We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

Seismic air guns are broadband sources that generate acoustic waves at many frequencies. The low frequency waves can be used for imaging while the high frequency waves are attenuated and/or scattered before they can reflect from targets of interest in the subsurface. It is desirable to reduce the amplitude of the high frequency acoustic waves as they are thought to be disruptive, and potentially damaging, to marine life and are not useful for geophysical purposes. The high frequency acoustic waves are primarily associated with the initial expansion of the air gun bubble and associated peak in the acoustic pressure time series, which is commonly referred to as the source signature of the air gun. Here, we develop a quasi-one-dimensional model of a seismic air gun coupled to a spherical bubble that accounts for gas dynamics and spatially variable depressurization inside the firing chamber in order to investigate controls on the initial peak of the source signature. The model is validated against data collected during field tests in Lake Seneca, New York. Simulations and field data both show that the initial peak is primarily dependent on the operating pressure. A lower gun pressure results in a smaller peak amplitude and a slower rise time. The slope, the amplitude of the initial peak divided by the rise time, is used as a proxy for environmental impact and can decrease by as much as 50% when the air gun pressure is reduced from 2000 psi to 1000 psi. The low frequencies are controlled by the total discharged mass, which is dependent upon gun volume and pressure. Decreasing the operating pressure while simultaneously increasing the gun volume will reduce the high frequencies while maintaining the desirable low frequency signals.

No abstract