THE METHOD

In the discussion to follow the predictability of linear events in the $({\omega},x)$ domain and a relation between the filters predicting the original and the interpolated data will be presented. This theoretical work is derived in the papers by Spitz and Canales (Spitz, 1991, Canales, 1984).

The data u(t,x) is assumed to be a collection of N equally spaced traces that is composed of linear events corresponding to L distinct dips, with L< N. In the case when several lines are parallel, they are treated as part of a single event with a unique dip. It is also assumed that there is no time aliasing and that the signals are invariant along the linear events. Following Canales (Canales, 1984) and with these assumptions, the data can be expressed in the (t,x) domain as

$$u(t,x)=\sum_{k=1}^{L}v_k(t-b_kx)$$ (1)

or in the $({\omega},x)$ domain after a Fourier transform over the time axis as

$$U(\omega,x)=\sum_{k=1}^{L}V_k(\omega)\exp(i{\omega}b_kx),$$ (2)

where v_k(t) is a wavelet, $V_k(\omega)$ its Fourier transform and b_k is a measure of the slope, all corresponding to the k^th linear event. If equation (2) is sampled in the x direction and we let b_k become the time shift per trace interval corresponding to the k^th linear event, then the data can be expressed in the following matrix form :

$$\pmatrix{U_{1}({\omega})\cr U_{2}({\omega})\cr \vdots\cr ... ...({\omega})\cr V_{2}({\omega})\cr \vdots\cr V_{L}(\omega)\cr}$$ (3)

or

$$U({\omega})={\bf Z}({\omega}){\times}V({\omega}),$$ (4)

where $U({\omega})$ is a frequency slice on the data $U({\omega},x)$, $z_{k}=\exp(i{\omega}b_k)$ and $V_k(\omega)$ is the wavelet from (2). The Vandermonde matrix involved in equation (3) is of full column rank since the dips are distinct. Therefore every row of this matrix may be expressed as a linear combination of the L previous rows :

$$\pmatrix{z_{1}^{k-1}&\ldots&z_{L}^{k-1}\cr}= \pmatrix{P_{L}(... ...-L-1}\cr \vdots&&\vdots\cr z_{1}^{k-2}&\ldots&z_{L}^{k-2}\cr}$$ (5)

for $k=L+1,\ldots,N$.Notice that the coefficients $P_{1}({\omega})$ to $P_{L}({\omega})$ are independent of the row number k because of the Vandermonde structure. Multiplying on both sides by $V({\omega})$, equation (5) becomes :

$$U_{k}({\omega})=\pmatrix{P_{L}({\omega})&P_{L-1}({\omega})&\... ...})\cr U_{k-L-1}({\omega})\cr \vdots\cr U_{k-1}({\omega})\cr}$$ (6)

or

$$U_{k}({\omega})=\sum_{j=1}^{L}P_{j}({\omega})U_{k-j}({\omega})$$ (7)

for $k=L+1,\ldots,N$. The meaning of the last equation is that in the case of strictly linear events, for each frequency, the data $U({\omega})$ is predictable along the x axis with a one-step-ahead prediction filter.

Considering equation (5) for k=L+1 it appears that z₁ to z_L are the roots of the polynomial $Q(z)=z^{L}-P_{1}({\omega})z^{L-1}-{\cdots}-P_{L}({\omega})$ since in this case equation (5) is equivalent to $(Q(z_{1}){\hspace*{.1in}}{\ldots}{\hspace*{.1in}}Q(z_{L}))=(0{\hspace*{.1in}}{\ldots}{\hspace*{.1in}}0)$. Therefore the relations between the roots z_j and the the coefficents $P_{k}({\omega})$ are given by

$$P_{1}({\omega})=\sum_{j=1}^{L}z_{j} {\hspace* {0.2in}} and {... ...{\leq}j_{1}<{\cdots}<j_{k}{\leq}L}{z_{j_{1}}{\cdots}z_{j_{k}}},$$ (8)

for $k=2,\ldots,L$.Multiplying those equations by the conjugate of $P_{L}({\omega})$ gives $P_{L}^{\ast}({\omega})P_{k}({\omega})=-P_{L-k}^{\ast}({\omega})$ which shows that the z_j are also the roots of $z^{k-1}P_{L}^{\ast}({\omega})Q(z)=P_{L}^{\ast}({\omega})z^{k+L-1}+P_{L-1}^{\ast}({\omega})z^{k+L-2}+{\cdots}+P_{1}^{\ast}({\omega})z^{k}-z^{k-1}$ for $k=1,{\ldots},N-L$.This fact can be expressed in a form equivalent to equation (5)

$$\pmatrix{z_{1}^{k-1}&\ldots&z_{L}^{k-1}\cr}= \pmatrix{P_{1}^... ...}\cr \vdots&&\vdots\cr z_{1}^{k+L-1}&\ldots&z_{L}^{k+L-1}\cr}$$ (9)

and after multiplication by $V({\omega})$ as

$$U_{k}({\omega})=\pmatrix{P_{1}^{\ast}({\omega})&P_{2}^{\ast}... ...ga})\cr U_{k+2}({\omega})\cr \vdots\cr U_{k+L}({\omega})\cr}$$ (10)

or

$$U_{k}({\omega})=\sum_{j=1}^{L}P_{j}^{\ast}({\omega})U_{k+j}({\omega})$$ (11)

for $k=1,{\ldots},N-L$.Like equation (7), equation (11) states that in the case of strictly linear events, for each frequency, the data $U({\omega})$ is also predictable along the x axis with a one-step-backward prediction filter.

Let $({P'}_{L}({\omega}),{\ldots},{P'}_{1}({\omega}))$ be the prediction filter and b'_k with $k=1,{\ldots},L$ be the time shift per trace interval for the interpolated data ${U'}({\omega},x)$; also let $(P_{L}({\omega}),{\ldots},P_{1}({\omega}))$ and b_k with $k=1,{\ldots},L$ be the same parameters for the original data $U({\omega},x)$. The roots of ${Q'}(z)=z^{L}-{P'}_{1}({\omega})z^{L-1}-{\cdots}-{P'}_{L}({\omega})$ are

$$z_{k}=\exp(i{\omega}{b'}_k)=\exp(i{\omega}{b_{k}\over{n}})=\exp(i{{\omega}\over{n}}b_{k})$$ (12)

if a n^th order interpolation is to be considered (the trace interval before interpolation equals n times the trace interval after interpolation). Since

$${P'}_{1}({\omega})=\sum_{j=1}^{L}z_{j} {\hspace* {0.2in}} an... ...\leq}j_{1}<{\cdots}<j_{k}{\leq}L}{z_{j_{1}}{\cdots}z_{j_{k}}},$$ (13)

replacing z_j in equation (13) with $\exp(i{{\omega}\over{n}}b_{k})$ from equation (12) gives

$${P'}_{k}({\omega})=P_{k}({{\omega}\over{n}})$$ (14)

for $k=1,{\ldots},L$.

As stated at the beginning of this section, equations (7) and (11) show that linear events can be one-step forward-backward predicted and that the coefficients of the prediction filters for the original and the interpolated data are related through equation (14). Those features can be used to implement an interpolation program, which is further discussed in the following section.