Showing papers in "Radio Science in 1981"

Scattering of electromagnetic waves from random media with strong permittivity fluctuations

Abstract: By taking into account the singularity of the dyadic Green's function in the renormalization method, a theory is derived for vector electromagnetic wave propagation in a random medium with large permittivity fluctuations and with anisotropic correlation function. The strong fluctuation theory is then applied to a discrete scatterer problem in which the permittivity can assume only two values. The results are found to be consistent with those derived from discrete scatterer theory for all values of dielectric constants of the scatterers.

Numerical inversion of Laplace transform and some applications to wave optics

Abstract: First, a new method for numerical inversion of the Laplace transform is proposed. The essential point of this method is to approximate the exponential function in the Bromwich integral by the function Eec(s, a) d= exp (a)/2 cosh (a – s). With this we can get an infinite series which gives a good approximation to the inverse transform. In practice, we accelerate the convergence of the series by the Euler transformation. Thus in ordinary cases, only twenty to thirty terms give satisfactory results, and even a $100 pocket calculator can solve practical problems which are not so easy by the usual method. Second, the wave propagation in dispersive media is studied by using the above method. This problem has been investigated analytically by many authors. The problem is so difficult that there has been no detailed solution available other than asymptotic expressions. Some detailed numerical results by our method are given for the transient phenomena in a waveguide and the medium discussed by Sommerfeld and Brillouin. Our numerical results show that Brillouin's results should not be taken quantitatively except the first forerunner whose approximate variation is predicted by Sommerfeld correctly.

Electromagnetic backscattering from a sparse distribution of lossy dielectric scatterers

Abstract: Electromagnetic backscattering from a sparse distribution of lossy dielectric particles having random orientation and position is studied. The paper begins by using the Foldy approximation to find an equation for the mean field. From this equation, an effective permittivity for the scattering medium is obtained. The correlation of the scattered field is found by employing the distorted Born approximation, i.e., particles embedded in the effective medium are assumed to be single scatterers. The above method is then used to find the backscattering coefficients from a leaf canopy. The leaf canopy is modeled by a half space of dielectric discs that are small in comparison to a wavelength. Numerical results show that the depolarized cross section is a sensitive function of leaf inclination angle statistics.

Structures of atmospheric precipitation systems: A global survey

Abstract: A survey of atmospheric precipitation systems, ranging from mid-latitude cyclones and thunderstorms to tropical cloud clusters, hurricanes, and monsoons, shows that all these systems are well described in terms of the rather traditional concepts of stratiform and convective precipitation. In stratiform precipitation, ice particles grow as they drift downward from high levels and pass through a well-defined melting layer. In convective precipitation, particles begin growing at low levels and are carried upward by strong updrafts and fall out in intense vertically oriented showers. Modern observations show that all the major types of precipitation observed over the globe can be and often are combinations of these two basic types of precipitation. Extratropical cyclonic precipitation is basically stratiform. However, it is typically intensified in regions called rainbands. Some rainbands are highly convective features which move through the basic stratiform precipitation. In other rainbands, shallow convective cells occur aloft and help to enhance the basic stratiform precipitation. Mid-latitude thunderstorms and tropical precipitation systems are basically convective. However, stratiform precipitation can develop in the middle to late stages of development. This type of stratiform precipitation, which can become quite extensive in both tropical and mid-latitude systems, apparently arises as groups or successions of active convective cells leave ice particles aloft to settle downward gradually after the cells' updrafts die out.

Modeling attenuation and phase of radio waves in air at frequencies below 1000 GHz

Abstract: Moist air is characterized for the frequency range 1–1000 GHz as a nonturbulent propagation medium described by meteorological parameters An adequate spectroscopic data base for air consists of three terms: (1) resonance information for 29 H2O lines up to 1097 GHz and 44 O2 lines up to 834 GHz in the form of intensity coefficients and center frequency for each line; (2) an empirical water vapor continuum spectrum; and (3) a liquid water attenuation term for haze and cloud conditions This data base is the heart of two computer programs which calculate and plot attenuation rates (in decibels per kilometer), refractivity (in parts per million), and refractive dispersion (in parts per million) The first covers the troposphere and requires pressure, temperature, and relative humidity as input data The second addresses isolated line behavior in the mesosphere wherein the geomagnetic field strength H is an additional input parameter due to the Zeeman effect of the O2 molecules Each oxygen line splits proportionally with H into numerous sublines, which are juxtaposed to form Zeeman patterns spread over a megahertz scale Patterns of three main polarization cases are considered Various typical examples for a model atmosphere demonstrate the utility of the approach, provide new information, and underline the serious role that water vapor plays above 120 GHz

