van Leeuwen & Mulder 2010). Cross-correlation computes the phase shift correctly only when the source spectrum of the modelled data and the real data are in phase.

Since WET is based on the misfit in finite-frequency traveltime, the inversion part is based on identifying regions in the model that could have contributed to this misfit. Mapping such regions is commonly referred to as the traveltime sensitivity kernel. The first step is to use first-order Born approximation (Beydoun & Tarantola 1988; Woodward 1992; Bleistein et al.2000; Aki & Richards 2002) to derive an equation for the wavefield perturbation (Woodward 1992; Marquering et al.1998). Then, cross-correlation of the background wavefield with the total wavefield is applied to compute the traveltime sensitivity kernels (Marquering et al.1998; Dahlen et al.2000; Hung et al.2000; Tromp et al.2004). Another approach utilizes the Rytov approximation, which gives the phase perturbation as a function of the Born wavefield perturbation (Snieder & Lomax 1996; Jocker et al.2006; Liu et al.2009). For both methods, a non-intuitive observation was first made by Woodward (1992), that the finite-frequency traveltime has zero sensitivity to slowness perturbation along the ray path. Marquering et al. (1999) and many other authors confirmed the same observations. Such sensitivity kernels are based on the Born approximation, which yields poor approximation for transmitted waves (Bleistein et al.2000). For the Rytov approximation, a linear relation between phase and traveltime is used to derive the sensitivity kernels. In fact, for real inversion problems, the Rytov approximation for phase perturbation should be unwrapped in order to compute accurate misfits (Gélis et al.2007). Thus, limitations in the Born and the Rytov approximations raise serious questions to the accuracy and applicability of the banana–doughnut kernels.

We can also extract the traveltime information from wavefield by using the formula dedicated to unwrap the phase (Stoffa et al.1974; Shin et al.2003; Choi & Alkhalifah 2011; Choi et al.2011). Shin et al. (2003) used the derivative of the wavefield with respect to the angular frequency to estimate traveltimes. A similar idea was utilized by Choi et al. (2011) to automatically pick the first arrival events by applying a damping factor. We refer to the computed traveltime through measuring the frequency derivative of the wavefield as the instantaneous traveltime. It provides us with an alternative finite-frequency traveltime measurement. The question we raise and address in this paper is whether this is a good alternative measure of finite-frequency traveltime than those associated with the lag of the cross-correlation of the measured and simulated wavefields.

Based on our previous work (Djebbi & Alkhalifah 2012, 2013), we investigate the prospect of using the instantaneous traveltime as a source for finite-frequency traveltimes. We analyse the traveltime sensitivity kernels and compare them to the banana–doughnut kernels. In this paper, we only consider the kernels for the direct P wave. We begin by reviewing Born and Rytov approximations and the resulting sensitivity kernels. We also introduce the instantaneous traveltime and derive the associated sensitivity kernels. We show that the instantaneous traveltime reduces to the unwrapped phase version of the Rytov approximation. Through numerical examples, we show how the instantaneous traveltime is more suitable to resolve small scatterers compared to the cross-correlation approach, even when the source spectrum of the modelled and observed data are not identical. We also present the sensitivity kernels obtained for a simple velocity model, as well as for a more complex one and we compare them with the banana–doughnut kernels. We show how the unwrapped phase is a good alternative approach for WET through analysis of the sensitivity kernels’ properties. Finally, we consider the effect of a small spherical anomaly embedded inside a 3-D homogeneous velocity model. According to the banana–doughnut kernel observations, such anomalies are hidden inside the banana–doughnut hole for certain frequencies. Based on the unwrapped phase, we will show that a small anomaly affects the wavefield and thus the measured traveltimes at the receivers. This final test confirms that instantaneous traveltime is a good alternative approach for WET.

THEORY

The traveltime sensitivity kernels or the Fréchet derivative are computed by mapping the region of the velocity model that contributes to the traveltime recorded at a specific receiver. Since the traveltime residual in WET is back-projected on that region, the kernels estimation affects the accuracy of the inversion. In this section, we review the theoretical background behind the finite-frequency traveltime sensitivity kernels.

Assuming that the squared slowness is the sum of a smoothly varying background, s0, and a weak perturbation, δs, the first-order Born approximation could be used to model the scattered wavefield.

Thus, the total wavefield is the sum of a background wavefield, U0, and a scattered wavefield, δUB as follows:  

(1)
\begin{equation} U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)= U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)+\delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right). \end{equation}
The first-order Born approximation of the wavefield perturbation (Beydoun & Tarantola 1988; Woodward 1992; Bleistein et al.2000; Aki & Richards 2002; Snieder & Lomax 1996) is given by  
(2)
\begin{equation} \delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)= \omega ^2 \int _{\mathbf {x}} G_0\left(\mathbf {x_r},\omega ;\mathbf {x} \right)G_0\left(\mathbf {x},\omega ;\mathbf {x_s} \right)\delta {\rm s}\left(\mathbf {x} \right){\rm d}\mathbf {x}^3, \end{equation}
where G0(x, ω; xs) is the Green's function for a source located at xs and recorded at the scatterer point x. G0(xr, ω; x) is the Green's function ignited at the scatterer x and recorded at the receiver xr. δs(x) is the squared slowness perturbation.

