THE PURPOSE of this paper and its companion [1] is to

Transcription

1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY Incoherent Scatter Spectral Theories Part I: A General Framework and Results for Small Magnetic Aspect Angles Erhan Kudeki, Member, IEEE, and Marco A. Milla, Member, IEEE Abstract A general framework for the incoherent scatter radar spectral theories is presented in terms of the generalized Nyquist theorem, Kramers Kronig relations, and the characteristic functions of random charge-carrier displacements in the ionosphere. Specific spectral models for typical ionospheric conditions are derived and discussed. The discussions focus on the effect of Coulomb collisions in magnetized plasmas, which is a topic of current interest treated in further detail in the companion paper by the authors, as well as on different combinations of physical principles that can be invoked in the derivation of incoherent scatter spectral models. Index Terms Incoherent scatter, ionosphere, radar. I. INTRODUCTION THE PURPOSE of this paper and its companion [1] is to present a unified description of the spectral signal models pertinent to ionospheric incoherent scatter radars (ISRs), including the latest model corrections for the F-region incoherent scatter at small magnetic aspect angles. In the magnetized F-region ionosphere, and particularly at small magnetic aspect angles, the ISR spectrum has been found to be highly sensitive to charged particle collisions [2] despite the smallness of the pertinent collision frequencies as compared with the ionospheric gyrofrequencies. A quantitative model of the collision effects applicable at all magnetic aspect angles has been developed and will be described in [1] based on a general framework presented and described in this paper. The new collisional model in [1] extends the 1-D collisional results of Sulzer and González [2] into three dimensions and to zero aspect angle and should enable the proper inversions of incoherent scatter spectral measurements conducted at equatorial latitudes by the Jicamarca and ALTAIR radars. The general framework presented in this paper provides a new perspective to understand how the different regimes of ISR response relate to one another and how the new collisional model relates to the less accurate F-region models utilized in the past. Manuscript received March 30, 2009; revised November 26, 2009; accepted January 24, Date of publication October 7, 2010; date of current version December 27, This work was supported by the National Science Foundation Grant ATM to the University of Illinois. E. Kudeki is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Urbana, IL USA. M. A. Milla is with the Remote Sensing and Space Sciences Group, Department of Electrical and Computer Engineering, University of Illinois, Urbana USA. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TGRS This paper is organized as follows. In Section II, we provide a brief review of the Thomson-scatter-based ISR technique and subsequently develop a general framework of incoherent scatter spectral theories that can be expressed in terms of the characteristic functions of charged particle displacements in thermal equilibrium. In Section III, we illustrate how the framework can be utilized to develop the specific spectral models pertinent to different types of ionospheric plasmas. Examples of the new spectral results representing the collisional and magnetized F-region plasmas are also presented in Section III as a prelude to the more detailed descriptions provided in [1]. This paper concludes with further discussions of our results in Section IV. A general discussion contrasting the different approaches based on different sets of physical principles to the incoherent scatter problem is included in Section IV. II. THOMSON SCATTER AND INCOHERENT SCATTER SIGNAL SPECTRUM Oscillating free electrons radiate like Hertzian dipoles to give rise to a phenomenon known as Thomson scatter. The so-called ionospheric incoherent scatter consists of Thomson scattering of radar pulses at radio frequencies from collections of ionospheric free electrons. The incoherent scatter signal spectrum depends not only on the motion of ionospheric free electrons but also on the dynamics of ionospheric ions interacting with the electrons. Therefore, a wealth of information about ionospheric charged particles can