A detailed analytical description of a Doppler-free probe response (absorption and dispersion) is presented for a coherently prepared four-level atomic tripod driven by two coupling fields. Under the density matrix formalism the field-dependent and field-independent coherence are introduced through the off-diagonal matrix elements. Field-dependent coherence appears due to the interference between the probability amplitudes of finding the atom in energy levels associated with the allowed transition, while field-independent phases measure the atomic coherence for a coherently prepared lower state triplet. The double-control coherent effects on the probe response are investigated by controlling the relative driving contribution (coherence) of the two coupling fields, whereas the driving contribution can be controlled by the use of field-independent coherence, along with two coupling parameters like Rabi frequency and detuning of the respective coupling field. We report on the possibility of modifying the Rabi splitting into a perfect electromagnetically induced transparent window at probe resonance by a coherent double-control mechanism in an on-resonance coupling situation. Moreover, for off-resonance coupling, we exhibit the enhancement of multiphoton (two- or three-photon) absorption via coherent control of multiresonance processes.