The Born approximation could be written in the form (Woodward 1992; Marquering et al.1998),  

(3)
\begin{equation} \delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)= \int _{\mathbf {x}} K_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x}, \mathbf {x_s}, \omega \right)\delta {\rm s}\left(\mathbf {x} \right) {\rm d}\mathbf {x}^3, \end{equation}
where $$K_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x}, \mathbf {x_s}, \omega \right)= \omega ^2 G_0\left(\mathbf {x_r}, \omega ; \mathbf {x} \right) G_0\left(\mathbf {x}, \omega ; \mathbf {x_s}\right)$$ is the waveform sensitivity kernel.

The Born approximation is a scattering location perturbation based on small scatterers in size and magnitude. This condition is more crucial when the Born approximation is used to represent transmitted waves, as any phase shift induced by the scatterer will not be reflected in the phase term. Considering a small scatterer located on the ray path and Green's functions of the form $$G_0\left(\mathbf {x},\omega ;\mathbf {x_s}\right) = A_{\mathbf {x_sx}} e^{i\varphi _{\mathbf {x_sx}}}$$, the phase of Born perturbation equation (3) is $$\varphi _{\mathbf {x_sx}}+\varphi _{\mathbf {xx_r}}$$. The phase of the direct wave is exactly the same as the perturbation phase: $$\varphi _{\mathbf {x_sx_r}} = \varphi _{\mathbf {x_sx}}+\varphi _{\mathbf {xx_r}}$$. As a result, the effect of the scatterer on the perturbed wavefield is visible on the amplitude and not on the phase. As such, Bleistein et al. (2000) demonstrated that Born approximation is poor for transmitted waves. In spite that the size of the effective scatterer can be larger for low frequencies as Born is also inherently a low-frequency approximation, the problem still exists.

Cross-correlation and Rytov traveltime sensitivity kernels

The time-domain cross-correlation, eq. (4), is used to measure the similarity between the observed wavefield U(t), and the modelled wavefield U0(t). The observed wavefield differs from the modelled wavefield by a small Born perturbation δUB(t).  

(4)
\begin{equation} C \left( t \right) = \int _{\tau _1}^{\tau _2 } U_0\left( \tau - t \right) U\left( \tau \right) \mathrm{d}\tau . \end{equation}
It typically has a maximum at δt measured from the zero lag, given by  
(5)
\begin{equation} \delta t ={\rm argmax}_t \, C \left( t \right). \end{equation}
Using the cross-correlation property equation (5), and based on the first-order Born wavefield perturbation, Marquering et al. (1999) derived an equation to compute the traveltime perturbation δt,  
(6)
\begin{equation} \delta t =\frac{\int _{\tau _1}^{\tau _2} \dot{U_0}(\tau )\delta U_{\rm B}(\tau ) {\rm d}\tau }{\int _{\tau _1}^{\tau _2} \ddot{U_0}(\tau )U_0(\tau )\,\mathrm{d}\tau }, \end{equation}
where $$\dot{U_0}( \tau )$$ and $$\ddot{U_0}(\tau )$$ denote the first- and second-order derivatives with respect to time.

The cross-correlation-based traveltime sensitivity kernel, or as Marquering et al. (1999) suggested the Born kernel, is given by  

(7)
\begin{equation} K_{{\rm t}}^{{\rm Cross}}\left( \mathbf {x_r},\mathbf {x}, {\bf x_s}\right) =\frac{\int _{\tau _1}^{\tau _2} \dot{U}_0(\tau )K_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},\tau \right)\,\mathrm{d}\tau }{\int _{\tau _1}^{\tau _2} \ddot{U}_0(\tau )U_0(\tau )\,\mathrm{d}\tau }. \end{equation}
To compute the traveltime sensitivity kernel $$K_{{\rm t}}^{{\rm Cross}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s}\right)$$, Marquering et al. (1999) used surface mode summation to evaluate seismograms and the wavefield sensitivity kernel. Accordingly, the resulting kernel is band-limited and it includes the effect of all frequencies of the source wavelet.

To evaluate the monochromatic version of the cross-correlation sensitivity kernel, we compute the Green's functions and the wavefield single-frequency sensitivity kernel $$K_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right)$$ in the frequency domain. Then, we apply an inverse Fourier transform.

Considering $$U^1_0(t)$$ the inverse Fourier transform of the Green's function G0(xr, ω; xs), and the wavefield kernel $$K^1_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},t\right)$$ the inverse Fourier transform of $$K_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right)$$, the monochromatic sensitivity kernel is given by  

(8)
\begin{equation} K_{{\rm t}}^{{\rm Cross}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right) =\frac{\int _{\tau _1}^{\tau _2} \dot{U}^1_0(\tau )K^1_{U_{\rm B}}\left(\mathbf {x_r},\mathbf {x},\mathbf {x_s},\tau \right)\,\mathrm{d}\tau }{\int _{\tau _1}^{\tau _2} \ddot{U}^1_0(\tau )U^1_0(\tau )\,\mathrm{d}\tau }. \end{equation}
We add a dependency on the angular frequency ω in the monochromatic sensitivity kernel's notation to differentiate it from the band-limited one. We will show in section 2.4 how we estimate the band-limited kernels based on a weighted summation over monochromatic kernels (Rickett 2000).

