An alternative method for modeling close-range interactions between air guns Daniel Barker 1 and Martin Landrø 1 ABSTRACT 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 incom- pressibility. 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 clus- tered air guns at any separation distance, including small dis- tances 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. INTRODUCTION The act of placing two or more air guns in close vicinity to each other remains a popular way of shaping the seismic signal emitted in marine seismic acquisition for several reasons. Historically, the main reason has been the improved primary-to-bubble ratio ob- served due to the interactions between the guns, but more recently, the increased bubble time period has been used to create a source signature, with enhanced low-frequent output (Hopperstad et al., 2012). A side effect of this is that the primary amplitude of the clus- ter will degrade compared to the primary amplitude of the single guns alone, and it may be more important now than before to under- stand exactly how the primary signal of the cluster will be affected by separation distance and the number of guns used to optimize these broadband sources by use of modeling. At the same time, modeling the interactions by assuming spherical bubbles and modi- fying the pressure surrounding the bubble by adding the acoustic field propagated by other bubbles (Ziolkowski et al., 1982; Li et al., 2011) will introduce problems at very close range, especially for coalescing bubbles. For instance, by using spherical spreading, one may experience that the hydrostatic pressure effect of another bubble may be more than the pressure inside the bubble itself if its radius is bigger than the original separation distance between the guns. While this certainly could be improved by some form of cor- rection, the general problem coming from the spherical assumption and a fixed separation distance is not trivial to solve, and another approach may be needed. Strandenes and Vaage (1992) analyze by experiment how clus- tering of air guns will change the primary amplitude, bubble time period, and primary-to-bubble ratio compared to firing single guns and introduce a dimensionless constant, defined by the separation distance divided by an equilibrium radius, the latter being the theo- retical radius at which the air bubble has hydrostatic pressure and the temperature of the surrounding fluid. When comparing clusters with the same constant, they find the relative results compared to single guns to be equal, suggesting that this is a key parameter for the dynamics. Barker and Landrø (2013) use this dimensionless parameter, and a representation based on isosurfaces of the velocity potential, to estimate the relative bubble time period, which matched the general trend of the recorded data. Because their esti- mate was based on an assumption of incompressibility, it seems that the most important contributor to cluster dynamics may be the fluid movement, rather than the traditional pressure field interaction, when it comes to the bubble time period. This might also be the case for the degradation of primary amplitudes due to clustering. In this paper, we will concentrate on the primary amplitude, and we investigate how it will decrease as clustered air guns are moved closer toward each other. To be able to investigate this for coalesc- ing bubbles, we will use an approach based on the isosurfaces that Barker and Landrø (2013) present, but we extend the notion for use in modeling. To investigate the effect on primaries, we also need a Manuscript received by the Editor 5 April 2013; revised manuscript received 9 September 2013; published online 18 December 2013. 1 Norwegian University of Science and Technology (NTNU), Department of Petroleum Engineering and Applied Geophysics, Trondheim, Norway. E-mail: daniel.barker@ntnu.no; mlan@ipt.ntnu.no. © 2013 Society of Exploration Geophysicists. All rights reserved. P1 GEOPHYSICS, VOL. 79, NO. 2 (MARCH-APRIL 2014); P. P1–P7, 12 FIGS. 10.1190/GEO2013-0141.1 Downloaded 12/20/13 to 129.241.27.113. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/