Geometrical optics provides an accurate description of electromagnetic phase fluctuations under a wide range of conditions. The phase variance computed in this way is a benchmark parameter for describing propagation in random media. One can calculate this quantity for most situations of practical interest. We shall find that it is proportional to the first moment of the spectrum of irregularities and is therefore sensitive to the small-wavenumber portion of the spectrum. This is the region where energy is fed into the turbulent cascade process. We have no universal physical model for the spectrum in this wavenumber range and phase measurements provide an important way of exploring that region.

In analyzing these situations, we must recognize the anisotropic nature of irregularities in the troposphere and ionosphere. Large structures are highly elongated in both regions and exert a strong influence on phase. These measurements are also sensitive to trends in the data that are caused by nonstationary processes in the atmosphere. Sample length, filtering and other data-processing procedures thus have an important influence on the measured quantities. By contrast, aperture smoothing has a negligible effect.

Single-path phase measurements have been made primarily at microwave frequencies because phase-stable transmitters and receivers were available in these bands. Early experiments were performed on horizontal paths using signals in the frequency range 1–10 GHz. At least one experiment has measured the single-path phase variance at optical wavelengths. Phase-stable signals from navigation satellites and other spacecraft are beginning to provide information about the upper atmosphere.

The first step in studying electromagnetic scintillation is to establish a firm physical foundation. This chapter attempts to do so for the entire work and it will not be repeated in subsequent volumes. We proceed cautiously because the issues are complex and the measured effects are often quite subtle. Section 2.1 explores the way in which Maxwell's equations for the electromagnetic field are modified when the dielectric constant experiences small changes. Because atmospheric fluctuations are much slower than the electromagnetic frequencies employed, their influence can be condensed into a single relationship: the wave equation for random media. This equation is the starting point for all developments in this field.

To proceed further one must characterize the dielectric fluctuations. We want to do so in ways that accurately reflect atmospheric conditions. Because we are dealing with a random medium, we must use statistical methods to describe them and their influence on electromagnetic signals. For instance, we want to know how dielectric fluctuations measured at a single point vary with time. Even more important, we need to describe the way in which fluctuations at separated points in the medium are correlated. There are several ways to do so and they are developed in Section 2.2. These descriptions assume that the random medium is isotropic and homogeneous. Those convenient assumptions are seldom realized in nature and we show how to remove them at the end of this section. Turbulence theory now gives an important but incomplete physical description of these fluctuations.

History

Quivering of stellar images can be observed with the naked eye and was noted by ancient peoples. Aristotle tried but failed to explain it. A related phenomenon noted by early civilizations was the appearance of shadow bands on white walls just before solar eclipses. When telescopes were introduced, scintillation was observed for stars but not for large planets. Newton correctly identified these effects with atmospheric phenomena and recommended that observatories be located on the highest mountains practicable. Despite these occasional observations, the problem did not receive serious attention until modern times.

How it began

Electromagnetic scintillation emerged as an important branch of applied physics after the Second World War. This interest developed in response to the needs of astronomy, communication systems, military applications and atmospheric forecasting. The last fifty years have witnessed a growing and widespread interest in this field, with considerable resources being made available for measurement programs and theoretical research.

Radio signals coming from distant galaxies were detected as this era began, thereby creating the new field of radio astronomy. Microwave receivers developed by the military radar program were used with large apertures to detect these faint signals. Their amplitude varied randomly with time and it was initially suggested that the galactic sources themselves might be changing. Comparison of signals measured at widely separated receivers showed that the scintillation was uncorrelated, indicating that the random modulation was imposed by ionized layers high in the earth's atmosphere.

The operational deployment of WSR–88D radars throughout the USA can be considered as a major milestone in the application of the Doppler principle to radar meteorology. Throughout the world, the operational use of Doppler radars is considered to be an indispensable tool for monitoring the development of hazardous storms. Doppler radar theory and techniques in meteorology have reached a level of maturity where even the non-specialist can begin to use and apply the data with little formal training. This chapter attempts to provide the interested reader with a fairly rigorous approach to Doppler radar principles. Following a very brief review of signal and system theory, the received voltage from a random distribution of precipitation particles is formulated for an arbitrary transmitted waveform. The expression for mean power is formulated in terms of the intrinsic reflectivity and a three-dimensional weighting function. The range–time autocorrelation function of the received voltage is formulated in terms of the range–time profile of the time-correlated scattering cross section of the particles and of the time correlation of the transmitted waveform. Next, the sample–time autocorrelation function is derived and the concept of coherency time of the precipitation medium is introduced, which dictates the pulse repetition time for coherent phase measurement. The concept of a spaced-time–spaced-frequency coherency function is introduced, to illustrate how frequency diversity can be used to obtain nearly uncorrelated voltage samples from the same resolution volume.

