  Recently a high precision spectroscopic investigation of the EF^1\Sigma_g^+ -
X^1\Sigma_g^+ system of molecular hydrogen was reported yielding information on
QED and relativistic effects in a sequence of rotational quantum states in the
X^1\Sigma^+_g$ ground state of the H_2 molecule [E.J. Salumbides et al., Phys.
Rev. Lett. 107, 043005 (2011)]. The present paper presents a more detailed
description of the methods and results. Furthermore, the paper serves as a
stepping stone towards a continuation of the previous study by extending the
known level structure of the EF^1\Sigma^+_g state to highly excited
rovibrational levels through Doppler-free two photon spectroscopy. Based on
combination differences between vibrational levels in the ground state, and
between three rotational branches (O, Q and S branches) assignments of excited
EF^1\Sigma^+_g levels, involving high vibrational and rotational quantum
numbers, can be unambiguously made. For the higher EF^1\Sigma^+_g levels, where
no combination differences are available, calculations were performed using the
multi-channel quantum defect method, for a broad class of vibrational and
rotational levels up to J=19. These predictions were used for assigning high-J
EF-levels and are found to be accurate within 5 cm^-1.