Alternatively, the Rytov approximation (Woodward 1992; Snieder & Lomax 1996; Jocker et al.2006; Liu et al.2009) could be used to derive the monochromatic traveltime sensitivity kernels. It is more adapted for the representation of transmitted waves than Born approximation. The first-order Rytov approximation provides a relationship between the slowness squared perturbation δs(x) and the complex phase perturbation δϕR(xr, ω; xs),  

(9)
\begin{eqnarray} &&{\delta \phi _R\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) = \frac{\delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) }} \nonumber \\ &&{\qquad= \frac{\omega ^2}{ U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \int _{\mathbf {x}} G_0\left(\mathbf {x_r},\omega ;\mathbf {x} \right)G_0\left(\mathbf {x},\omega ;\mathbf {x_s} \right)\delta s\left(\mathbf {x} \right)d\mathbf {x}^3.} \end{eqnarray}
Assuming that the phase is a linear function of traveltime (i.e. the high-frequency approximation), ϕR = iωt. The traveltime perturbation is the imaginary part of the phase perturbation divided by the angular frequency ω (Woodward 1992; Snieder & Lomax 1996), 
(10)
\begin{equation} \delta t=\frac{\Im [\delta \phi _R\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)] }{\omega } = \frac{ \Im \left( \frac{\delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) }{\omega } . \end{equation}
The symbol ℑ(.) stands for the imaginary part.

Finally, the single-frequency Rytov traveltime sensitivity kernel is computed as  

(11)
\begin{equation} K_{\rm t}^{{\rm Rytov}}\left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s}, \omega \right) = \frac{ \Im \left( \frac{K_{U_{\rm B}}\left( \mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right)}{U_0\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) }{\omega }. \end{equation}

Unwrapped phase sensitivity kernels

Traveltime measurement using cross-correlation is only valid under certain circumstances (Hörmann et al.2002). In fact, De Hoop & Van Der Hilst (2005) showed that cross-correlation computes the traveltime difference correctly only when the source spectrum of the modelled data and the real data is identical. For the Rytov approximation, a high-frequency assumption of the phase is used to separate the traveltime term. Thus, for both methods, there exist some limitations which may lead to inaccurate results when used for traveltime sensitivity kernels derivation and traveltime inversion.

Alternatively, we use another approach to compute the traveltime misfit. Taking the derivative of the wavefield with respect to the angular frequency, dividing by the wavefield, and finally taking the imaginary part can be a good alternative to cross-correlation-based traveltimes. The result is referred to as the instantaneous traveltime (Choi & Alkhalifah 2011; Choi et al.2011), 

(12)
\begin{equation} t=\Im \left( \frac{\frac{\partial U \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right). \end{equation}
Instantaneous traveltime measurements, like the cross-correlation or the Rytov approach, is based on wavefields; however, there is no assumption on the data which results in a more practical method. Using this approach, we compute traveltimes automatically without the need for picking. The instantaneous traveltime, able to unwrap the phase, gives accurate estimates of the traveltime delays. It is the unwrapped phase version of Rytov approximation, and could be used to derive the sensitivity kernels.

To compute the derivative of the wavefield, we utilize the Helmholtz wave equation:  

(13)
\begin{equation} \mathbf {S}(\omega ) \mathbf {U}(\omega ) = \mathbf {f}, \end{equation}
where S is the impedance matrix, U is the wavefield and f the source. Taking the derivative of eq. (13) with respect to ω yields,  
(14)
\begin{equation} \mathbf {S}(\omega ) \frac{\partial \mathbf {U}(\omega )}{ \partial \omega } =- \frac{\partial \mathbf {S}(\omega )}{\partial \omega } \mathbf {U}(\omega ). \end{equation}
Since we solve for the Green's functions, the source f is frequency independent and the term ∂f/∂ω is not included in the right-hand side of eq. (14). Note that the derivative is obtained by solving the same Helmholtz equation, but with a virtual source $$-\frac{\partial \mathbf {S}(\omega )}{\partial \omega } \mathbf {U}(\omega )$$. Assuming that we use a direct solver, a considerable part of the computational cost is for the impedance matrix decomposition. The same impedance matrix is used for both eqs (13) and (14). As a result, the computational cost is not affected heavily by computing the wavefield derivative.

Considering a wavefield of the form of eq. (1), the instantaneous traveltime is (Djebbi & Alkhalifah 2012)  

(15)
\begin{eqnarray} t=\Im \left( \frac{\frac{\partial U \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right) \!=\! \Im \left( \frac{\frac{\partial U_0 \left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right)+\Im \left( \frac{\frac{\partial \left( \delta U_{\rm B} \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) \right)}{\partial \omega }}{U \left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right).\nonumber\\ \end{eqnarray}
For small perturbations, we use the Born approximation giving U(xr, ω; xs) ≈ U0(xr, ω; xs). The instantaneous traveltime becomes  
(16)
\begin{equation} t \approx \Im \left( \frac{\frac{\partial U_0 \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U_0 \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right)+\Im \left( \frac{\frac{\partial \left( \delta U_{\rm B} \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) \right)}{\partial \omega }}{U_0 \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right) = t_0 + \delta t. \end{equation}
Finally, using the instantaneous traveltime, we derived an equation for the traveltime perturbation for a weak wavefield perturbation,  
(17)
\begin{equation} \delta t = \Im \left( \frac{\frac{\partial \left( \delta U_{\rm B} \left(\mathbf {x_r},\omega ; \mathbf {x_s} \right) \right)}{\partial \omega }}{U_0 \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right). \end{equation}
The monochromatic traveltime sensitivity kernel using the instantaneous traveltime is given by 
(18)
\begin{equation} K_{\rm t}^{{\rm Inst}}\left( \mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right) = \Im \left( \frac{\frac{\partial \left( K_{U_{\rm B}}\left( \mathbf {x_r},\mathbf {x},\mathbf {x_s},\omega \right) \right)}{\partial \omega }}{U_0 \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right). \end{equation}
The derivative of the wavefield kernel with respect to ω is 
(19)
\begin{eqnarray} \frac{\partial \left( K_{U_{\rm B}} \left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s},\omega \right) \right)}{\partial \omega } &=& 2\omega G_0\left(\mathbf {x_r},\omega ;\mathbf {x} \right)G_0\left(\mathbf {x},\omega ;\mathbf {x_s}\right) \nonumber \\ &&+\; \omega ^2 \Big[ \frac{\partial G_0\left(\mathbf {x_r},\omega ;\mathbf {x} \right)}{\partial \omega } G_0\left(\mathbf {x},\omega ;\mathbf {x_s}\right)\nonumber\\ && +\; G_0\left(\mathbf {x_r},\omega ;\mathbf {x} \right)\frac{\partial G_0\left(\mathbf {x},\omega ;\mathbf {x_s}\right)}{\partial \omega }\Big], \end{eqnarray}
which, as we saw earlier, is computed using a Helmholtz solver for a virtual source as described in eq. (14).

