Interbed (internal) multiple prediction between two horizons

Theory

IMP predicts a model of the interbed (internal) multiples generated between two subsurface reflectors: a shallower generator horizon at zero-offset time T1 and a deep reflector at zero-offset time T2. Energy reflected upward by the deep reflector is reflected back down at the underside of the generator and returns from the deep reflector again, producing a train of reverberations that repeats every ΔT = T2 − T1.

The module only predicts the multiples; it does not remove them. The predicted multiple gather is meant to be subtracted from the input afterwards, normally with Adaptive subtraction. Surface-related multiples are not handled here; use SRMP (Surface related multiple prediction) or SRME 2D/3D for those.

Kinematics. The zero-offset time of reverberation order n is

t0(n) = T2 + n · (T2 − T1) (order 1 arrives at 2·T2 − T1)

Each order adds two extra passes through the interval between the horizons only, so a multiple does not move out with the velocity of the deep primary. IMP derives the interval velocity v2 of the reverberating layer from the Vrms values at T1 and T2 (Dix equation) and gives each order its own moveout velocity:

Vrms(n)² = ( Vrms2² · T2 + n · v2² · (T2 − T1) ) / t0(n)

Using the correct per-order velocity is what keeps the prediction aligned with the real multiple at far offsets, where adaptive subtraction otherwise fails.

Amplitude. Every extra bounce multiplies the amplitude by −R1·R2 (R1 at the generator, R2 at the deep reflector), so successive orders alternate in polarity and decay geometrically. R1·R2 can be entered directly or estimated from the data.

Processing flow per trace:

• The trace is NMO-flattened with the input Vrms field (pseudo-depth domain).
• A mute is applied on the prediction path only (mute above the generator and/or NMO stretch mute), with a cosine taper.
• In HorizonDriven mode the deep primary waveform is extracted from the data in a window around T2, so the phase and bandwidth of the prediction match the data. The reverberation train is then built from this waveform and moved out to the recorded offsets with the per-order velocities.
• In DataDriven mode the multiples are predicted with the third term of the inverse scattering series (ISS) and returned with the primary velocity field.

The operator is 1.5D: it assumes one midpoint per gather, so the input should be sorted to CMP. Other sortings are processed, but a warning is written to the log (see Information).

Input data

Input gather (CMP)

Prestack seismic gather, preferably a CMP gather in time. Trace offset and CDP headers must be populated: the offset drives the moveout, and the CDP number is used to select the velocity function.

Vrms velocity

RMS velocity field in time, as a velocity gather. For each input trace the velocity trace with the nearest CDP is used. The velocity is read on the same time axis as the input gather (same sample interval and start time), so make sure both share the same time sampling.

Parameters

Prediction mode

Selects how the multiples are predicted. Default: HorizonDriven.

• HorizonDriven - reverberates the picked deep reflector between T1 and T2 with exact per-order moveout. Recommended whenever the two horizons are known.
• DataDriven - uses the inverse scattering series to find interbed multiples from all generators in the data, without picked horizons. Predicted events keep the primary moveout, so some residual moveout remains on the model at far offsets.

Reverberation train

How the train of orders is built in HorizonDriven mode. Both options model the same physics. Default: ClosedForm.

• ClosedForm - all orders are generated at once in the frequency domain and moved out in a single pass, with each sample given the velocity of the order it belongs to. Faster and not truncated by Max reverberation order. Not exact where successive orders overlap, that is when T2 − T1 is shorter than the Extraction window.
• PerOrder - each order 1..Max reverberation order is built and moved out separately with its own velocity. No overlap problem, but the series stops at Max reverberation order, and it is slower.

Generator horizon time

Zero-offset two-way time T1 of the shallower (generator) reflector, in seconds. Default: 0.5 s. Must be smaller than Deep reflector time.

Deep reflector time

Zero-offset two-way time T2 of the deep reflector whose reverberation is predicted, in seconds. Default: 1.0 s. The first-order multiple will appear at 2·T2 − T1.

Max reverberation order

Highest reverberation order predicted when Reverberation train = PerOrder. Range 1-10, default 3. Orders that fall beyond the end of the trace are skipped. With R1·R2 = 0.2 the fourth order is already below 1% of the deep primary, so 3 is usually enough.

R1*R2

Product of the reflection coefficients at the generator and the deep reflector, used to scale each order (amplitude factor (−R1·R2)ⁿ). Range 0-0.9, default 0.2. Ignored when Estimate R1*R2 from data is enabled. Because the model is adaptively subtracted later, a reasonable estimate is sufficient.

