In the context of inverse problems, a mathematical model of the measurements is called the direct problem. To understand this term, let us consider a model for obstetric sonography. An ultrasonic scanner produces a sound wave by using a transducer, the wave propagates in the body and echoes back to the transducer that records the echo. The propagation is mathematically modelled by the wave equation which gives a connection between the wave speed (a property of the medium) and the sound wave (oscillation in the medium). If we are given the wave speed as function of the position in the body and the vibrations of the transducer as a function of time, then we can solve the wave equation for the received echoes. In other words, the wave equation gives a model for ultasound measurements. Solving this equation with a wave speed corresponding to the mixture of different tissues in the human body requires computationally demanding simulations and is indeed a problem, the direct problem.
In addition to sonography, reflection seismology gives an example of an imaging method where the wave equation can be used as a model of measurements. Seismic reflection method seeks to create an image of the Earths crust from recording of echoes stimulated for example by explosions. In contrast to medical sonography, seismic reflection method reconstructs an image of the wave speed.
The inverse problem for the wave equation is closely related to several inverse problems of geometric nature. By a geometric inverse problem we mean a problem to recover a Riemannian manifold from a geometric data set, that is, a data set derived from the Riemannian structure only. A widely studied example is the boundary rigidity problem for which the data set is the distances between each pair of boundary points.