be extracted from the measured signal spectra of ISRs. In this section, we first outline how the incoherent scatter signal spectrum can be related to the k-ω spectrum of electron density fluctuations in the ionosphere and then present a general framework relating the electron density fluctuation spectrum to random charge-carrier dynamics in an ionosphere in thermal equilibrium. A. Thomson Scattering From a Single Free Electron Consider a free electron at the origin which has been exposed to a ẑ-polarized time-harmonic TEM wave with a wavenumber k ω/c and a field phasor E i =ẑe i e jkx (1) arriving from the ˆx direction, as shown in Fig. 1. Forcing ee i due to the incident field will accelerate the electron and cause /$ IEEE

2 316 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 Fig. 1. Free electron exposed to an incident TEM wave will radiate like a Hertzian dipole antenna (see text). it to radiate a scattered TEM wave. The equation of motion (Newton s second law) of the electron is, in phasor form, mjωv z = ee i (2) from which we find v z = e mjω E i. (3) An oscillating electron with a velocity phasor ẑv z is effectively a Hertzian dipole with a dipole moment I o dz = ev z. Its radiation field is obtained from standard result for Hertzian dipoles (e.g., [3]) as E r (r) =jkη o I o dz e jkr 4πr sin θˆθ = ˆθ sin θ r e r E ie jkr, (4) where η o μ o /ɛ o is the intrinsic impedance of free space e 2 r e 4πɛ o mc m (5) is the classical electron radius, and ˆθ is the unit vector in the direction of increasing θ (see Fig. 1). For θ = π/2 and ˆθ = ẑ representing the backscatter case, we have where E r (r) =E r (rˆx) ẑe s, (6) E s = r e r E ie jkr (7) will be referred to as the backscattered field amplitude (complex). Consider next a radar antenna radiating from the origin and illuminating a free electron located at some radar range r and a vector position r such that r = r. Taking the origin as our phase reference, we can express the incident field phasor on the electron as E i = E o (r)e jk or, (8) where k o ω o /c is the incident wavenumber at the radar operation frequency ω o and E o (r) denotes a slowly varying function of r representing the incident field amplitude illuminating the electron. The backscattered field phasor from the electron propagated back to the origin can then be expressed, with the help of (7), as E s = r e r E ie jkor = r e r E o(r)e j2kor. (9) We will next use (9) as a building block to model the complex envelope of the signal received by a backscatter radar Fig. 2. (a) Cartoon depicting radar scattering volume defined by the radar antenna beam and (b) the geometry of a subvolume ΔV of the scattering volume shown in (a). Blue and red dots represent the scattering particles (electrons, mainly, since the scattering cross section of ions is negligibly small compared with that of the electrons) moving toward and away from the radar antenna, respectively. responding to a collection of a large number of ionospheric free electrons filling a radar antenna beam as shown in Fig. 2. B. Thomson Scattering From a Collection of Free Electrons Focus first on the signal received from a small subvolume ΔV of a radar beam shown in Fig. 2(a); the subvolume is illustrated in the blowup shown in Fig. 2(b). We will consider the subvolume to be small enough so that a spherical wave incident from the radar antenna can be well approximated as a plane wave within ΔV. Superposing the field contributions of the form (9) from each one of P N o ΔV electrons contained within ΔV, we can then model the backscattered signal from the entire subvolume as N o ΔV E s = r e E op e j2k or p r N e r p r E o ΔV o e jk r p. (10) The aforementioned r p is the radar range of each electron and E op is the incident field amplitude at each electron location, whereas r and E o are the same parameters at the center of the subvolume. The rightmost expression in (10) invokes the planewave (or paraxial) approximation idea discussed previously and can be justified for r>4δv 2/3 /λ o, effectively the far-field condition for an antenna of size ΔV 1/3 and operating wavelength λ λ o /2; in the same expression, vector r p denotes the positions of individual electrons and k 2k oˆr, (11) known as the Bragg vector, is the scattered minus the incident wave vector relevant for subvolume ΔV. Also, N o is the average density of the electrons in the subvolume.