Augmented electric‐ and magnetic‐field integral equations

Abstract: Augmented electric- and magnetic-field integral equations, which preserve the basic simplicity, solution capability, and pure electric- and magnetic-field character of Maue's original integral equations, are introduced to eliminate the spurious resonances from the exterior solution of the original integral equations. The exact dependence of the original and augmented integral equations on the geometry of the principal area (self patch) which excludes the singularity of their kernels is also determined, and alternate forms for the integral equations are provided that avoid integrals dependent upon the geometry of the principal area. Numerical results obtained for scattering from the perfectly conducting cube, sphere, and infinite circular cylinder confirm the theoretically predicted result that the augmented integral equations eliminate the spurious resonances for all perfectly conducting scatterers except the sphere.

Clear weather meteorological effects on propagation at frequencies above 1 GHz

Abstract: This paper reviews the various effects the clear atmosphere may have on the propagation of radio waves above 1 GHz. The role of the transfer processes at the lower boundary of the atmosphere in modifying the refractive index structure of air masses is reviewed, and some results of modification of refractive index and its structure constant by the lower boundary are shown. The formation of various classes of elevated layers is discussed and examples are shown of subsidence layer formation. The formation of refractive structures capping the convective boundary layer is discussed and examples of monsoonal refractive layers are shown. Examples of refractive index gradients and structure constant distributions in the neighborhood of frontal interfaces are also shown and discussed.

Radiative transfer theory for active remote sensing of a layer of small ellipsoidal scatterers

Abstract: The radiative transfer theory is applied within the Rayleigh approximation to calculate the backscattering cross section of a layer of randomly positioned and oriented small ellipsoids. The orientation of the ellipsoids is characterized by a probability density function of the Eulerian angles of rotation. The radiative transfer equations are solved by an iterative approach to first order in albedo. In the half space limit the results are identical to those obtained via the approach of Foldy's and distorted Born approximation. Numerical results of the theory are illustrated using parameters encountered in active remote sensing of vegetation layers. A distinctive characteristic is the strong depolarization shown by vertically aligned leaves.

Microwave equatorial scintillation intensity during solar maximum

Abstract: A comparison of scintillation levels at 1.5 GHz made from the Appleton anomaly region of the magnetic equator and from the region close to the magnetic equator (termed the electrojet latitudes) showed increased F region irregularity intensity over the anomaly region during years of high sunspot number. Peak to peak fading greater than 27 dB was noted from Ascension Island (through a dip latitude of 17°) in the anomaly region while only 7–9 dB from Natal, Brazil, and Huancayo, Peru, were noted, the last two paths being close to the magnetic equator. The hypothesis advanced is that the dominant factor responsible for the intense gigahertz scintillation is the traversal of the propagation path through the anomaly region. During years of high sunspot numbers the high levels of ΔN constituting the F region irregularity structure are due to (1) very high electron density in the anomaly region (compared to the electrojet region) and (2) the late appearance of these high electron densities (to 2200 local time) in the anomaly region. The patches or plumes of irregularities seen in the postsunset time period then produce high ΔN; scintillation excursions are proportional to this parameter. The postulation of vertical irregularity sheets in the patches was examined to determine the possibility of this being an important factor in the difference between electrojet and anomaly scintillation levels. Older gigahertz data from the sunspot maximum years 1969–1970 were reanalyzed, and more recent observations from other studies were also reviewed. It was found that through the anomaly region, high scintillation indices were noted at a variety of azimuths of the propagation path rather than just along a path closely aligned with the magnetic meridian. A more complete evaluation of the geometrical factor, which must be of considerable importance in determining the absolute value of the scintillation intensity, awaits further observations.

