To compensate for interchannel variability, the SDIMs were normalized utilizing an external reference [Eq. 1], an internal reference [Eq. 2], or both [Eq. 3].
3where i and j = 1,2,…,11 are indices for the source and detector, respectively; M is the raw signal arranged in a matrix; R is the SDIM from the external reference surface, and d is the measuring distance ranging from 6 to 30 mm in steps of 1 mm. The prime in Eq. 3 indicates that measurements were performed at a time different from that of the acquisition of M and R. Model ear measurements were normalized using Eqs. 1- 2, whereas the tympanic membrane measurements were normalized using Eqs. 1- 3. SDIMext relates the shape of the sample to the shape of the external reference surface, whereas SDIMint relates the sample to itself in the neutral position, Mneutral. The external reference surface was a sheared cylinder made from Delrin, 20 mm in diameter with thickness ranging from 11 to 18 mm, yielding a slope of 19 deg from the horizontal plane. The external reference surface was made to mimic the neutral position of the tympanic membrane relative to the optical axis and the array distribution. In Eq. 3, C is introduced to correct for differences in the measuring distance between the tympanic membrane and reference phantom measurements, see Sec. 4. The distance from the otoscope to the tympanic membrane was measured once for each temporal bone. However, multiple series of measurements were performed on each temporal bone, and for measurements with an unknown measuring distance, the correction factor, R′(d) in Eq. 3, was determined by use of the mean SDIMc(d) from neutrally positioned tympanic membranes with a known measuring distance, ⟨SDIMc,neutral⟩, the SDIMc from a neutrally positioned tympanic membrane with an unknown measuring distance and least squares fitting: the measuring distance, d, was estimated by finding d in Eq. 3 that minimized the relative error between SDIMc(d) and ⟨SDIMc,neutral⟩ in a least square sense. Analogous to internal normalization in Eq. 2, a fourth normalization approach was performed by relating SDIMc to ⟨SDIMc, neutral⟩ so that SDIMcr = SDIMc/⟨SDIMc, neutral⟩.