3 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 317 C. Particle Trajectories r p (t) and Density Waves n e (k,t) Let r p (t) denote the trajectories of the individual electrons located in subvolume ΔV. Assuming nonrelativistic electron speeds, the scattered field envelope (10) can then be modified as E s (t) r N e r E o ΔV i e jk r p(t r c ) r e r E i n e (k,t r ) (12) c to account for electron motions, with r/c corresponding to the propagation time delay from the center of ΔV to the radar antenna 1 and n e (k,t) N o ΔV e jk r p(t) (13) denoting the 3-D spatial Fourier transform dr n e (r,t)e jk r (14) of n e (r,t) N o ΔV δ (r r p (t)) (15) representing the microscopic number density function of the electrons within ΔV. Hence, (12) indicates that the scattered field amplitude from ΔV varies in time like a scaled and delayed replica of the 3-D spatial Fourier transform of the electron density n e (r,t) across ΔV. This will ultimately provide the link between the radar signal statistics and the statistics of electron density variations in the region probed by the radar. Note that the density function n e (r,t) and its Fourier transform n e (k,t) should be modeled as random processes to the extent that the individual particle trajectories r p (t) are, in general, unpredictable. In that case, our formulation of radar signals will have to rely on probability distribution functions (pdfs) of various functions of r p (t) and related statistical measures, including the following. 1) Normalized variance 1 ΔV n e (k,t) 2 (16) of the electron-density Fourier transform n e (k,t), which is, in ΔV limit, the spatial power spectrum of n e (r,t). Here and in the following, and denote the expected value of the bracketed random variable calculated using the relevant pdf. We will assume that the expected value is independent of time t, which is the same as assuming n e (r,t) to be a wide-sense stationary random process. 1 Note that r p(t r/c) is an acceptable approximation for the actual electron locations contributing to E s(t) so long as v p /c λ/δv 1/3, compatible with nonrelativistic electron motions. Quantum mechanical fuzziness of electron trajectories can also be ignored when the de Broglie wavelength of the electrons is negligible compared with practical radar wavelengths λ = λ o/2. 2) Normalized autocorrelation function (ACF) 1 ΔV n e(k,t)n e (k,t+ τ) (17) of n e (k,t), another expected value, which can be Fourier transformed over time lag τ to obtain the space time or k-ω spectrum of n e (r,t), namely n e (k,ω) 2 dτe jωτ 1 ΔV ( n e k,t r c ) n e (k,t r c +τ ). (18) In view of these definitions, we find that (12) implies a baseband signal spectrum E s (ω) 2 = r2 e r 2 E i 2 n e (k,ω) 2 ΔV (19) for the scattered field from subvolume ΔV, where E s (ω) 2 dτe jωτ Es(t)E s (t + τ) (20) and, conversely E s(t)e s (t + τ) = dω 2π ejωτ E s (ω) 2. (21) Note that (19) is obtained from (18) by first forming the ACF of E s (t) E i n e (k,t (r/c)) and then Fourier transforming the ACFs in accordance with (20). D. Soft-Target Radar Equation and Electron Density Spectrum By Parseval s theorem, and as implied by (21), the total power E s (t) 2 backscattered from a subvolume ΔV must be the integral of E s (ω) 2 over all ω/2π. The total power detected by a backscatter radar is then a subsequent sum over all subvolumes ΔV contained within the radar beam and pulse, with each term of the sum given a proper weighting that accounts for the effective area of the receiving antenna. Assuming an open radar bandwidth see [4] for a discussion of receiver filter effects this leads to a soft-target radar equation of the form dω P r = dv E i 2 /2η o 2π r 2 A e re 2 n e (k,ω) 2 (22) representing the total power collected by a radar having an effective antenna area denoted by A e and a finite-duration transmitted pulse E i 2 that limits the extent of the integration volume. This backscatter radar equation is applicable not only to ISRs but also to all types of ionospheric backscatter radars having operation frequencies far exceeding the ionospheric plasma frequency. The distinction between ISRs and smaller backscatter radars is just a matter of sensitivity smaller radars can collect detectable levels of P r only when n e (k,ω) 2 magnitudes substantially exceed the minimal incoherent scatter levels (discussed in Section III) encountered in stable ionospheres.