Scattering from hydrometeors: A survey

Abstract: Investigation of the scattering properties of hydrometeors, especially those of nonspherical hydrometeors, has become increasingly important in connection with the estimation of depolarization due to hydrometeors in terrestrial and earth-space microwave communication systems. Scattering from nonspherical hydrometeors is also important in remote sensing of precipitation parameters. This paper first reviews various analytical-numerical approaches for the calculation of single-scattering properties of nonspherical hydrometeors. Approaches for the analysis of the practical problems, such as depolarization phenomena and rain scatter interference, are outlined in order to see how the single-scattering calculations are related to these problems. The present knowledge of incoherent scattering effects on rain attenuation, cross polarization, and channel transfer characteristics at the microwave and millimeter wave region is also reviewed.

Fresnel scattering model for the specular echoes observed by VHF radar

Abstract: In recent years several groups using VHF radars have reported highly anisotropic echoes originating from stable regions of the troposphere and stratosphere The echoes are most intense looking vertically, and their occurrence is well correlated with regions of hydrostatic stability Observations also show that the echoes are usually much stronger when observed with coarse spatial resolution There is no commonly used scattering or reflection model that can account for these features In this paper we present a new model to explain the enhanced vertical echoes The model pertains to scattering from the half-wavelength Fourier component of horizontally coherent fine structure in the radio refractive index within a region of finite vertical extent We refer to this echoing mechanism as ‘Fresnel scattering’ The new model is applied to simulate the magnitude of VHF radar echoes by using radiosonde temperature soundings as input Good agreement is found between these simulations and observations at the Sunset VHF radar in Colorado

Scattering from a perfectly reflecting arbitrary periodic surface: An exact theory

Abstract: We extend our exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface. Both Dirichlet (transverse electric polarization) and Neumann (transverse magnetic polarization) boundary conditions are considered. The method uses Green's theorem and our previous idea of expanding the surface fields in Fourier series multiplied by the Kirchhoff or physical optics approximation. The expansion coefficients solve a set of linear equations. The field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as a summation over these expansion coefficients multiplied by integrals over the surface function. The Rayleigh hypothesis is not invoked. Some examples are presented. For an analytic surface, steepest descent methods yield the asymptotic values of the amplitudes. Using this and other asymptotic results, the convergence of the scattered wave expansion is studied as it is analytically continued into the surface wells, and a simple and explicit confirmation of the conditions under which the Rayleigh hypothesis is valid is presented as well as new results for other examples. The periodic surface examples include a sinusoid, an echelette, a quadratic surface, a log cosine surface, a vortexlike surface, a cycloid, a trapezoid, a full-wave rectified surface, and a surface of semicircular cylinders (bosses). The method is general and applies to a very broad class of physical problems.

A comparison of radiative transfer models for predicting the microwave emission from soils

Abstract: Two general types of numerical models for predicting microwave emission from soils are compared-coherent and noncoherent. In the former, radiation in the soil is treated coherently, and the boundary conditions on the electric fields across the layer boundaries are used to calculate the radiation intensity. In the latter, the radiation is assumed to be noncoherent, and the intensities of the radiation are considered directly. The results of the two approaches may be different because of the effects of interference, which can cause the transmitted intensity at the surface (i.e., emissivity) to be sometimes higher and sometimes lower for the coherent case than for the noncoherent case, depending on the relative phases of reflected fields from the lower layers. This coupling between soil layers in the coherent models leads to greater soil moisture sampling depths observed with this type of model, and is the major difference that is found between the two types of models. In noncoherent models, the emissivity is determined by the dielectric constraint at the air/soil interface. The subsequent differences in the results are functions of both the frequency of the radiation being considered and the steepness of the moisture gradient near the surface. The calculations were performed at frequencies of 1.4 and 19.4 GHz and for two sets of soil profiles. Little difference was observed between the models at 19.4 GHz; and only at the lower frequency were differences apparent because of the greater soil moisture sampling depth at this frequency.

Total and partial reflection from the end of a parallel‐plate waveguide with an extended dielectric slab

Abstract: This investigation concerns a TEM wave obliquely incident onto the open end of a parallel plate wave guide with a truncated upper plate. The lower plate as well as the dielectric slab between the two plates is assumed to extend out indefinitely in order to support the propagation of at least one surface wave mode. Reflection from the wave guide opening is determined as a function of incident angle via a dual Wiener-Hopf formulation. It is found that total reflection, i.e., a reflection coefficient with magnitude equal to unity, is possible for incident angles greater than the critical angle associated with the lowest-order surface wave mode of the grounded dielectric slab. The field external to the parallel plate wave guide in this case becomes completely evanescent in the cross-sectional plane perpendicular to the plane of incidence, which is determined by the propagation direction of the incident wave and the edge of the upper plate. It is shown that the phase of the reflection coefficient thus obtained can be used to construct the modal equation for the fundamental mode(s) of a wide microstrip.