Doppler radars are now considered to be an indispensable tool in the measurement and forecasting of atmospheric phenomena. The deployment of WSR–88D radars, the terminal Doppler weather radar (TDWR) at major airports, and wind-profiling radars in the USA can all be considered as major milestones in the operational application of the Doppler principle. Another milestone is the successful deployment of the first precipitation radar in space as part of the Tropical Rainfall Measurement Mission (TRMM). Measurement of the reflectivity and velocity of precipitation particles basically exploits the information contained in the amplitude and phase of the scattered electromagnetic wave. In the last two or three decades, it has become increasingly clear that significant information is also contained in the polarization state of the scattered wave. The physical and experimental basis for the application of radar polarimetry to the study of precipitation is the main subject of this book.

The evaluation of polarimetric measurement options for operational WSR–88D radars is gaining momentum and, if realized, will result in widespread application of polarimetric techniques by a broad segment of the meteorological community. Basic and applied research in polarimetric radar meteorology continues to be strong world-wide, especially in Europe and Japan. We believe that the time has arrived for a detailed treatment of the physical principles underlying coherent polarimetric radar for meteorological applications.

This book is based in part on a graduate class taught by the authors at Colorado State University since the mid-1980s.

Dual-polarized radar systems can be configured in different ways depending on the measurement goals and the choice of orthogonal polarization states. From a theoretical perspective, the 3 × 3 covariance matrix (see Section 3.11) forms a complete set, but only a few research meteorological radars exist at the present time that are configured for this measurement. The circularly polarized radars built at the National Research Council of Canada were essentially configured for coherency matrix measurements (see Section 3.9). In the early 1980s, a number of single-polarized research Doppler radars were upgraded for limited dual-polarization measurements in the linear h/v-basis (for measurement of differential reflectivity and differential propagation phase). Because only copolar signals were involved, the system requirements were much less stringent and significant practical results (e.g. rain rate estimation, hail detection) were obtained fairly quickly (Hall et al. 1980; Bringi et al. 1984; Sachidananda and Zrnić 1986). This chapter discusses a number of dual-polarized radar configurations from a systems perspective. Since antenna performance is critical for achieving high accuracy in the measurement of the “weak” cross-polar signal, both antenna performance characteristics and formulation of radar observables in the presence of system polarization errors are treated. Calibration issues relevant to polarization diversity systems are also discussed. A significant portion of this chapter is devoted to estimation of the elements of the covariance matrix from signal samples under three different pulsing schemes. The accuracy of these covariance matrix estimates is also treated in some detail.

The conventional single-polarized Doppler radar uses the measurement of radar reflectivity, radial velocity, and storm structure to infer some aspects of hydrometeor types and amounts. With the advent of dual-polarized radar techniques it is generally possible to achieve significantly higher accuracies in the estimation of hydrometeor types, and in some cases of hydrometeor amounts. A description of these techniques and their rationale, within the scattering and propagation theory presented in Chapters 3 and 4, is the main subject of this chapter. For certain important classes of hydrometeors, the information that can be obtained from dual-polarized radar is so dramatic that it is now considered to be an indispensable tool for the study of the formation and evolution of precipitation.

The determination of hydrometeor types and amounts can be formulated in terms of the elements of the covariance matrix. The covariance matrix defined in (3.183) is averaged over the particle size distribution N(D) (here D is the characteristic size of a hydrometeor) and over the joint probability density function of particle states (typically inclusive of shape, orientation, and density). The radar elevation angle is an independent variable and the radar operating frequency is fixed. Thus, if N(D) and the particle state distributions are assumed, then the covariance matrix (and radar observables derived from the matrix, such as differential reflectivity and linear depolarization ratio) can be computed as a function of elevation angle and operating frequency.