Relation between unwrapped phase and Rytov sensitivity kernels

The phase perturbation given by the Rytov approximation (eq. 9) is accurate if the traveltime residual is within a half cycle. Otherwise, we will need to unwrap the Rytov to obtain the accurate traveltime difference. When deriving the sensitivity kernels based on the Rytov approximation, a linear relation between the phase perturbation and traveltime perturbation is considered. The sensitivity kernels based on the instantaneous traveltime have similar forms as the Rytov ones; however, the phase is now unwrapped.

If we consider the Rytov wavefield approximation  

(20)
\begin{equation} U_{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) = U_0\left( \mathbf {x_r}, \omega ; \mathbf {x_s} \right) {\rm e}^{\delta \phi _{\rm R} \left( \mathbf {x_r}, \omega ; \mathbf {x_s} \right)}, \end{equation}
we can use the instantaneous traveltime given by eq. (12) to derive an equation for the traveltime perturbation δt, as follows: 
\begin{eqnarray*} t &=& \Im \left( \frac{\frac{\partial U_{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) \end{eqnarray*}
 
(21)
\begin{eqnarray} \nonumber \;&=& \Im \left( \frac{\frac{\partial U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) {\rm e}^{\delta \phi _{\rm R} \left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}}{\partial \omega }}{U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) \nonumber \\ &=& \Im \left( \frac{\frac{\partial U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega } {\rm e}^{\delta \phi _{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}}{U\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) \nonumber\\ &&+\; \Im \left( \frac{U_0\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right) {\rm e}^{\delta \phi _{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \frac{\partial \delta \phi _{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) }{\partial \omega } }{U\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right)\nonumber \\ &=& \Im \left( \frac{\frac{\partial U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right) + \Im \left(\frac{\partial \delta \phi _R\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right) }{\partial \omega } \right)\nonumber \\ &=& t_0 + \delta t. \end{eqnarray}
The traveltime perturbation using the unwrapped phase is, then, given by 
(22)
\begin{equation} \delta t_{{\rm inst}}= \Im \left(\frac{\partial \delta \phi _{\rm R}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) }{\partial \omega } \right). \end{equation}
In the Rytov traveltime perturbation equation (10), a linear relation between phase and traveltime is considered, δϕ = iωδt. Then, for this special case, we have  
(23)
\begin{equation} \delta t_{\rm Rytov} = \delta t_{\rm inst}. \end{equation}
As a result, the instantaneous traveltime sensitivity kernels are the unwrapped versions of the Rytov ones. For the general case, we consider an unknown relation between the phase and traveltime. Then, phase unwrapping through the instantaneous traveltime is necessary to get the traveltime perturbation,  
(24)
\begin{eqnarray} \delta t_{\rm inst} &=& \Im \left(\frac{\partial \delta \phi _{\rm R} \left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega } \right) = \Im \left(\frac{\partial \left(\frac{\delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}\right) }{\partial \omega } \right) \nonumber \\ &=& \Im \left( \frac{\frac{\partial \delta U_{\rm B}\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} - \frac{ \delta U_B\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) \frac{\partial U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U_0^2\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right). \end{eqnarray}
For weak phase perturbation, the contribution of the second term in eq. (24) is negligible. This was verified numerically for a band-limited kernel in a 3-D v(z) model. Accordingly, we neglect the term $${ \delta U_{\rm B}\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right) [{\partial U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}/{\partial \omega }]}/{U_0^2\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)}$$ and the instantaneous traveltime perturbation corresponds to the approximation 17, 
(25)
\begin{equation} \delta t_{\rm inst} \approx \Im \left( \frac{\frac{\partial \delta U_{\rm B}\left(\mathbf {x_r},\omega ; \mathbf {x_s} \right)}{\partial \omega }}{U_0\left( \mathbf {x_r},\omega ; \mathbf {x_s} \right)} \right). \end{equation}
Eq. (25) is the general form of Rytov phase approximation. It unwraps the phase and is used to compute the sensitivity kernels given by eq. (18).

Broad-band kernels

The monochromatic sensitivity kernels, $$K_{\rm t}^{{\rm Cross}}\left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s},\omega \right)$$, $$K_{\rm t}^{{\rm Rytov}}\left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s},\omega \right)$$ and $$K_{\rm t}^{{\rm Inst}}\left(\mathbf {x_r},\mathbf {x}, \mathbf {x_s},\omega \right)$$ are used to compute the band-limited kernels (Woodward 1992; Snieder & Lomax 1996; Marquering et al.1999; Djebbi & Alkhalifah 2013). A weighted summation over frequencies is applied. The weighting function could be a Gaussian distribution or the amplitude spectrum of the Ricker wavelet.