Electromagnetic scattering from a buried cylindrical inhomogeneity inside a lossy earth

Abstract: A solution is given for electromagnetic wave scattering from an infinitely long circular cylinder with a possibly radially stratified internal structure and which is buried in a lossy half space. The fields of the primary source are allowed to vary along the cylinder's axis. The scattered fields result from induced multipole line sources of electric and magnetic types located at the cylinder's axis. Explicit expressions are obtained for the source amplitudes up to the dipole terms that take into account the presence of the air-earth interface. The dipole sources may contribute significantly to the scattered fields even when the cylinder's radius is small relative to the wavelength in the surrounding medium. Some specific results are given for a buried tunnel and a conducting cylinder surrounded by an air gap and for an incident plane wave from the air region.

A new technique for the analysis of the dispersion characteristics of planar waveguides and its application to microstrips with tuning septums

Abstract: An efficient method for calculating the dispersion characteristics of planar waveguide structures is presented, of which the principle is known as the ‘method of lines’ in mathematical literature The wave equation is discretized in only one direction, and the resulting differential-difference equation can be solved analytically As an example of application the phase constant and the characteristic impedance of microstrips with tuning septums are calculated

Scattering cross sections for random rough surfaces: Full wave analysis

Abstract: The full wave approach developed earlier to evaluate the radiation fields scattered by deterministic two dimensionally rough surfaces is used here to determine the scattering cross sections for random rough surfaces. The medium below the irregular boundary is characterized by complex permittivity and permeability. For slightly rough surfaces, the full wave solutions for the incoherent scattered fields are shown to be in agreement with the perturbation solution. However, when the major contributions to the scattered fields come from the region of the rough surface around the stationary phase (specular) points, the full wave solutions are in agreement with the physical optics solutions. Thus the full wave solutions which reduce to the perturbation, the physical optics, and the geometrical optics approximations in special cases precisely determine the limitations of these approximations and reconcile the differences between them. The full wave solutions satisfy duality, reciprocity, and realizability relations in electromagnetic theory, and they are invariant under coordinate transformations.

Depolarization of radio waves by atmospheric hydrometeors in earth‐space paths: A review

Abstract: Many measurements of earth-space radio wave depolarization from 10 to 30 GHz have been made in recent years. This review relates the results from these measurements within the framework of theoretical relationships derived from rain and ice depolarization models. The review includes measured statistical relationships between depolarization and rain attenuation and contains cumulative statistics showing the relative frequencies of occurrence of rain and ice depolarization. Depolarization is usually strongest for circular polarizations and for fixed linear polarizations oriented 45° from vertical and horizontal. Depolarization is usually minimized for fixed vertical and horizontal polarizations. For frequencies up to 30 GHz and elevation angles ranging from 10° to 50°, amplitude ratios of ice and rain depolarization are approximately proportional to frequency and to cos2 θ/sin θ, where θ is the path elevation angle.

An analysis of short pulse and dual frequency radar techniques for measuring ocean wave spectra from satellites

Abstract: Scanning beam microwave radars were used to measure ocean wave directional spectra from satellites. In principle, surface wave spectral resolution in wave number can be obtained using either short pulse (SP) or dual frequency (DF) techniques; in either case, directional resolution obtains naturally as a consequence of a Bragg-like wave front matching. A four frequency moment characterization of backscatter from the near vertical using physical optics in the high frequency limit was applied to an analysis of the SP and DF measurement techniques. The intrinsic electromagnetic modulation spectrum was to the first order in wave steepness proportional to the large wave directional slope spectrum. Harmonic distortion was small and was a minimum near 10 deg incidence. NonGaussian wave statistics can have an effect comparable to that in the second order of scattering from a normally distributed sea surface. The SP technique is superior to the DF technique in terms of measurement signal to noise ratio and contrast ratio.

