We generate independent Gaussian random variables on a regular grid and use a spatial filter to smooth the independent random variables to obtain a spatially correlated Gaussian random field. The FFT is used to speed up the smoothing since convolution is a simple cell by-cell multiplication in the Fourier domain. A representation of the spatial convolution filter in the Fourier domain is efficiently obtained from the FFT of any stationary correlation function. Since FFT is cyclic, the grid must be padded to ensure that opposite sides are uncorrelated. The size of the padding is discussed in detail. Most standard covariance functions fail to be positive definite on finite cyclic domains. This causes striping artifacts in the final simulated realizations and failure to meet statistical properties such as variogram reproduction in the simulated realizations. These problems are addressed and solutions are provided to ensure near perfect statistical properties of the generated realizations. The method is fast and can generate a hundred million grid cell realization in approximately 1.5 minutes on a standard laptop PC. The method scales approximately linearly in the number of grid cells.

Summary In this paper we discuss the behaviour of air gun source arrays in marine seismic acquisition. We comment the fact that the source configuration and depth are changing continually with the combined actions of surface waves, sea currents and general towing conditions. This has a direct effect on the emitted source pressure wave field and thus on the signature in the seismic data. We describe the method of backpropagation with relative motion that allows an efficient and robust estimation of notional and far-field signatures from near-field measurements at every shot point. The derived shot by shot signatures show very good correlation with sea-state and sea-currents, as we would expect. We show that the variation of the signature can affect the quality of seismic data. We demonstrate that the estimated far-field signatures describe the real variation of the signature in the data and we show how the estimated shot by shot signatures can be used to mitigate the effect of signature variations and thereby improve the quality of the seismic data.

Summary The current implementations of marine source modeling theory have been calibrated and adjusted against measured signatures with a goal of high modeling accuracy within a limited frequency band. As multicomponent streamers and source de-ghosting allows for utilizing a significantly broader range of frequencies in seismic imaging, adjustments to the modeling are necessary in order to achieve a better match between measured and modelled signatures over the expanded frequency band. This includes significant changes to the calibration process such as considering de-ghosted measurements and avoiding the historically rooted standard DFS V filtering. The modeling results after applying the improved calibration show a very good match with measured array signatures over a wide frequency range.

We evaluated the problem of modeling the decay of the primary pulse amplitudes of air-gun clusters caused by the traditional assumption of sphericality. This was done by generalizing the Rayleigh equation to work with arbitrary bubble shapes, while retaining the assumption of incompressibility. To approximate the coalescence of the bubbles, we let the shapes be isosurfaces of the velocity potential. With this method, it is possible to model the firing of clustered air guns at any separation distance, including small distances that would cause two spherical bubbles to overlap. In this way, we obtained results matching the relative decay shown to be present for air-gun clusters. In addition, this method also allowed a way of calibrating the model such that effects created by the presence of the gun, compared to just a single spherical air bubble, may be estimated and included.

We evaluated a method of estimating the relative bubble time period of air-gun clusters with an arbitrary number of guns. This was done by assuming incompressible flow and representing the bubbles as isosurfaces of the potential field to account for coalescence. The kinetic energy at the equilibrium radius was then compared to the equivalent energy of the single gun to estimate the relative change. The results agreed well with two-gun cluster measurements, but the lack of data does not allow us to compare with clusters containing more guns than that. We found that more compact configurations, such as a triangle instead of three guns in a line, gave a more rapid increase in the bubble time period as the gun separation decreased. This indicated that compact configurations were attractive for enhancing the low-frequency output from an air-gun cluster.

We describe an alternative way of modeling clustered air guns, by allowing the shape of the air bubbles to be described by isosurfaces of an incompressible velocity potential, as opposed to the traditional way of considering them as perfect spheres. This method solves some close-range interaction problems, and may possibly be used as a basis for a more complete modeling of clustered air guns.

We have developed a simple method of estimating the bubble-time period of clustered air guns from the bubble-time period of a single air gun of the same type and volume. This was done by deriving a characteristic time scale for the normal Rayleigh equation, and then deriving the same scale from a modified Rayleigh system for clusters. Comparing the value for clusters with the value for a single-gun, we then estimate their relative bubble period, which gives a reasonable match (less than 4% relative error in the appropriate domain) to field data.

ABSTRACTFor a 4D seismic operation to be successful, it is important to know what kind of 4D signal we expect to observe, as well as its magnitude. Normally, in a 4D feasibility study, we use rock physics models to quantify the effect of fluid or pressure changes within the reservoir and calculate the corresponding effects to the seismogram. However, to find if the predicted changes are actually observable at a given field, a dedicated calibration procedure might give valuable insight. One such procedure for marine seismics is to gradually change the source strength by varying the firing pressure in order to detect the sensitivity threshold for a given subsurface reflection. This procedure would be practical and feasible if the change of the source signature changes linearly with the source pressure. However, non‐linear effects will lead to minor changes in the later arrivals of the source signature, the so‐called bubble. By investigating these introduced errors for a reasonable air‐gun array we conclude that the method is still feasible since we find it possible to control and diminish the impact of the introduced errors.

We suggest two different mechanisms for generation of high-frequency signals from seismic sources: one type that we interpret as being caused by high-frequency effects close to and within each individual air gun and another type caused by an effect that we refer to as ghost cavitation. The former one is found to have a steep decreasing amplitude trend with frequency, while the latter has a close to 1=f attenuation for frequencies above 1 kHz. A thorough understanding of the effects is of significant importance to quantify and estimate any environmental impact of marine seismic air-gun arrays. The proposed ghost-cavitation mechanism needs further experimental testing. However, given that the suggested model is proven, we think it is possible to attenuate the high-frequency noise generated by compact air-gun arrays by increasing the areal extent of the gun array.

A direct way to calibrate 4D signals is to gradually change the source strength, and detect the sensitivity threshold for a given subsurface reflection. However, varying the source strength in a controlled manner by changing the firing ressure of an air g