# Dual-sensor wavelet estimation

Other languages:
Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 9 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

In the layered model (Robinson and Treitel, 1977[1], 1978[2]), the boundary layers are the air (on the top) and the basement rock (on the bottom). The reflectivity is the sequence of reflection coefficients of the interfaces. The reflectivity represents the internal structure of the earth. The specific problem that we wish to consider is the case of a buried receiver. The source also can be buried, but we require that the source be placed at a level above the level of the receiver. The layered-earth system (lying above the basement rock) then is divided into two subsystems by a horizontal plane passing through the receiver. The upper subsystem of layers is called the “shallow system,” and the lower subsystem is called the “deep system.” The shallow system is an active system because it contains the energy source. The deep system is a passive system because it does not contain an energy source. The deep system has two inputs and two outputs (Figure 16).

For the deep system, one input is the downgoing wave at the receiver, and one output is the upgoing wave at the receiver. The other input is the upgoing wave at the lowermost interface, and the other output is the downgoing wave at the lowermost interface. We assume that the basement rock is below the lowermost interface and that no reflected energy is produced from the basement. As a result, the upgoing wave at the lowermost interface is null, and hence, in effect, the other input to the deep system is null. Thus, the deep system is passive, with a single input (the downgoing wave at the receiver). If we consider the output given by the upgoing wave at the receiver, then we have a conventional convolutional model with one input and one output (Figure 17).

Figure 16.  The deep, layered, passive system, which lies below the shallow, active system. The two systems are separated from each other at the location of the receiver.
Figure 17.  The convolutional model for the deep, layered system.

The input of the convolutional model for the deep system is the downgoing wave at the receiver, and the output is the upgoing wave at the receiver. The impulse-response function is the reflection response of the deep system. This reflection response involves just the reflection coefficients of the deep system and in no way depends on the reflection coefficients of the shallow system. The reflectivity of the deep system is the object of interest.

To use this convolutional model, two wavelets have to be estimated: the input wavelet and the output wavelet. Both occur at the receiver location. But what is measured when either a geophone or a hydrophone is used as the receiver? A geophone measures the particle-velocity attribute of the seismic disturbance, and a hydrophone measures the pressure attribute of the seismic disturbance. All of seismic processing is based on the availability of downgoing and upgoing traveling waves. However, a traveling wave never is recorded as such in seismic acquisition. The signal recorded by a geophone is the sum of the particle velocity attributes of the downgoing and upgoing waves. The signal recorded by a hydrophone is the sum of the pressure attributes of the downgoing and upgoing waves.

A traveling-wave assumption that links the pressure attribute and the particle-velocity attribute of a wavefield must be made before a geophysicist can process conventional seismic data acquired in the field. Let us give an example. Look at the event on the left side of Figure 18. There is no way to tell whether that event is upgoing or downgoing. Now introduce the traveling-wave assumption as specified on the right side of Figure 18 (also see below). The traveling-wave assumption is that the event is a primary reflection. This additional information makes it possible to say that this event is upgoing.

Figure 18.  The geophone signal by itself is not enough to determine whether the event is upgoing or downgoing. However, using the traveling-wave assumption that the event is a primary reflection, it can be concluded that the event is upgoing.
Figure 19.  Neither the geophone signal by itself nor the hydrophone signal by itself suffices to determine whether the event is upgoing or downgoing. However, we see that the geophone and the hydrophone signals are out of phase, so (by the convention used here) the event is upgoing.

What are the d’Alembert equations? Augustin Jean Fresnel (1788-1827) showed that the definition of the reflection coefficient of an interface requires consideration of both the particle-velocity attribute and the pressure attribute of the wave motion on each side of the interface. Both of these attributes must be continuous across the interface. Jean Le Rond d’Alembert (1717-1783) showed that a disturbance that satisfies the wave equation is equal to the sum of two traveling waves, one of which travels downward and the other upward. The downgoing and upgoing waves of d’Alembert transport the energy to and from a reflecting horizon, and the particle velocity and the pressure attributes of Fresnel determine the partition of energy at that horizon.