4 318 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 Using the spatial Fourier transform (13), the k-ω electron density spectrum (18) needed in radar (22) can be expanded as n e (k,ω) 2 = dτe jωτ 1 ΔV No ΔV e jk r p(t r c ) N oδv e jk r p(t r +τ) c = dτe jωτ 1 ΔV N o ΔV N o ΔV e jk r p(t r c ) e jk r q(t r +τ) c. q=1 (23) Assume for a moment that the individual electrons in ΔV follow independent random trajectories so that the expected value within (23) is zero except when q = p. Then, and only then, (23) would simplify as n e (k,ω) 2 =N o dτe jωτ e jk Δr n te (k,ω) 2 (24) with ( Δr r p t r ) ( c + τ r p t r ) c (25) denoting the random displacement vector of individual electrons over time lags τ along their independent trajectories r p (t). We have labeled this special result, namely, N o dτe jωτ e jk Δr, with a special symbol n te (k,ω) 2 in anticipation of its future use, since, as we will find out, N o dτe jωτ e jk Δr computed for hypothetical particles with independent trajectories plays a crucial role in describing n e (k,ω) 2 even when the independent trajectory assumption is not valid (as in a real ionosphere). Note that if we were to assume the validity of (24) for a moment, we would obtain from (19) E s (ω) 2 n te (k,ω) 2 = N o dτe jωτ e jk vτ e ω 2 2k 2 C e 2 (26) for a collisionless plasma in thermal equilibrium in which the electrons are assumed to follow straight-line trajectories so that Δr = vτ in terms of Maxwellian-distributed electron velocities v. This result clearly predicts a Gaussian signal spectrum shape with a width proportional to electron thermal speed C e KT/m, as shown in Fig. 3(a), and it represents the original expectation of Gordon [5] for ionospheric backscatter radars operating under thermal equilibrium conditions. However, the assumption of independent electron trajectories is not applicable in the real ionosphere because of collective interactions in the plasma arising from macroscopic field fluctuations. Because of collective interactions, ionospheric ISR spectra turn out to be narrower and have double-humped shapes, as in Fig. 3(b). The spectra shown in Fig. 3 are representative of a 41-MHz backscatter radar like the one used by Bowles [6] in his initial ISR measurements. We will next describe how collective interaction effects can be included in the incoherent scatter spectral models. Fig. 3. (a) Gaussian spectrum from a hypothetical ionosphere of independent electrons and (b) a double-humped spectrum typical of ISR observations, assuming an operation frequency of 41 MHz (and a collisionless and nonmagnetized O + ionosphere with T e = T i = 1000 K). E. Collective Interaction Effects on Ionospheric Backscatter Ionospheric plasmas consist of charged particles exhibiting random motions of thermal origin. Thermally driven electron motions can be described in terms of statistically independent trajectories, but those trajectories are invariably perturbed by the correlated response of the electrons to electrostatic fields generated by space-charge imbalances at macroscopic scales. As a result of such perturbations, the electron density in a volume ΔV of the ionosphere deviates from n te (r,t) with the spectrum given by (24) and needs to be viewed as some n e (r,t)=n te (r,t)+δn e (r,t) (27) where δn e (r,t) accounts for the density changes caused by macroscopic forces a so-called collective interaction effect. Likewise, positive ion density within ΔV can be expressed by some n i (r,t)=n ti (r,t)+δn i (r,t) (28) where δn i (r,t) is due to collective interactions while n ti (r,t) denotes the thermally impressed component of ion density with a spectrum n ti (k,ω) 2 of the same form as (24). We will next show that the spectrum n e (k,ω) 2 of the actual electron density fluctuations n e (r,t) in a plasma, including the collective interaction effects, can be expressed as a weighted superposition of n te (k,ω) 2 and n ti (k,ω) 2, representing the fluctuation spectra of electrons and ions computed in the absence of collective interactions.

