Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Abundance estimation of solid and liquid mixtures in hyperspectral imagery with albedo-based and kernel-based approaches



David W. Allen, Ronald G. Resmini, Robert S. Rand


This study investigates methods for characterizing materials that are mixtures of granular solids, or mixtures of liquids, which may be linear or non-linear. Linear mixtures of materials in a scene are often the result of areal mixing, where the pixel size of a sensor is relatively large so they contain patches of different materials within them. Non-linear mixtures are likely to occur with microscopic mixtures of solids, such as mixtures of powders, or mixtures of liquids, or wherever complex scattering of light occurs. This study considers two approaches for use as generalized methods for un-mixing pixels in a scene that may be linear or non-linear. One method is based on earlier studies that indicate non-linear mixtures in reflectance space are approximately linear in albedo space. This method converts reflectance to single-scattering albedo (SSA) according to Hapke theory assuming bidirectional scattering at nadir look angles and uses a constrained linear model on the computed albedo values. The other method is motivated by the same idea, but uses a kernel that seeks to capture the linear behavior of albedo in non-linear mixtures of materials. The behavior of the kernel method can be highly dependent on the value of a parameter, gamma, which provides flexibility for the kernel method to respond to both linear and non-linear phenomena. Our study pays particular attention to this parameter for responding to linear and non-linear mixtures. Laboratory experiments on both granular solids and liquid solutions are performed with scenes of hyperspectral data.
Proceedings Title
Proceedings of SPIE
Proc. SPIE 9976
Conference Dates
August 29-September 2, 2016
Conference Location
San Diego, CA
Conference Title
Imaging Spectrometry XXI


solid and liquid mixtures , hyperspectral imagery , albedo , Hapke theory


Allen, D. , Resmini, R. and Rand, R. (2016), Abundance estimation of solid and liquid mixtures in hyperspectral imagery with albedo-based and kernel-based approaches, Proceedings of SPIE, San Diego, CA, [online], (Accessed June 21, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created September 19, 2016, Updated November 10, 2018