Context. Recognizing the transport of Energetic Neutral Atoms (ENA), from their place of birth to Earth orbit, has become an important issue in light of the forthcoming launch of the NASA SMEX mission IBEX, which is devoted to imaging of the heliospheric interface by in-situ detection of ENAs. Aims. We investigate the modifications of both energy of survival probability of the hydrogen ENA (H ENA) detectable by IBEX (0.01-6 keV), between the termination shock and Earth orbit. We take into account the influence of the variable and anisotropic solar wind and of solar EUV radiation. Methods. Energy changes of the atoms are calculated by numerical simulations of the orbits of H ENA between ∼100 AU from the Sun and Earth orbit, taking into account solar gravity and Lyman-α radiation pressure, which is variable in time and depends on the radial velocity of the atom. To calculate the survival probabilities of the atoms against ionization, a detailed observation-based 3D and time-dependent model of H ENA ionization is constructed, and with the use of this model the probabilities of survival of the atoms are calculated by numerical integration along the previously-calculated orbits. Results. Due to radiation pressure, H ENA reach the Earth orbit practically without energy and direction change, apart from the atoms of energy lower than 0.1 keV, during high solar activity. The survival probability of H ENA increases from just ∼2% for the slowest detectable ENA at solar minimum to ∼80% for the fastest ENA. For a given energy at Earth orbit we expect fluctuations in the survival probability of amplitude between ∼20 percent at 0.01 keV to just a few percent at 6 keV and a modulation of survival probability as a function of the location at Earth orbit, ecliptic latitude of the arrival direction, and phase of solar cycle with an amplitude of a few dozen percent for 0.1 keV atoms at solar minimum to a few percent for 6 keV atoms at solar maximum. Conclusions. With appropriate account of local transport effects IBEX should be able to discover departures from symmetry in the flux of H ENA from the heliospheric interface at a level of a few percent.