Occurrence of nighttime VHF scintillations near the equatorial anomaly crest in the Indian sector

Abstract: The behavior of nighttime F region irregularities near the northern crest of the equatorial anomaly in the Indian sector has been investigated by using VHF amplitude scintillation measurements made at Calcutta (27°N dip subionospheric) during the period April 1977 through February 1980. With the increase in solar activity the occurrence of scintillations increases remarkably during the equinoxes and to a lesser extent during the December solstice, while the local summer occurrence shows little change. The observed patterns are assessed in terms of the variation of the F layer height with solar activity and upwelling motion of the depleted flux tubes associated with small scale irregularities. It is shown that the propagation path is more likely to intercept these equatorial irregularities during periods of high solar activity, while during periods of low solar activity the equatorial irregularities lie south of the propagation path.

Scattering cross sections for composite random surfaces: Full wave analysis

Abstract: The full wave approach to rough surface scattering is applied to composite models of rough surfaces. In this work the principal distinguishing feature of the individual rough surface is its correlation distance. Thus this model can be applied to scattering by rough seas as well as by hilly terrain. It is shown that the full wave approach accounts for both specular scatter and Bragg scattering. The scattering cross section for the composite surface, with two or more roughness scales, is shown to be a weighted sum of the scattering cross sections for the individual rough surface heights. Shadowing effects are accounted for explicitly in the analysis. The full wave solutions satisfy reciprocity, duality, and realizability relationships in electromagnetic theory.

Real Hamilton equations of geometric optics for media with moderate absorption

Abstract: For lossless media, Hamilton's equations of geometrical optics can be derived from the dispersion equation either by the method of characteristics or by its combination with Sommerfeld-Runge's refraction law, Whitham's conservation law, and the expression ∂ω / ∂ k for the group velocity. The formal generalization to media with absorption leads to characteristics with complex space-time coordinates due to the now complex coefficients of the dispersion equation. For media with moderate absorption a real-valued generalization of Hamilton's equations is proposed. It is based on a dispersion equation with complex coefficients, on Sommerfeld-Runge's and Whitham's laws for the real parts of k, ω, on Connor and Felsen's condition for the imaginary part of k, and on the expression Re (∂ ω/∂k) for the velocity of a wave packet. These real-valued equations have been shown to hold in homogeneous media with moderate absorption.

Time domain scattering from open thin conducting surfaces

Abstract: The computation of transient electromagnetic scattering from open thin surfaces using a time domain E field integral equation is described. The integral equation is solved by a process of marching in time, eliminating the need for matrix inversion. A Gaussian incident pulse is used, avoiding discontinuities, and yielding the ‘smoothed impulse response’ of the target. Responses are plotted for several target shapes.

Observations of cloud‐produced amplitude scintillation on 19‐ and 28‐GHz earth‐space paths

Abstract: The amplitudes of satellite signals sometimes scintillate ± several decibels when heavy cumulus clouds pass through the radio path on hot summer days. These scintillations have been measured on a 19-GHz and 28-GHz earth-space path using 7-m and 0.6-m antennas at Crawford Hill, New Jersey. Scintillation intensity at 28 GHz is 1.2 times that at 19 GHz, consistent with the ƒ7/12 frequency dependence produced by a thin turbulent layer. The scintillation process is polarization-independent and has a low-pass power spectrum with a cutoff frequency of about 0.3 Hz. Rain attenuation often accompanies the more intense scintillation. The mean duration of scintillation-produced fades is short (about 1.3 s for fades greater than 1 dB), but over 1000 fades of over 1 dB at 28 GHz were observed in two summer months. Because of their weak frequency dependence these short but frequent events could produce repeated outages on 4- and 6-GHz earth-space links having low attenuation margins.

Use of polarization in electromagnetic inverse scattering

Abstract: The complete description of electromagnetic scattering processes implies polarization and since an electromagnetic scatterer acts like a polarization transformer we require measurements for the complete description of the target scattering matrices so that the descriptive parameters of a scatterer can be uniquely recovered from the measured field data. For the purpose of introducing the concept of polarization utilization in practice, the radar case is chosen and generalizations to other electromagnetic inverse problems are presented in the conclusions. For example, in radar target discrimination, identification and imaging use of measurement data available over the entire spatial frequency domain of the radar cross section must be made leading to various approximate frequency domain related approaches. Mainly for historical reasons of having had amplitude data available only, in most cases the approximations had been simplified to purely scalar nature; i.e., polarization-dependent properties were discarded, and the resulting theories are no longer valid or unique. It is the main objective of this analysis to show that because of the vector nature of electromagnetic inverse scattering, theories, if applicable in practice, require incorporation of complete polarization information into their formulation. By applying this approach to existing theories, it is shown that remarkable improvements in fidelity and quality of the reconstructed images are obtained and that indeed there is ample justification for continuing efforts in developing methods and theories of inverse scattering applicable to all those fields of physical sciences where information on the characteristic parameters of a scattering process is to be drawn from remote measurements, be it the electromagnetic vector case or the even more complicated seismic case of s and p wave interactions in elastic media.