5 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 319 Fig. 4. Equivalent circuit implied by constraint (30). As a prelude to our derivation of n e (k,ω) 2,wefirstnote that the component of the total electric current density J along a wave vector k will be a superposition of the following: 1) thermally impressed random current component J t =(ω/k)e(n ti n te ) obtained from a) the plane-wave form of current continuity equation jωρ t jk J t =0, and b) thermally driven space-charge density ρ t e(n ti n te ) expressed in terms of random variables n te and n ti ; 2) macroscopic-scale conduction currents σ e E and σ i E carried by the deviation components of particle motions, where a) σ e and σ i denote the electron and ion conductivities, respectively, and b) E is the amplitude of the macroscopic polarization electric field developed in the k-direction as a consequence of the overall mismatch between electron and ion densities n e and n i. Thus, the scalar component of the plane-wave form of Ampere s law, jk H = J + jωɛ o E, (29) in the direction of wave vector k, can be written as 0=(σ i + σ e )E + ω k e(n ti n te )+jωɛ o E. (30) Note that (30) has the form of Kirchhoff s current law applied at the top node of a lumped-element equivalent circuit shown in Fig. 4, consisting of a capacitance ɛ o shunting a pair of electron and ion branches with admittances σ e,i and parallel current sources n te,ti representing the independent thermal drivers within the plasma. The branch currents ωk 1 en i and ωk 1 en e indicated on the diagram refer to the total ion and electron current density components J i and J e, including those introduced by collective effects. Solving the equivalent circuit problem for voltage E and then using E in the electron current expression ωk 1 en e = Eσ e ωk 1 en te, we find that the electron density fluctuation amplitude including the collective effects is n e (k,ω)= (jωɛ o + σ i )n te (k,ω)+σ e n ti (k,ω) jωɛ o + σ e + σ i. (31) Fig. 5. Sketch of the electron-line and ion-line components of density spectrum (32) calculated for a 41-MHz backscatter radar assuming a collisionless and nonmagnetized O + ionosphere with T e = T i = 1000 K. Since n te and n ti are independent random variables, we also find, by squaring and averaging (31), the electron density spectrum n e (k,ω) 2 jωɛ o + σ i 2 n te (k,ω) 2 = jωɛ o + σ e + σ i 2 σ e 2 n ti (k,ω) 2 + jωɛ o + σ e + σ i 2 (32) which is a sum (on the right) of the so-called electronand ion-line components proportional to n te (k,ω) 2 and n ti (k,ω) 2, respectively. Fig. 5 shows the electron- and ionline components of an incoherent scatter spectrum for a possible radar configuration and an ionosphere with Maxwellian electrons and ions. The spectrum formula (32) is a very general result which is valid with any type of velocity distribution Maxwellian or not of the plasma particles so long as the plasma is stable, i.e., so long as the poles of (31) are confined to the left half of the complex jω plane. Also, (32) can be modified in a straightforward way to treat the multi-ion case by rewriting the ion current in (30) as a sum over multiple ion species. It is also valid for magnetized plasmas in the electrostatic approximation i.e., for nearly longitudinal modes with phase speeds ω/k c for which the longitudinal currents produced by transverse field components are negligible with σ e,i (k,ω) denoting just the longitudinal components of particle conductivities in that case. Clearly, the use of (32) for spectral calculations requires having the appropriate conductivity expressions σ e,i (k,ω).the latter can be derived by employing the standard tools of plasma kinetic theory for ionospheric plasmas or, alternatively, by taking advantage of some fundamental links that exist between conductivities σ e,i (k,ω) and e jk Δr -statistics of the plasma particles under thermal equilibrium conditions.

