Chapter 4 presents a number of examples of the applications of the synergistically unified method to compute the single-scattering properties of ice crystals and dust aerosols and the relevant downstream applications to remote sensing and climate modeling. We first discuss the refractive indices of ice and mineral compositions of dust and the particle size distribution required for computing the bulk optical properties of a polydisperse medium. Then, we present the bulk optical property models associated with ice clouds and dust aerosols and show the comparison of the linear polarization properties of the optical property models with satellite observations. We also discuss the optical properties of surface snow using the present light-scattering computational capabilities. In the case of ice clouds, we also show the optical properties of specifically oriented ice crystals. We then introduce three satellite remote sensing techniques for ice clouds and demonstrate the constraints in terms of spectral consistency and passive-active remote sensing in retrieving ice cloud optical thickness to evaluate the adequacy of an ice cloud optical property model. The remaining portion of this chapter is devoted to the application of the optical properties of ice crystals to climate modeling. In addition, we also discuss the importance of ice cloud long-wave scattering in climate studies.

Chapters 1–4 give a comprehensive and detailed description of the physical-geometric optics method (PGOM). As the full name of PGOM implies, the method combines the theories and techniques of geometric optics and physical optics. The development of PGOM is inspired by previous research efforts on geometric optics and endeavors to improve the accuracy of geometric optics methods in light scattering computations by incorporating the effects of physical optics. Chapter 5 first presents a summary of PGOM from the perspectives of theory, technique, and applications. Then, we present our view of future efforts toward improving PGOM and enhancing its downstream applications.

In Chapter 3, the improved geometric optics method based on electromagnetic integral relations is introduced. Both the surface/volume integral equations linking the near field on the particle surface/volume internal field to the far field are derived. A proof is presented to show the equivalence of the surface and volume integral equations. The surface (or volume) integral equation is then employed to map the near field computed by the geometric optics method to the far field. To improve the computational efficiency of the mapping of the near field to the far field as well as ray tracing, a broad-beam method is presented. A beam-splitting technique based on computer graphics is presented to facilitate efficient beam tracing processes. The performance of PGOM is evaluated via comparing the PGOM simulations and the benchmarks provided by the invariant-imbedding T-matrix method (IITM). Furthermore, a simplified physical-geometric optics method, which considers the interference of emerging waves through the “ray-spreading effect,” is illustrated. A number of examples for the physical-geometric optics method and its simplified version are presented. Finally, this chapter presents a synergistically unified method based on a combination of IITM for small-to-moderate size parameters and PGOM for moderate-to-large size parameters.

Chapter 2 discusses the concept of rays as localized plane waves and elucidates the criteria for the validity of defining a ray. This chapter also presents the conventional ray-tracing technique for light scattering by a nonabsorptive particle. In particular, the incident rays are specified with the Monte Carlo method or in a deterministic form. The ray directions in the ray-tracing process are specified in a closed set of equations and a vector form without referring to specific coordinate systems. Furthermore, the contributions of emerging (scattered) rays and diffraction to the amplitude scattering matrix are explicitly derived, followed by the formulas for averaging the single-scattering properties over particle orientations with respect to three Euler angles. For randomly oriented particles, a simplified ray-tracing method based on Stokes parameters is presented. The remaining portion of this chapter focuses on the ray-tracing process involving an absorptive particle, within which the electromagnetic waves may be inhomogeneous. Furthermore, the scattering of light by a particle with surface roughness is also discussed. This chapter ends with summarizing the inherent shortcomings of the conventional ray-tracing technique.

Chapter 1 presents a summary of the brief history of the geometric optics method for light-scattering computation. Then, basic physical variables and constants of quantifying light scattering by a particle are introduced. In particular, Maxwell’s equations are briefly discussed to introduce the Poynting vector, which is used in later discussions of electromagnetic energy conservation concerning extinction, scattering, and absorption. We also introduce the concepts of the Stokes parameters, amplitude scattering matrix, optical theorem, phase matrix, and scattering/absorption/extinction cross sections. The simplification of the scattering matrix is discussed with respect to particle orientations in both the particle and laboratory coordinate systems. We also discuss chirality and mirror symmetry associated with light scattering by nonspherical particles. In addition, we discuss the extinction matrix associated with a specifically oriented particle. The remaining portion of this chapter is devoted to the discussion of Snell’s law, Fresnel equations, and Fraunhofer diffraction. The scalar and vector Kirchhoff diffraction formulas are discussed in detail. Furthermore, we discuss the applicability of Babinet’s principle to the diffraction computation of a particle.

In the previous chapter we learned how satellite data to estimate various water targets such as precipitation and surface water, can be combined in a model-reservoir system to track a reservoir’s dynamic state and understand river regulation. In this chapter we will cover how satellite data can be used to manage crops and irrigation. We will learn how satellite data can be used to estimate an area under a specific crop using classification techniques, which then helps us understand the water need for that area. Next we will learn methods to estimate crop water demand and actual crop water consumption.

Chapter 3 details the kinematics of satellite orbits and their use in InSAR processing and its automation. It covers the six parameters needed to describe an orbit (Kepler elements or Cartesian state vector), transforming coordinates from an Earth-fixed frame to the satellite frame, and methods to calculate a centimeter-accuracy satellite trajectory from a sequence of state vectors.

In this chapter, we will cover the remote sensing of precipitation to understand how precipitation is tracked. Precipitation is considered one of the most important components of the water cycle that drives the availability of water and its management. For example, precipitation leads to runoff and streamflow, irrigates a field of crops and provides the water for crop growth, fills up lakes, reservoirs and ponds that are a key source for water management. The understanding of precipitation remote sensing will pave the way for learning more complex water management applications that are being increasingly carried out around the world today using satellite water data. We will first cover the history of precipitation remote sensing that began with using active sensing and ground radar. Next, we will cover satellite-based sensing where the challenges and complexities are different. The pros and cons of using various electromagnetic wavelengths will be covered. Finally, we will cover the topic of multi-sensor precipitation estimation based on the synergistic use of multiple satellite sensors spanning different wavelengths of the electromagnetic spectrum.