_Applied_  
_sciences_  
_Article_  
**Numerical Simulation on Spatial-Frequency Domain Imaging**  
**for Estimating Optical Absorption and Scattering Properties of Two-Layered Horticultural Products**  
**[PERSON] 1,*, [PERSON] 2, [PERSON] 3,** **[PERSON] 1, [PERSON] 1 and [PERSON] 1,4,5,*  
safety evaluation (e.g., firmness, soluble solids content, titratable acidity, etc.) of horticultural products. In the last two decades, measuring optical properties (i.e., \(\mu_{a}\) and \(\mu_{s}\)') of horticultural products (e.g., apple, citrus and tomato) have been widely studied by different researchers in the field of food and agricultural engineering [1; 2; 3]. It was reported that the multiplication of absorption and reduced scattering coefficients of tomato tissues measured by spatially-resolved techniques were highly correlated with flesh firmness, with a correlation coefficient of 0.835 [1]. In the study of [PERSON] et al. (2020) [3], absorption and reduced scattering coefficients of 'Braeburn' apples were determined, which were used to evaluate the ripening processes during the shelf life period, as absorption phenomena were related to changes in pigments present in the fruit flesh and peel, while scattering events mirrored changes in the flesh texture. Most research teams treated the samples as homogeneous media and neglected the difference of optical properties among different layers for simplifying parameter estimation procedure. However, this simplification could bring errors in studying the optical properties, as well as the loss of critical physicochemical information for individual layers. Efforts have been made on developing two- and multi-layered models for measuring \(\mu_{a}\) and \(\mu_{s}\)' of each layer in the traverse direction (i.e., along the surface of a turbid sample) [4; 5; 6; 7], but the estimation errors reported in most studies are still too large and unacceptable, especially for the second or bottom layer. For example, [PERSON] and [PERSON] (2009) estimated the optical properties of two-layered turbid materials simultaneously by using spatially-resolved techniques [5]. The results showed that absorption and reduced scattering coefficients of the top layer of the model samples were determined with errors within 23.0 and 18.4%, respectively, while the inverse algorithm did not give acceptable estimations for the bottom layer. [PERSON] et al. (2009) applied the technique of spatial-frequency domain imaging (SFDI) to estimate optical properties of layered tissues, and reported average accuracies of \(\pm\)2 and \(\pm\)17% for absorption and reduced scattering coefficients of the top layer, respectively, by using the four-parameter fit [8]. However, the estimation errors for absorption coefficients of the bottom layer were as large as \(\pm\)25%, and no acceptable estimations for the bottom-layer's reduced scattering coefficients, even with the two-parameter fit. The major reason causing the large estimation error is the much more complex inverse algorithm for a two-layered model since it has five optical parameters (i.e., \(\mu_{a}\) and \(\mu_{s}\)' of each layer, plus the unknown thickness of the top layer). It is, therefore, desirable to understand the intrinsic properties of the two-layered model prior to implementing an inverse algorithm for optical property estimation. [PERSON] and [PERSON] (2009) conducted sensitivity analysis to study the effects of optical parameters (\(\mu_{a1}\), \(\mu_{s1}\)', \(\mu_{a2}\), \(\mu_{s2}\)' and \(R_{d}\), where subscripts 1 and 2 refer to the top and bottom layer, and \(R_{d}\) is diffuse reflectance) in a two-layered model on optical property estimation by using spatially-resolved technique [5], while [PERSON] et al. (2019) studied the effects of five optical parameters (\(\mu_{a1}\), \(\mu_{s1}\)', \(\mu_{a2}\), \(\mu_{s2}\)' and top-layer thickness \(d\)) on reflectance prediction with the technique of spatially-resolved [7].