New image representation for dipoles near a dissipative earth 1. Discrete images

Abstract: The image theory for dipoles above an imperfectly conducting earth which has been developed by Wait, among others, is used to compute the change in the input impedance of a vertical magnetic dipole (VMD) due to the presence of the earth. Comparison with a previously developed rigorous solution is made and a good agreement is obtained, particularly when the conduction currents in the earth dominate over the displacement currents. Extension of the-single image theory is made in two ways. First, multi-images which are spread in space are used to represent the effect of the earth. Second, multipole sources having the same location are used. Both representations have shown a noticeable improvement over the single image in relation to computational results on the input impedance of a VMD.

Estimation of the propagation delay through the troposphere from microwave radiometer data

Abstract: Microwave propagation delay through the troposphere is studied as a means of estimating the adverse effect of atmospheric water vapor on the accuracy of measurements made with very long baseline interferometry. Vertical profiles of temperature and water vapor density were obtained from a total of 240 radiosonde launches taken simultaneously at three New England locations in 1974; all studies were made at the 19 and 22.2 GHz frequency operating range of the radiometers. The wet path length in the zenith direction could be estimated from the surface water vapor density to an accuracy of 5 cm for the summer data and 2 cm for winter data, and with the brightness temperatures, it could be estimated to an accuracy of 0.3 cm. Two dual-frequency radiometers were also used to determine the accuracy of prediction of the path length from real radiometry data. The rms error for the estimation of wet path length from surface meteorological parameters was 3.2 cm, and for the radiometer brightness temperatures the rms error was 1.5 cm.

Emission from an inhomogeneous layer with irregular interfaces

Abstract: A theoretical model for microwave emission from an inhomogeneous layer has been developed using the radiative transfer method. The top and bottom boundaries of the layer are assumed to be randomly rough surfaces, and the scatter characteristics are assumed to be describable by the Kirchhoff scatter model. The presence of an irregular top boundary is found to cause a slower angular trend than the case of a plane layer and causes, in addition, a rise in temperature if the layer permittivity is large. On the other hand, if only the bottom boundary of the layer is rough, a rise in emission is observed relative to the plane layer. When both the top and the bottom boundaries are rough, the emissions at small nadir angles are closer to those of a layer with a rough bottom boundary, and the emissions at large nadir angles are closer to those of a layer with a rough top boundary. A comparison of the layer model with reported angular measurements from a snow layer at three different frequencies and polarizations shows promise for such a model. An additional comparison of emission versus the depth of a snow pile indicates that an irregular snow ground interface causes higher emissions at small depths. This fact is necessary to explain the observed measurements.

Cross polarization during precipitation on terrestrial links: A review

Abstract: This paper reviews recent theoretical and experimental research on cross polarization that occurs during precipitation on terrestrial microwave links. Consideration is first given to the physical mechanisms, such as raindrop canting, that cause precipitation depolarization. Coherent rain depolarization models are then discussed and recent investigations of incoherent depolarization effects summarized. Particular attention is given to semi-empirical models since they can be used most readily for statistical predictions in design applications. ‘Model oriented’ cross polarization measurements aimed at characterizing the physical properties of the depolarizing medium for use in model development are also given particular attention. Effective parameters of the raindrop canting angle distribution derived from these measurements are tabulated. Finally, some mention is made of the relative importance of rain depolarization and attenuation in causing outages on digital radio systems.

Using radar to estimate dissipation rates in thin layers of turbulence

Abstract: A method is described for estimating the energy dissipation rate, e, from radar measurements of the structure parameter, C2n, in stably stratified layers of turbulence. This estimate applies to horizontal layers of turbulence whose thickness, Lt, is less than the vertical resolution, 2Δ, of the radar (Lt, < 2Δ). Both e within the turbulence layer and the average of e over the radar volume, are determined. The derived expression for e is based on a theoretical relation of turbulence layer thicknesses to the buoyancy length.