An isotropic elastic medium is given by a density, \(\rho\), and two Lamé parameters, \(\lambda\) and \(\mu\). To determine the spatial dependency of these parameters one can probe the medium with elastic waves. The authors consider the scattering of time-harmonic elastic waves by perturbations \(\delta\rho(x)\), \(\delta\lambda(x)\) and \(\delta\mu(x)\) of a homogeneous background medium, \((\rho^0,\lambda^0,\mu^0)\). Thus \(\rho(x)=\rho^0+\delta\rho(x)\), etc. The perturbations are integrable and compactly supported. Reflection (or transmission) scattering data are measured on planes which are distant to the scatterer and orthogonal to the propagation direction of a given incoming wave. Assuming validity of the Born approximation, the outgoing wave is represented as the convolution of the background Green’s function with a scattering potential. Associated with the decomposition of the incoming waves into shear and pressure waves, there are two scattering potentials, \(f_s\) and \(f_p\), both linear combinations of the perturbations and their first order derivatives. Generalizing from the scalar case treated in [C. Kirisits et al., Inverse Probl. 37, No. 11, Article ID 115002, 37 p. (2021; Zbl 1505.65125)], the authors prove a Fourier diffraction theorem for the elastic case. The theorem relates the 2D Fourier transform of the measured data to the Fourier transforms \(\hat f_{s/p}\), more accurately, to their restrictions to certain 2D hemispheres \(\subset\mathbb{R}^3\). Since \(\hat f_{s/p}\) are continuous functions, even entire analytic, restrictions to surfaces are defined. Using elastic modes, SS, PS, SP, and PP, the formulae obtained for \(\hat f_{s/p}\) are separated into a system of linear equations for the perturbations \(\hat{\delta\rho}\), etc. To enlarge \(k\)-space coverage, the size of the subsets \(Y\subset\mathbb{R}^3\) on which \(\hat f_{s/p}(k)\) is recovered, measurement data over intervals of incoming propagation directions or over intervals of frequencies are assumed and used. Backprojections, the inverse Fourier transforms of \(1_Y\hat f_{s/p}\), \(1_Y\) the indicator function of \(Y\), are taken as approximations to the scattering potentials.