We choose the amplitude spectrum of a Ricker wavelet as our weighting function. The broad-band sensitivity kernel is given by  

(26)
\begin{eqnarray} K_{\rm t}^{i}\left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s}\right) = \sum _{\omega } P(\omega ) K_{\rm t}^{i}\left( \mathbf {x_r},\mathbf {x}, \mathbf {x_s},\omega \right) ;\nonumber\\ i = {\rm Cross},\; {\rm Rytov},\; {\rm Inst}. \end{eqnarray}
The weighting function is  
(27)
\begin{equation} P(\omega ) = 4 \sqrt{\pi }\frac{\omega ^2}{\omega _0^3} {\rm e}^{-\frac{\omega ^2}{\omega _0^2} }, \end{equation}
where ω = 2πf is the angular frequency, ω0 = 2πf0 and f0 is the peak frequency of the Ricker wavelet.

Since both Rytov and the cross-correlation give the same type of sensitivity kernels (Rickett 2000; Liu et al.2009), we will use the term banana–doughnut kernels instead of Rytov or cross-correlation in our analysis.

CROSS-CORRELATION LIMITATIONS IN TRAVELTIME DELAY MEASUREMENT

In conventional WET implementations, cross-correlation is used to compute the traveltime lag δt. Thus, finding δt measured from the zero lag to the maximum energy of the cross-correlation represents the shifts required for the synthetic seismogram to give the best match to the observed seismogram (eq. 5).

Hörmann et al. (2002) and De Hoop & Van Der Hilst (2005) showed that cross-correlation computes the traveltime difference correctly only when the source spectrum of the modelled data and the real data are in phase, which is not always the case even with wavelet estimation techniques. Van Leeuwen & Mulder (2010) derived a new misfit based on a weighted norm of the cross-correlation. Such regularization makes the misfit function less sensitive to the difference in wavelet spectrum.

Fig. 1 shows two different scenarios for traveltime lag measurement. In Fig. 1(a), the two wavelets have the same shape and time difference between the two is δt = 0.1 s. However, in Fig. 1(b), one of the wavelets is subjected to a phase rotation equal to π/2. In this numerical test, we measure the traveltime difference by picking the lag corresponding to the maximum of the cross-correlation. Also, we use the instantaneous traveltime given by eq. (12). The traveltime lag is basically the difference between the measured instantaneous traveltimes for both.

Figure 1.

Two different scenarios. (a) Two Ricker wavelets, one of the two is advanced with δt = 0.1 s with no phase rotation. (b) Two Ricker wavelets, one of the two wavelets is advanced with δt = 0.1 s with a phase rotation π/2.

Figure 1.

Two different scenarios. (a) Two Ricker wavelets, one of the two is advanced with δt = 0.1 s with no phase rotation. (b) Two Ricker wavelets, one of the two wavelets is advanced with δt = 0.1 s with a phase rotation π/2.

Fig. 2 shows the cross-correlation for the two scenarios presented in Fig. 1. The maximum of the cross-correlation corresponds to the measured traveltime lag. When the two wavelets have the same signature, the maximum of the cross-correlation is exactly the true traveltime time lag δt = 0.1 s. However, when one of the two wavelets is phase rotated, cross-correlation estimates an inaccurate traveltime delay. The maximum of the cross-correlation corresponds to δt = 0.08 s.

Figure 2.

Cross-correlation of the two scenarios depicted in Fig. 1. (a) No phase rotation, the maximum of the cross-correlation corresponds to δt = 0.1 s. (b) π/2 phase rotation, the maximum of the cross-correlation corresponds to δt = 0.08 s and cross-correlation is inaccurate.

Figure 2.

Cross-correlation of the two scenarios depicted in Fig. 1. (a) No phase rotation, the maximum of the cross-correlation corresponds to δt = 0.1 s. (b) π/2 phase rotation, the maximum of the cross-correlation corresponds to δt = 0.08 s and cross-correlation is inaccurate.

In Fig. 3 , we show the traveltime difference measured using the instantaneous traveltime. The two figures present the measured δt as a function of the frequency. For all the frequencies equal or larger than 2 Hz, the instantaneous traveltime is able to predict the exact values of the delay for both cases. We conclude that the instantaneous traveltime is less sensitive to wavelet distortions. As a result, the instantaneous traveltime is an accurate alternative measure of traveltime misfit to the cross-correlation approach. Thus, it could be used to derive the traveltime sensitivity kernels instead of the conventional approaches.

Figure 3.

Traveltime difference measurement using the instantaneous traveltime (unwrapped phase). (a) No phase rotation. (b) π/2 phase rotation. For both cases, the traveltime lag δt is given as a function of the frequency, for most of the frequencies, the instantaneous traveltime is able to compute the exact traveltime difference. We can pick the lag for the wavelet peak frequency f = 10 Hz.

Figure 3.

Traveltime difference measurement using the instantaneous traveltime (unwrapped phase). (a) No phase rotation. (b) π/2 phase rotation. For both cases, the traveltime lag δt is given as a function of the frequency, for most of the frequencies, the instantaneous traveltime is able to compute the exact traveltime difference. We can pick the lag for the wavelet peak frequency f = 10 Hz.

TRAVELTIME SENSITIVITY KERNELS

In this section, we compare the sensitivity kernels using the instantaneous traveltime and the banana–doughnut ones through numerical examples. We consider a simple 3-D model with an increasing velocity with depth and a more complex Marmousi 2-D model.

