Laser light propagation in a turbid medium: solution including multiple scattering effects

THE EUROPEAN PHYSICAL JOURNAL D Eur. Phys. J. D (2023) 77 :110 https://doi.org/10.1140/epjd/s10053-023-00694-6 Regular Article – Optical Phenomena and Photonics Laser light propagation in a turbid medium: solution including multiple scattering effects Knut Stamnes1,a , Wei Li1,b , Snorre Stamnes2,c , Yong Hu2,d , Yingzhen Zhou1,e , Nan Chen1,f , Yongzhen Fan3,g , Børge Hamre4,h , Xiaomei Lu2,i , Yuping Huang1,j , Carl Weimer5,k , Jennifer Lee5,l , Xubin Zeng6,m , and Jakob Stamnes4,n 1 Department of Physics, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, NJ 07030, USA Langley Research Center, NASA, Hampton, VA, USA 3 Earth System Science Interdisciplinary Center (ESSIC), University of Maryland, College Park, MD, USA 4 Department of Physics, University of Bergen, Bergen, Hordaland, Norway 5 Ball Aerospace, Boulder, CO, USA 6 Department of Hydrology and Atmospheric Science, The University of Arizona, Tucson, AZ, USA 2 Received 29 December 2022 / Accepted 30 May 2023 / Published online 19 June 2023 © The Author(s) 2023 Abstract. We have shown that solutions to the radiative transfer equation for a homogeneous slab yield a zenith radiance reflectance that for collimated beam incidence in the nadir direction can be expressed in terms of the lidar ratio, defined as the extinction coefficient divided by the 180◦ backscattering coefficient. The recently developed QlblC method, which allows one to quantify layer-by-layer contributions to radiances emerging from a slab illuminated with a collimated beam of radiation, was used to show explicitly that in the single-scattering approximation the attenuated backscatter coefficient estimated by the new QlblC method gives the same result as the lidar equation. Originally developed for the continuous wave (CW) lidar problem, we have extended the new QlblC method to apply to the pulsed lidar problem. A specific example is provided to illustrate the challenge encountered for ocean property retrievals from space observations due to the fact that a very significant fraction of the signal is due to aerosol scattering/absorption; typically only about 10% (or less) comes from the ocean. 1 Introduction The United States National Academy of Sciences, Engineering, and Medicine 2017 Decadal Survey, ‘Thriving Wei Li, Snorre Stamnes, Yong Hu,Yingzhen Zhou, Nan Chen,Yongzhen Fan, Børge Hamre, Xiaomei Lu, Yuping Huang, Carl Weimer, Jennifer Lee, Xubin Zeng and Jakob Stamnes have contributed equally to this work. a e-mail: (corresponding author) e-mail: c e-mail: d e-mail: e e-mail: f e-mail: g e-mail: h e-mail: i e-mail: j e-mail: k e-mail: l e-mail: m e-mail: n e-mail: b on Our Changing Planet: A Decadal Strategy for Earth Observations from Space’ calls for a lidar and polarimeter to accurately characterize vertically resolved absorbing and scattering properties of aerosols. The lidar is expected to be at least as capable as the Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument, which has been shown to enable retrievals of aerosol, cloud, and ocean products from space. In addition, the next generation of airborne lidars includes high spectral resolution lidars (HSRL) with the capability for aerosol measurements at 1064, 532, and 355 nm [1] and ocean measurements at 532 and 355 nm [2]. These advanced, powerful lidar systems require new algorithms in order to accurately and efficiently account for multiple scattering in cirrus and water clouds as well as in open ocean waters and coastal waters with high levels of particulate scattering. For spaceborne lidar systems, multiple scattering leads to a significant contribution to lidar returns [3]. Although Monte Carlo (MC) algorithms exist for elastic backscatter lidar like CALIOP, it is highly desirable to have access to an efficient yet accurate forward radiative transfer model that will be significantly faster 123 110 Page 2 of 12 Eur. Phys. J. D (2023) 77 :110 than existing MC algorithms in simulating lidar signals due to multiple scattering in cirrus and water clouds as well as coastal water. Reliable interpretation of lidar/radar returns from the atmosphere (or lidar returns from the ocean) relies on accurate solutions of a forward/inverse problem. To solve the inverse problem, it is very useful to have access to a fast yet accurate forward model. For continuouswave lidar/radar beam illumination, one needs to solve the steady-state (time-independent) radiative transfer equation (RTE) for beam propagation including multiple scattering throughout the medium (atmosphere or ocean). However, most active remote sensing techniques rely on illumination using lidar/radar pulses, implying that one must solve the time-dependent RTE. For a space-based lidar, such as the CALIOP instrument, which has operated in space since 2006 on the Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) platform, one needs to solve the time-dependent RTE for the coupled atmosphereocean system in order to infer water parameters from space [4–6]. Many retrieval algorithms ignore or treat multiple scattering in an approximate manner, which may yield unreliable results. For example, it has been shown that if lidar multiple-scattering effects were omitted in combined radar-lidar retrievals of ice clouds from space, the retrieved optical depth might be underestimated by as much as 40% [7]. Therefore, to fully understand and exploit the information contained in received lidar/radar returns, one should have access to accurate methods to assess the error incurred by ignoring or not including multiple scattering effects in an accurate manner. Exploring to what extent multiple scattering can be used as an additional source of retrievable information about medium properties is also an important issue that deserves serious attention [8]. Radiative transfer involving lidar/radar (finite) beam illumination is a three-dimensional (3-D) problem. The solution of the 3-D RTE for a narrow finite laser beam (i.e., the so-called searchlight problem) is quite challenging and computationally demanding. Therefore, it has become customary to use a one-dimensional (1-D) approach instead, and most treatments of the lidar/radar problem rely on solving a 1-D RTE for both atmospheric [9,10] and oceanic [11] applications. where the lidar ratio S (i.e., the extinction coefficient divided by the 180◦ backscattering coefficient) depends on the scattering phase function, p(cos Θ): S = Siso = 4π for isotropic scattering, piso (cos Θ) = 1.0;S = SRay = 8π 3 for Rayleigh scattering, pRay (cos Θ) = 3 Results 3.1 The QlblC method Let’s consider a medium consisting of two adjacent, horizontal, multilayered coupled slabs illuminated from the top of the upper slab by a collimated beam of irradiance F0 in direction θ0 with respect to the normal to the two adjacent slabs. As discussed elsewhere [12], the radiative transfer equation (RTE) for such a problem can be solved by integrating the source function S± i (...truncated)