  We present a seismological method to probe the solar atmosphere above sunspot
umbrae with three-minute oscillations. Our technique allows us to estimate both
the vertical distance between atmospheric layers and the wave-propagation speed
without specifying any extra parameters, in particular, the phase speed of the
wave or the emission formation heights. Our method uses the projected wave
paths of slow MHD waves propagating through the atmospheric layers of different
heights and guided by the magnetic field. The length of the projected wave path
depends upon the distance between the layers and the inclination angle of the
magnetic field with respect to the line of sight, allowing us to estimate the
distance between the layers from measured projected wave paths and the local
magnetic-field vector. In turn, the wave-propagation delay registered at
different heights allows for the calculation of the phase speed. We estimated
the vertical distance between the emission layers at the temperature minimum
(1600 {\AA}) and transition region (304 {\AA}), as well as the average phase
speed above the sunspot umbrae, for three active regions. We found that the
distance between the 1600 {\AA} emission layer and the transition region above
the sunspot umbrae lies in the range of 500 -- 800 km. The average phase speed
between these layers was found to be about 30 km/s giving the sound speed of 6
km/s. The temperature between the layers has been roughly estimated as 3000 K
and corresponds to the region of the temperature minimum. The results obtained
are consistent with the semiempirical model of the sunspot umbrae atmosphere by
Fontenla, 2009.
  The manuscript has been accepted for publication in Solar Physics journal.