3-D simple model

We consider a linearly increasing velocity model with depth, v(z) = v0 + αz, where v0 = 2.0 km s−1 and α = 0.5 s−1. For this type of velocity model a simple two-point ray tracing could be used to compute approximate Green's functions (Rickett 2000). We consider a source and a receiver placed on the surface to mimic the behaviour of diving waves, which are useful to invert for near surface structures in WET.

Fig. 4 shows the single-frequency kernels for f = 30 Hz. In order to compute the broad-band kernels, we apply a weighted summation of the monochromatic kernels. The weighting function consists of the normalized amplitude spectrum of a Ricker wavelet with peak frequency equal to 30 Hz (eq. 28).

Figure 4.

The 30-Hz single-frequency traveltime sensitivity kernels. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Figure 4.

The 30-Hz single-frequency traveltime sensitivity kernels. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

We show in Fig. 5 the resulting, broad-band 30 Hz, traveltime sensitivity kernels based on cross-correlation and the instantaneous traveltime. A constructive interference of the monochromatic kernels can be noted in the banana-shape zone around the ray path. In the rest of the model, a destructive interference removes the effect of any slowness perturbation on traveltimes. Fig. 5(a) shows the sensitivity kernel obtained using the Born approximation with cross-correlation lags. We observe the same features obtained first by Woodward (1992) and then confirmed by many other authors (Marquering et al.1999; Dahlen et al.2000; Hung et al.2000; Rickett 2000). The sensitivity kernel has a hollow banana shape. This is the case even if we use the Rytov approximation (Snieder & Lomax 1996; Rickett 2000; Jocker et al.2006; Liu et al.2009). In the cross-section, it has a doughnut shape. The observation is counter-intuitive since the traveltime is insensitive to velocity perturbations along the ray path. In the high-frequency limit, these sensitivity kernels are expected to thin out and reduce to the ray path. However, the doughnut effect collapse is not clear, which raises a possible singularity in this definition, suggesting that the traveltime defined by the high-frequency asymptotic approximation is different than that associated with correlation lag in the high-frequency limit.

Figure 5.

The 30-Hz broad-band traveltime sensitivity kernels for v(z). (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Figure 5.

The 30-Hz broad-band traveltime sensitivity kernels for v(z). (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

On the other hand, the instantaneous traveltime yields the kernel as shown in Fig. 5(b). The traveltime kernel has a banana shape but now the banana is a plain one. Traveltimes are sensitive to slowness perturbation along the geometric ray path. The hollow banana shape is the direct result of using cross-correlation traveltimes. Alternative methods for traveltime measurement result in a different kernel shape. Van Leeuwen & Mulder (2010) developed a WET by regularizing the cross-correlation. This regularization makes the sensitivity kernels more stable and has a plain banana shape. Here, in our instantaneous traveltime time method, we get directly a realistic plain kernel with lower sensitivity to the wavelet's spectrum distortion. For the Rytov approximation, it has limitations caused by assuming a linear relation between phase and traveltime. This approximation suffers from the phase wrapping phenomena; thus, the instantaneous traveltime is a good alternative, since it unwraps the phase of the Rytov. For realistic models, the instantaneous traveltime includes a derivative operator. When applied to the wavefield, it is able to overcome some of the existing non-linearities in the phase.

Fig. 6 shows a slice at x = 4 km of the two kernels. The broad-band slice is overlaid onto the monochromatic one to get a better understanding of the sensitivity kernels’ properties.

Figure 6.

Slices at x = 4 km of the 30 Hz traveltime sensitivity kernels. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime). The broad-band slices in continuous blue line are overlaid to the monochromatic 30 Hz slices represented in dashed red line.

Figure 6.

Slices at x = 4 km of the 30 Hz traveltime sensitivity kernels. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime). The broad-band slices in continuous blue line are overlaid to the monochromatic 30 Hz slices represented in dashed red line.

The instantaneous traveltime provides a good alternative to measure finite-frequency traveltime misfit. It is less sensitive to wavelet distortion and has smooth kernels. Thus, using the instantaneous traveltime for WET is a good option to examine.

Marmousi model

The Marmousi model has served the seismic exploration for long time as a benchmark for imaging and inversion algorithms. We thus compute the sensitivity kernels for its smoothed version given by Fig. 7. The original Marmousi model is smoothed using triangular smoothing with a radius of 40 samples in both directions. The source is located $$(x_{\rm s}=3.0 \,\mathrm{km}, z_{\rm s}=0.1 \,\mathrm{km})$$ and the receiver is located at $$(x_{\rm r}=7.0 \,\mathrm{km}, z_{\rm r}=0.1 \,\mathrm{km})$$. The Green's functions are computed using a Helmholtz equation solver based on lower-upper (LU) decomposition.

Figure 7.

Smoothed Marmousi model.

Figure 7.

Smoothed Marmousi model.

Fig. 8 shows the sensitivity kernels for the Marmousi model. The kernels using the unwrapped phase and cross-correlation cover the same parts of the model. They have a similar behaviour with some differences in their amplitude. A closer look given by Fig. 9 reveals the smooth nature of the instantaneous traveltime sensitivity kernel.

Figure 8.

The 30-Hz broad-band traveltime sensitivity kernels for the Marmousi model. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Figure 8.

The 30-Hz broad-band traveltime sensitivity kernels for the Marmousi model. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Figure 9.

Slices at x = 4 km of the 30-Hz broad-band traveltime sensitivity kernels for the Marmousi model. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Figure 9.

