VELOCITY SPECTRA

An important transformation in exploration geophysics is from data as a function of shot-receiver offset to data as a function of apparent velocity. To go from offset to velocity, the transformation sums along hyperbolas of many velocities. The adjoint is a superposition of hyperbolas of all the different velocities. Pseudocode for these transformations is

		do v 
		 		do $\tau$ 
		 		 		do x 
		 		 		 		$t = \sqrt{ \tau^2 + x^2/v^2 }$ 
		 		 		 		if hyperbola superposition
		 		 		 		 		data(t,x)= data(t,x) + vspace$(\tau,v)$ 
		 		 		 		else if velocity analysis
		 		 		 		 		vspace$(\tau,v)$=vspace$(\tau,v)$+data(t,x)