Looking for a analytical formula to compute the central frequency of a signal analyzed by a discrete wavelet at a given scale
They do mean the pseudo-frequency of the wavelet which is not dependent on the signal being analyzed. The misleading terminology that they use seems to come from from one of the references, Han, P. (2013), Investigation of ULF seismo-magnetic phenomena in Kanto, Japan during 2000–2010, PhD thesis, Chiba University, Chiba, Japan. Quoting an article of a similar name and where Han is one of the authors, Hattori et al., Investigation of ULF Seismo-Magnetic Phenomena in Kanto, Japan During 2000–2010: Case Studies
and Statistical Studies, Surv Geophys (2013) 34:293–316,
DOI 10.1007/s10712-012-9215-x, directly answers your question:


  In this paper, we yield discrete wavelet transform (DWT) and decompose the original
  signature into six levels. The central frequency ($F_a$) of signals in each level is given as
  
  $$F_a = \frac{F_c}{a\Delta} \tag{4}$$
  
  where $F_c$ is the frequency of basic wavelet; $a$ is the scale of the
  level, and $\Delta$ is the sampling rate of original data. After evaluating
  different kinds of wavelets, we adopted Daubechies 5 (db5) as the
  mother wavelet. Moreover, the db5 wavelet has been proved to be
  effective in the previous study (Jach et al. 2006). According to Eq.
  4, for db5 wavelet, the central frequency of signals in the sixth
  level is 0.01 Hz.