Slices at x = 4 km of the 30-Hz broad-band traveltime sensitivity kernels for the Marmousi model. (a) Cross-correlation. (b) Unwrapped phase (instantaneous traveltime).

Furthermore, we show the kernel for a larger offset. In this case, the kernel will cover a deeper area of the Marmousi model. Since the deep velocity is more complex, the kernel has a more complicated shape. We show in Fig. 10 the sensitivity kernel computed using the instantaneous traveltime. The figure shows the single 30 Hz frequency kernel and the broad-band one obtained by the weighted summation over frequencies. The broad-band kernel has less oscillations compared to the single-frequency one. However, because of the complex velocity, oscillations in the area far from the ray path do not cancel perfectly. The sensitivity kernel shows the large area that could be updated using finite-frequency traveltimes compared to ray-theory-based updates.

Figure 10.

Traveltime sensitivity kernels using the unwrapped phase for a source–receiver offset h = 8.2 km. (a) 30 Hz single frequency kernel. (b) 30 Hz broad-band kernel.

Figure 10.

Traveltime sensitivity kernels using the unwrapped phase for a source–receiver offset h = 8.2 km. (a) 30 Hz single frequency kernel. (b) 30 Hz broad-band kernel.

TRAVELTIME MEASUREMENT FOR A HOMOGENEOUS BACKGROUND MEDIUM WITH A 3-D SPHERICAL ANOMALY

In this section, we consider a homogeneous background velocity model with a small 3-D smooth spherical anomaly. The background velocity is v0 = 2.0 km s−1. The model size is (2.0 × 1.0 × 1.0) km3. The anomaly is positive with 10 per cent velocity perturbation. The diameter of the anomaly is da = λ = 0.2 km. The distance between the source and the anomaly centre is xa = 0.5 km. We show in Fig. 11 a slice at y = 0.5 km of the velocity model. The source is placed at {xs, ys, zs} = {0.0, 0.5, 0.5} and the data are recorded along one horizontal and two vertical arrays of equidistant receivers, as shown with the red lines. We use a finite-difference time-domain scheme to solve the wave equation and model the wavefields. The source function is a Ricker wavelet with a peak frequency f0 = 10 Hz.

Figure 11.

Velocity model slice at y = 0.5 km. The anomaly is placed at a distance xa = 0.5 km from the source. The source is shown with the blue star. The red lines show receivers’ location. We place receivers on one horizontal and two vertical lines.

Figure 11.

Velocity model slice at y = 0.5 km. The anomaly is placed at a distance xa = 0.5 km from the source. The source is shown with the blue star. The red lines show receivers’ location. We place receivers on one horizontal and two vertical lines.

The objective of this simple test is the analysis of the anomaly's effect on the measured traveltimes. According to the banana–doughnut theory, a small anomaly, compared to the wavelength, cannot be detected using cross-correlation. A small anomaly can be hidden inside the hole in the banana, as shown in Fig. 12. Actually, for a small distance between the source and the receiver, the traveltime is sensitive to some parts of the velocity anomaly (Fig. 12a). However, as long as we increase the source–receiver offset to 2.0 km (Fig. 12c), the anomaly is hidden and could not be resolved. For the last case, there is zero traveltime difference which explains the hollow banana shape of the sensitivity kernel.

Figure 12.

Banana–doughnut traveltime sensitivity kernels slices overlaid to a small anomaly for different receiver locations. The anomaly diameter is da = λ = 0.2 km, where λ is the wavelength corresponding to the peak frequency f0 = 10 Hz.

Figure 12.

Banana–doughnut traveltime sensitivity kernels slices overlaid to a small anomaly for different receiver locations. The anomaly diameter is da = λ = 0.2 km, where λ is the wavelength corresponding to the peak frequency f0 = 10 Hz.

Based on the instantaneous traveltime equation (12), we use the methodology described by Choi et al. (2011) to automatically compute traveltimes. A damping factor is applied to the instantaneous traveltime. The traveltime difference is basically the difference between the instantaneous traveltimes for the homogeneous model and the one with anomaly. We compare the measured traveltimes with the ones computed using cross-correlation.

Fig. 13 shows the traveltime delay measured along the horizontal array of receivers. The traveltime difference, measured using the Eikonal solver, in black, reflect the conventional ray time perturbation and has larger values compared to the two other methods. On the other hand, cross-correlation (blue) and unwrapped phase (red) methods show smaller traveltime perturbations. They both decrease with offset reflecting the finite-frequency spreading effect (Fresnel zone), where the traveltime perturbation is spread over a region. However, when cross-correlation traveltime converges to zero delay for receivers far from the anomaly, the unwrapped phase is still detecting the effect of the anomaly.

Figure 13.

Measured traveltimes along the horizontal array of receivers. Black line represents traveltime difference computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Delay measured using the Eikonal solver is constant after the anomaly, but is over-estimated. For both cross-correlation and instantaneous traveltime, we observe the same decreasing tendency. Cross-correlation traveltimes converge to zero for receivers far from the anomaly; however, the instantaneous traveltime is still detecting the effect of the anomaly. Note that the inflection point near 1.8 km for the cross-correlation curve is a sampling artefact.

Figure 13.

Measured traveltimes along the horizontal array of receivers. Black line represents traveltime difference computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Delay measured using the Eikonal solver is constant after the anomaly, but is over-estimated. For both cross-correlation and instantaneous traveltime, we observe the same decreasing tendency. Cross-correlation traveltimes converge to zero for receivers far from the anomaly; however, the instantaneous traveltime is still detecting the effect of the anomaly. Note that the inflection point near 1.8 km for the cross-correlation curve is a sampling artefact.

