## 1 Introduction  
The application of near-infrared spectroscopy (NIRS) is well known in brain health monitoring, and has been used in many clinical settings to monitor cerebral oxygenation during cardiac surgery [1; 2]. The most commonly used NIRS instrumentation is a continuous wave (CW) system that is based on the light intensity measurements, i.e., near-infrared light is sent into the tissue and the intensity of the diffusely reflected light is measured. These are of low cost, robust, and have high signal-to-noise ratio, but due to the limited available information of the light propagation, these systems rely on approximations of scattering properties and using simpler homogenous models for parameter recovery. There are other sophisticated systems such as frequency domain (FD) and time domain (TD) NIRS systems, which, additionally to the intensity measurements, can also measure the average time of flight and the distribution of time of flight of photons through the tissue. However, due to the low SNR of TD and FD systems at a relatively high cost, CW systems are widely preferred for cerebral health monitoring. The parameters of interest are mainly hemoglobin concentration and tissue oxygenation index. Most of the parameter recovery algorithms of CW systems assume a homogenous tissue for the investigated region [3], which is not true for understanding cerebral hemodynamics as the head consists of extra-cerebal tissue between the source-detectors and the cerebral tissue and this can lead to quantification errors. Multi-distance algorithms have been proposed using gradient of intensity attenuation measurements [4; 5; 6] based on a flat semi-infinite homogenous model. Since the gradient is more sensitive to deeper regions than raw intensity, this makes the recovered parameters less sensitive to changes in extra-cerebal tissue. However, the accuracy of the recovered cerebral tissue parameters still remains low due to the homogenous approximation of the head and due to the curvature and complex internal structure of the head. The assumption of scattering properties of the tissue is another source of error in these gradient-based algorithms. This uncertainty due to scatter approximation has been investigated and an alternative method was proposed [7] to recover scattering parameters and scaled hemoglobin concentrations for a more accurate estimate of tissue oxygenation. Other works that are aimed at the removal of signal contamination due to extra-cerebal layer include: signal regression-based methods [8; 9] for recovering changes rather than absolute values. Frequency domain-based methods [10; 11] have been proposed that can recover both absorption and scattering properties of extra-cerebal and cerebral tissues along with the thickness of the top layer. In CW-based methods however, phantom studies on a two layered model have been studied with a known superficial layer thickness. [12] The work reported to date has relied on spectral and spatial derivative measurements, where a theoretical non-uniqueness is unavoidable (as detailed in later sections). The unknown superficial layer thickness, and the performance of a two layered model in a realistic scenario with oxy and deoxyhemoglobin as the primary absorption constituents remains unexplored. This is a crucial point to address since the spectral characteristics play a major role in distinguishing the tissue parameters in any recovery algorithm. With no constraint on the spatial distribution of the optical properties, it has been shown that there can be more than one solutions to CW diffusion equation [13] and with optimal choice of wavelengths this non-uniqueness can be avoided by spectrally constraining the parameter recovery process [14]. In this work, the uniqueness/non-uniqueness problem of a CW system [13] in the context of a layered medium is addressed, by assuming regional homogeneity; it is shown that the solution set is unique if the measured boundary data is intensity-based only.