We explore the effect of sampling rates when measuring data given by Mf for special operators M arising in Thermoacoustic Tomography. We start with sampling requirements on Mf given f satisfying certain conditions. After this we discuss the resolution limit on f posed by the sampling rate of Mf without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when Mf is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing.

We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.

We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.

We explore the effect of sampling rates when measuring data given by M f for special operators M arising in Thermoacoustic Tomography. We start with sampling requirements on M f given f satisfying certain conditions. After this we discuss the resolution limit on f posed by the sampling rate of M f without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when M f is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing.Here, c(x) > 0 is the wave speed, which we take to be identically 1 outside of K ⊂⊂ Ω. We assume that c is a smooth function of x. In addition, g 0 is the Riemannian metric on the space Ω, assumed to be Euclidean on ∂Ω. We define g := c −2 g 0 , which is the metric form which determines the geometry of this problem. Assume u(t, x) is a solution to (1) for all (t, x) ∈ [0, ∞) × R n . Further suppose that we have access to u(t, y) for (t, y) ∈ (0, T ) × Γ where T > 0 and Γ ⊂ ∂Ω is a relatively