We show in Fig. 14 seismograms recorded for different locations along the horizontal array of receivers. We show seismograms for x = {0.6; 1.0; 1.4; 1.8} km, the blue curve represents the seismogram for the homogeneous case, whereas the red curve represents the case with an embedded anomaly. We also show the measured traveltime shifts using cross-correlation and the instantaneous traveltime. From Fig. 14(a), the traveltime shift is visible and it is well approximated using both methods. As long as we move along the x direction, wave front healing affects the seismograms and the difference decreases. The instantaneous traveltime, more sensible to small scatterers, gives larger values of traveltime shifts compared to cross-correlation. We conclude that as predicted by the sensitivity kernels, the instantaneous traveltime is able to resolve small anomalies.

Figure 14.

Recorded pressure seismograms along the horizontal array of receivers for different locations: (a) x = 0.6 km, (b) x = 1.0 km, (c) x = 1.4 km and (d) x = 1.8 km. The blue and red curves correspond to the homogeneous and the case with positive anomaly, respectively. We show under each figure the measured traveltime shift using cross-correlation ΔtCross, and using the instantaneous traveltime ΔtInst.

Figure 14.

Recorded pressure seismograms along the horizontal array of receivers for different locations: (a) x = 0.6 km, (b) x = 1.0 km, (c) x = 1.4 km and (d) x = 1.8 km. The blue and red curves correspond to the homogeneous and the case with positive anomaly, respectively. We show under each figure the measured traveltime shift using cross-correlation ΔtCross, and using the instantaneous traveltime ΔtInst.

We display in Figs 15 and 16 traveltime lags computed for the vertical lines of receivers, at 1.0 and 1.8 km, respectively. Traveltime delays using the Eikonal, as shown before, give ray perturbation traveltimes. Traveltimes computed using cross-correlation have their minimum along the source–receiver horizontal axis, and converge to zero for receivers far from the anomaly. Though, like the cross-correlation, the instantaneous traveltime spreads the perturbation over a wider area, it maintains a maximum along the ray path.

Figure 15.

Measured traveltimes along the vertical array of receivers placed at x = 1.0 km. Black line represents traveltime computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Cross-correlation traveltimes minimum is along the source–receiver axis. For the unwrapped phase, it predicts traveltimes with maximum values along z = 0.5 km, the source–receiver axis.

Figure 15.

Measured traveltimes along the vertical array of receivers placed at x = 1.0 km. Black line represents traveltime computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Cross-correlation traveltimes minimum is along the source–receiver axis. For the unwrapped phase, it predicts traveltimes with maximum values along z = 0.5 km, the source–receiver axis.

Figure 16.

Measured traveltimes along the vertical array of receivers placed at x = 1.8 km. Black line represents traveltime computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Cross-correlation traveltimes minimum is along the source–receiver direction and is almost zero. For the unwrapped phase, it predicts traveltimes with maximum values along z = 0.5 km, the source–receiver axis.

Figure 16.

Measured traveltimes along the vertical array of receivers placed at x = 1.8 km. Black line represents traveltime computed using an Eikonal solver, blue is the cross-correlation traveltime and red is the traveltime measured using the unwrapped phase. Cross-correlation traveltimes minimum is along the source–receiver direction and is almost zero. For the unwrapped phase, it predicts traveltimes with maximum values along z = 0.5 km, the source–receiver axis.

In summary, in both finite-frequency traveltime misfit measurements, the misfit stretches over a larger region courtesy of the Fresnel zone, reflecting the milder traveltime sensitivity of low frequencies to small anomalies. However, the instantaneous traveltime spreads the misfit keeping the maximum sensitivity along the ray path.

CONCLUSIONS

WET shares the general same misfit basic measure as conventional traveltime tomography. However, the difference lies in the way traveltimes are computed and the model is updated. Each traveltime measurement method has its own sensitivity kernel or Fréchet derivatives, which shows the parts of the model that will be updated during WET.

Conventionally, first-order Born or Rytov approximation is used to derive these kernels, and examine their spatial distribution. The Born or Rytov kernels have the same banana–doughnut shape with zero sensitivity along the geometrical ray path.

We introduced an alternative approach to measure the traveltime misfit based on the unwrapped phase or the instantaneous traveltime. This traveltime is a function of frequency, and thus, admits finite-frequency features. We showed that the instantaneous traveltime is more sensitive to small anomalies compared to the cross-correlation-based traveltimes. We demonstrated that the instantaneous traveltime is the unwrapped phase version of the Rytov traveltime approximation. The unwrapped phase is used to derive the corresponding traveltime sensitivity kernels. The resulting kernels admit a plain banana shape with maximum sensitivity along the geometrical ray path. Observations on a simple 3-D velocity model and the more complex Marmousi model show smooth kernels. A simple test of traveltime lag measurement for a 3-D model with an embedded small spherical anomaly confirms the plain sensitivity for the unwrapped phase. At the end, the unwrapped phase kernels are smoother and share with the high-frequency asymptotic ray path kernels the maximum sensitivity along the ray path. Moreover, the instantaneous traveltime kernels account for real physical phenomena like dispersion and wave front healing. The unwrapped phase forms a good alternative for traveltime misfit measurements with finite-frequency information in WET.

We would like to thank King Abdullah University of Science and Technology for the financial support. We thank Yunseok Choi, Ru-Shan Wu and Maarten De Hoop for many useful discussions.

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