1-3 leptonic mixing and the neutrino oscillograms of the Earth

We develop a detailed and comprehensive description of neutrino oscillations driven by the 1-3 mixing in the matter of the Earth. The description is valid for the realistic (PREM) Earth density profile in the whole range of nadir angles and for neutrino energies above 1 GeV. It can be applied to oscillations of atmospheric and accelerator neutrinos. The results are presented in the form of neutrino oscillograms of the Earth, i.e. the contours of equal oscillation probabilities in the neutrino energy--nadir angle plane. A detailed physics interpretation of the oscillograms, which includes the MSW peaks, parametric ridges, local maxima, zeros and saddle points, is given in terms of the amplitude and phase conditions. Precise analytic formulas for the probabilities are obtained. We study the dependence of the oscillation pattern on theta_13 and find, in particular, that the transition probability P > 1/2 appears for sin^2(2theta_13) as small as 0.009. We consider the dependence of the oscillation pattern on the matter density profile and comment on the possibility of the oscillation tomography of the Earth.