The net result is that seismic processing simply cannot be done with conventionally recorded data unless one is willing to make traveling-wave assumptions. However, no traveling-wave assumption has to be made if a dual sensor (that is, both a geophone and a hydrophone) is used as the receiver. A dual sensor provides both the hydrophone signal and the geophone signal (Canales and Bell, 1996[3]). Let us give an example (Figure 19). The geophone and hydrophone signals are out of phase. As a result, it can be concluded that the event is a traveling upgoing wave. On the other hand, if the geophone and hydrophone signals were in phase, the event would be a traveling downgoing wave.

For the dual-sensor case, the traveling waves can be computed directly from the data by using the d’Alembert equations (Figure 20). The receiver is a dual sensor buried below the source of seismic energy. The dual sensor measures the particle-velocity signal and the pressure signal at the receiver location. By use of the d’Alembert equations, the particle-velocity signal and the pressure signal are converted into the downgoing wave and the upgoing wave at the receiver location. The downgoing wave is the input signal, and the upgoing wave is the output signal that occurs in the convolutional model of the deep system.

Figure 20.  The d’Alembert equations. The inputs are the particle velocity, pressure, and acoustic impedance, all measured at the receiver location. The outputs are the downgoing particle-velocity wave and the upgoing particle-velocity wave, both occurring at the receiver location.

Now we shall derive d’Alembert’s equations. For a given rock layer, denote the density by ${\displaystyle \rho }$ and the wave propagation velocity by v. The product ${\displaystyle Z=\rho v}$ is the acoustic impedance. A dual sensor consists of a geophone and a hydrophone. The geophone records the particle-velocity trace V, and the hydrophone records the pressure trace p. Each trace is equal to the sum of the downgoing wave motion plus the upcoming wave motion at the sensor. Let D denote the downgoing wave motion of the particle-velocity trace, and let U denote the upgoing wave motion of the particle-velocity trace. Similarly, let d denote the downgoing wave motion of the pressure trace, and let u denote the upgoing wave motion of the pressure trace. Thus, we have two equations (the first equation is for the particle-velocity trace and the second is for the pressure trace):

 {\displaystyle {\begin{aligned}V=D+U\mathrm {\;\;\;and\;\;\;} p=d+u.\end{aligned}}} (51)

Various conventions are used. Let us use the Berkhout convention (Berkhout, 1987[4]), which we already have presented in Chapter 8.

1) The downgoing wave motion d has the same polarity as does the downgoing wave motion D, and the two are related by a scale factor given by the acoustic impedance.

2) The upgoing wave motion u has a polarity opposite that of the upgoing wave motion U, and the same scale factor relates the two.

Thus, we have

 {\displaystyle {\begin{aligned}d=ZD\mathrm {\;\;\;\;and\;\;\;\;} u=-ZU.\end{aligned}}} (52)

The solution of the above equations yields the d’Alembert equations for the downgoing and upgoing particle-velocity wave motion

 {\displaystyle {\begin{aligned}D={\frac {V+p{\rm {/}}Z}{2}}\mathrm {\;\;\;\;and\;\;\;\;} U={\frac {V-p{\rm {/}}Z}{2}}.\end{aligned}}} (53)

Alternatively, we could solve for the downgoing and upgoing pressure wave motion to obtain the corresponding d’Alembert equations

 {\displaystyle {\begin{aligned}d={\frac {ZV+p}{2}}\mathrm {\;\;\;\;\;and\;\;\;\;} u={\frac {-ZV+p}{2}}.\end{aligned}}} (54)

## References

1. Robinson, E. A., and S. Treitel, 1977, The spectral function of a layered system and the determination of the waveforms at depth: Geophysical Prospecting, 25, 434-459.
2. Robinson, E. A., and S. Treitel, 1978, Fine structure of the normal incidence synthetic seismogram: Geophysical Journal of the Royal Astronomical Society, 53, 289-309.
3. Canales, L. L., and M. L. Bell, 1996, Ghost attenuation using dual sensor cable: 66th Annual International Meeting, SEG, Expanded Abstracts, 1591-1594.
4. Berkhout, A. J., 1987, Applied seismic wave theory: Elsevier.