Estimate R1*R2 from data

When enabled, R1·R2 is estimated for every trace from the autocorrelation of the flattened, muted trace at the reverberation lag T2 − T1 (clipped to 0-0.9). Default: off.

Extraction window

Length, in seconds, of the tapered window centred on T2 from which the deep primary waveform is extracted (HorizonDriven mode). Default: 0.2 s. It should contain the whole deep reflection wavelet, but not neighbouring reflections. If T2 − T1 is shorter than this window, consider PerOrder or a shorter window.

ISS ordering gate

Minimum time separation, in seconds, between the events combined by the inverse scattering series (DataDriven mode only). Default: 0.05 s. It prevents a single reflection from being combined with itself; set it to about the length of the wavelet.

Mute above generator

When enabled, all data above T1 minus Mute lead above T1 are muted on the prediction path, so shallower events do not contribute to the prediction. Default: on. The input gather itself is not changed; the muted version is shown in the Muted input output.

Mute lead above T1

How far above T1 the mute is placed, in seconds. Default: 0.06 s. The mute is deliberately kept above T1 so the generator reflection itself, which is needed for the prediction, is preserved.

Mute taper

Length of the cosine taper below the mute, in seconds. Default: 0.08 s. A smooth taper avoids ringing artefacts in the predicted model.

NMO stretch mute

NMO stretch limit in percent applied in the flattened domain. Default: 40%. Data stretched more than this (shallow times at far offsets) are muted on the prediction path. Set to 0 to disable.

Min T2 - T1

Quality-control threshold, in seconds. Default: 0.1 s. If T2 − T1 is smaller than this, a warning is written to the log, because the Dix interval velocity (and hence the far-offset moveout of the prediction) becomes very sensitive to velocity errors. The prediction is still computed.

Min interval velocity

Lower limit, in m/s, for the Dix interval velocity of the reverberating layer. Default: 1400 m/s.

Max interval velocity

Upper limit, in m/s, for the Dix interval velocity. Default: 6000 m/s. Values outside the minimum/maximum range are clamped; if Dix gives a non-physical result, Vrms at T2 is used instead. The number of affected traces is reported in the log.

Output data

Predicted multiples

Gather with the same geometry and headers as the input, containing only the predicted interbed multiples with their offset-dependent moveout. Use it as the multiple model in Adaptive subtraction.

Muted input (prediction path)

The NMO-flattened input after the prediction mutes. This is the data actually used to make the prediction; use it to check that the mutes keep the generator and deep reflector intact.

Mute function

Mute curve displayed over the input gather in the Input gather view, showing per trace where the prediction-path mute starts (in flattened time).

Information

The module writes the following warnings to the log. They do not stop processing:

• T2 − T1 is below Min T2 − T1 - the interval velocity may be unreliable.
• T2 − T1 is shorter than the extraction window - successive orders overlap; ClosedForm may assign the wrong velocity in the overlap. Use PerOrder or a shorter window.
• Input gather spans more than one CDP - the data are not a CMP gather; sort to CMP for accurate results.
• Dix hit a guard on N of M traces - the interval velocity was clamped or replaced by Vrms2; check the velocity field at T1 and T2.

An error is returned if Deep reflector time is not greater than Generator horizon time.

Examples

Typical workflow:

• Sort the data to CMP gathers and prepare an RMS velocity field on the same time sampling.
• Identify the generator and deep reflector on a stack and read their zero-offset times T1 and T2.
• Run IMP in HorizonDriven mode and compare the Predicted multiples with the input at 2·T2 − T1, 3·T2 − 2·T1, and so on. Check the Muted input to confirm the mutes.
• Subtract the predicted multiples from the input with Adaptive subtraction.

To test the parameters on a controlled data set, generate a synthetic CMP gather with Interbed multiple modeling: its Total field and Vrms velocity outputs can be connected directly to IMP, and its Interbed multiples output is the exact answer to compare the prediction with.

Related articles

• Interbed multiple modeling
• Multiple Modeling Marchenko 2D - data-driven alternative that needs no horizons
• Adaptive subtraction
• SRMP (Surface related multiple prediction)
• SRME 2D/3D

References

Weglein, A. B., Gasparotto, F. A., Carvalho, P. M., and Stolt, R. H., 1997, An inverse-scattering series method for attenuating multiples in seismic reflection data: Geophysics, 62, 1975-1989.

Yilmaz, O., 2001, Seismic data analysis: Society of Exploration Geophysicists.

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