The variability observed in many complex gamma-ray bursts (GRBs) is inconsistent with causally connected variations in a single, symmetric, relativistic shell interacting with the ambient material ("external shocks"). Rather, either the central site must produce ∼ 10 50 erg s −1 for hundreds of seconds ("internal shocks") or the local spherical symmetry of the shell must be broken on an angular scale much smaller than Γ −1 where Γ is the bulk Lorentz factor for the shell. The observed variability in the external shock models arises from the number of causally connected regions that (randomly) become active. We define the "surface filling factor" to be the ratio of the area of causally connected regions that become active to the observable area of the shell. From the observed variability in 52 BATSE bursts, we estimate the surface filling factor to be typically ∼ 5 × 10 −3 although some values are near unity. We find that the surface filling factor, f , is ∼ 0.1∆T /T in both the constant Γ phase (which probably produces the GRB) and the deaccelerating phase (which probably produces the x-ray afterglows). Here, ∆T is a typical time scale of variability and T is the time since the initial signal. We analyze the 2 hr flare seen by ASCA 36 hr after the GRB and conclude that the surface filling factor must be small (10 −3 ) in the x-ray afterglow phase as well. Compared to the energy required for an isotropic shell, E iso , explanations for low surface filling factor can either require more energy (f −1 E iso ∼ 10 56 erg) or less energy (( ∆T 4T ) 2 E iso ∼ 10 49 erg). Thus, the low filling factor cannot be used as a strong argument that GRBs must be internal shocks.
Using independent wave and photon approaches, the energy transfers from a finite duration radiation pulse normally incident from vacuum on a mirror moving uniformly in the pulse propagation direction are derived. The classical sinusoidal wavetrain approach yields the quantum-like result that the ratio of the total energy in the reflected pulse to that in the incident pulse is equal to the corresponding ratio of their frequencies. Both approaches reveal that although the conventional sinusoidal plane wave reflection coefficient for the moving mirror does not have unit magnitude, there is no violation of energy conservation for pulses of finite duration. Instead, the energy in the incident pulse is partly transferred to the stretched or compressed Doppler-shifted reflected pulse, and partly converted to work done on the mirror. This work is negative if the mirror is moving toward the source/detector, resulting in a reflected pulse more energetic and shorter than the incident one, much more so as the relative speed approaches lightspeed.
We present a case study that demonstrates the use of our robust Seismic-Well Tie (SWT) process and seismic attributes to validate the added resolution from Seismic Spectral Blueing (SSB) on the carbonate Mishrif reservoir in the Rumaila oil field. Our SWT process included Vertical Seismic Profile (VSP) corridor stack traces and Reflection Coefficient Modelling (RCM). Seismic attributes generated following the interpretation of the SSB data, revealed geological features that weren’t previously visible on the full- stack seismic. All of these provide validation that the extra wiggle from the SSB is real in this case study. SSB outputs bandlimited reflectivity traces derived from shaping the amplitude spectra of the input seismic to that of the well log-based reflectivity series. SSB adds seismic bandwidth to the full-stack data that is expressed as an extra trough within the Mishrif reservoir in certain parts of the field. Three-way SWTs, achieved by including a VSP corridor stack trace to a more conventional tie between well log synthetic and seismic trace, is typically seen as a thorough approach. It can help to reinforce confidence in seismic events observed in all three data types and to highlight events or intervals where well logs or seismic may contain significant anomalous data. Three-way SWTs tying full-stack synthetic, full-stack seismic and 8-12-30-45 Hz VSP corridor stack traces, as well as SSB synthetic, SSB and 8-12-50-75 Hz VSP corridor stack traces are of good-quality, with a comparable extra trough also identified on the broader bandwidth VSP corridor stack trace. Reflection Coefficient Modelling (RCM), a part of the SWT process, is a way of deconstructing a synthetic seismic trace by looking at the intermediate step in wavelet convolution to isolate the contributions of individual Reflectivity Coefficient (RC) contrasts to the resulting seismic event, often referred to as a ‘wiggle’. RCM suggests that the extra trough observed on the SSB data is associated with the development of a rudist-dominated grainstone shoal body. VSP data was used to generate both conventional primary reflectivity response, as well as multiple corridor stacks based on key interbed multiples to understand their generation and kinematics. Different wavefields were generated to allow the discrimination between surface and interbed multiples. This provides support for amplitude fidelity for multiple events and helped identify the adverse effect of multiples on a different reservoir interval trough. Due to the large well stock, with over 700 wells with porosity logs penetrating the Mishrif reservoir, this case study is peculiar in the sense that the previous Geomodel had no direct seismic attributes used in property distribution. Therefore, seismic attributes generated were compared to the Geomodel properties, such as porosity to see if geological features were identifiable on seismic. A grainstone shoal body on a Geomodel average porosity map, also clearly delineated on the SSB sections and attributes, was only subtlely expressed and not properly identifiable on the full-stack data. One of several sinuous features, interpreted as grainstone-dominated tidal channels, targeted using seismic attributes was recently drilled and encountered good reservoir quality channel facies. This case study shows how a SWT process (three-way tie, RCM), seismic attributes and results from a recently drilled well provide validation of the authenticity of the added SSB resolution.
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