Wave-equation Datuming Models

To illustrate the datuming algorithm I use several velocity models. I use the topographic surface displayed in Figure . The reflectivity model consists of a syncline in the middle of a horizontal layer. In order to analyze how the datuming algorithm affects the initial data I apply the conjugate pair of datuming operators and compare the resulting data with the initial data.

 

datum Figure 3 The surface topography (datum) used for the datuming examples.

The initial data is the result of zero-offset modeling using the syncline reflectivity model with different velocity models. The zero-offset data is obtained on a flat datum. Figure a shows an example for a constant velocity medium. The zero-offset data is then extrapolated to the uneven topographic datum shown in Figure to obtain the data in Figure b. Using the conjugate transpose algorithm, data is extrapolated back to the flat datum to obtain the third panel in Figure c. By comparing the first and the third panel in Figure we can estimate the effect of the datuming algorithm on the initial data.

The initial zero-offset data is migrated prior to the datuming step in Figure a and then again after the pair of conjugate datuming operators is applied. The migrated image after conjugate datuming is shown in Figure b.

I use several velocity models for assessing the effects of the datuming algorithm on zero-offset data. The first velocity model, used to obtain Figures and , is just constant. The next model, used to obtain Figures and , is a linearly increasing velocity in depth $v(z)=1000.+0.6 z \: [m/s]$. The third velocity model is linearly increasing both in depth and laterally $v(x,z)=1000.+0.6z+0.2x \: [m/s]$. For this case I use a PSPI and a Split-Step datuming algorithm. The last velocity model consists of three vertical areas of constant velocity $v(x)=(1000.,2000.,3000.) \: [m/s]$.

All the conjugate operators used in this paper (Gazdag migration and modeling, Gazdag datuming, PSPI migration and modeling, PSPI datuming, Split-Step migration and modeling, Split-Step datuming) pass the Dot-Product test.

 

data3datum
Figure 4 Effect on initial data of the datuming algorithm (constant velocity).
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datum
Figure 5 Migration before and after datuming in constant velocity.
a. Migration before datuming.
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum.

 

data3datumvz
Figure 6 Effect on initial data of the datuming algorithm v(z)=1000+0.6*z.
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datumvz
Figure 7 Migration before and after datuming in depth variable velocity v(z).
a. Migration before datuming.
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum.

 

data3datumsplit
Figure 8 Effect on initial data of the datuming algorithm v(x,z). Split-Step.
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datumsplit
Figure 9 Migration before and after datuming. v(x,z)=v₀+ax*x+az*z.
a. Migration before datuming (Split-Step).
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum (Split-Step).

 

data3datumpspi
Figure 10 Effect on initial data of the datuming algorithm v(x,z). PSPI datuming.
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datumpspi
Figure 11 Migration before and after datuming. v(x,z)=v₀+ax*x+az*z.
a. Migration before datuming (PSPI).
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum (PSPI).

 

data3datumxsplit
Figure 12 Effect on initial data of the datuming algorithm v(x,z). Split-Step.
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datumxsplit
Figure 13 Migration before and after datuming in v(x,z)=1000,2000,3000 velocity.
a. Migration before datuming (Split-Step).
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum (Split-Step).

 

data3datumxpspi
Figure 14 Effect on initial data of the datuming algorithm v(x,z). PSPI datuming.
a. Initial data at the flat surface.
b. Data on the topographic datum.
c. Data after datuming to the topographic surface and back.

 

Mig2Datumxpspi
Figure 15 Migration before and after datuming in v(x,z)=1000,2000,3000 velocity.
a. Migration before datuming (PSPI).
b. Migration after continuing the wavefield to the topographic surface and back to the flat datum (PSPI).