6 320 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 These links and procedures which are sufficient to specify σ e,i (k,ω) under thermal equilibrium conditions (without going through kinetic theory) are described next. F. Links Consider the following three relationships of very general applicability: 1) Nyquist noise theorem applicable to linear resistive circuits; 2) its generalization, known as fluctuation dissipation theorem, applicable to stable linear dissipative systems in thermal equilibrium; 3) Kramers Kronig relations applicable in all linear causal systems. It turns out that a joint use of these general relationships constrains the forms of thermally driven density spectra n te,ti (k,ω) 2 and longitudinal conductivities σ e,i (k,ω) needed in the computation of ISR spectral models applicable at thermal equilibrium. According to the Nyquist theorem [7], the open-circuit voltage v(t) and short-circuit current i(t) of an electrical circuit element with an input impedance Z(ω) =Y 1 (ω) will exhibit thermally driven fluctuations having the two-sided power spectra V (ω) 2 and I(ω) 2, respectively, with associated variances such that v(t) 2 = i(t) 2 = V (ω) 2 dω 2π I(ω) 2 dω 2π (33) (34) V (ω) 2 =2KTRe {Z(ω)} (35) I(ω) 2 =2KTRe {Y (ω)} (36) for frequencies f = ω/2π satisfying h f KT, where T is the equilibrium temperature of the element in kelvin while h and K are Planck s and Boltzmann constants, respectively. These power spectra can be used for specifying the thermal noise sources in equivalent circuit models of electrical systems constructed for noise analysis. Callen and Welton [8] showed that the Nyquist noise theorem is a particular case of a more general fluctuation dissipation theorem that applies to all linear dissipative systems sustained in thermal equilibrium. A simplified statement of the fluctuation dissipation theorem can be made as follows: Any linear dissipative system composed of components in thermal equilibrium will exhibit thermally driven fluctuations having the power spectra that can be obtained by applying the Nyquist noise theorem to an appropriately constructed equivalent circuit model of the system. According to the fluctuation dissipation theorem then, the current sources in the model circuit of ionospheric fluctuations shown in Fig. 4 (corresponding to the short-circuit currents of the electron and ion branches, respectively) must have the twosided spectra ω 2 k 2 e2 n te,i (k,ω) 2 =2KT e,i Re {σ e,i (k,ω)} (37) per unit bandwidth, per species (in species frames of reference), so long as each species is in thermal equilibrium (i.e., has a Maxwellian velocity distribution) at a temperature T e,i. This relation specifies Re{σ e,i (k,ω)} unambiguously when e jk Δr e,i dependent thermal spectra n te,i (k,ω) 2 expressed by (24) are known. 2 Note that, in specifying e jk Δr e,i in (24), the trajectories r e,i (t) should be computed, assuming the absence of macroscopic electric fields in the plasma. Kramers Kronig relations (e.g., [9]) finally identify Im{σ e,i (k,ω)} as the Hilbert transform of Re{σ e,i (k,ω)}. This relation is, in fact, a general rule that applies between the real and imaginary parts of Fourier transforms of all causal signals vanishing for t<0, as discussed in more detail in Appendix I. Electric conductivity σ e,i (k,ω) is a canonical example of such a Fourier transform, one that describes the frequency response pertinent to electric-field forcing of conduction currents in linear conductors (such as the stable ionosphere). Clearly, having the Kramers Kronig relations, we get full access to complex σ e,i (k,ω) once n te,i (k,ω) 2 and Re{σ e,i (k,ω)} are known. In summary then, as a consequence of the aforementioned relations, we find that, in a plasma in thermal equilibrium, all the inputs needed to compute the electron density fluctuation spectrum can be deduced from the expectations e jk Δr e,i representing the characteristic function of thermally impressed particle displacements Δr e,i in the absence of collective effects (i.e., with suppressed macroscopic fields). We will call these characteristic functions as single-particle ACFs, since the exponential e jk r e from which the ACF e jk Δr e is obtained represents, according to (9), the backscattered radar signal returned from a single electron. G. General Framework of ISR Spectral Models Let J s (ω) 0 dτ e jωτ e jk Δr s (38) denote a Gordeyev integral for species s (e or i for the singleion case) in a plasma in thermal equilibrium, where e jk Δr s is the characteristic function of the corresponding particle displacements Δr s over intervals τ in the absence of collective interactions. Then we have the following (see Appendix II for a 2 Also, when T e = T i = T, i.e., in case of full thermal equilibrium, ω 2 k 2 e 2 n e(k,ω) 2 =2KTRe{σ t(k,ω)}, with σ t representing the Thevenin admittance looking into the circuit from an open introduced in the electron branch.

7 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 321 detailed derivation based on the ideas introduced in the previous section): n ts (k,ω) 2 and N o =2Re{J s (ω s )} (39) σ s (k,ω) jωɛ o = 1 jω sj s (ω s ) k 2 h 2 s (40) where ω s ω k V s is a Doppler-shifted frequency in the radar frame due to the bulk velocity V s of species s, h s ɛo KT s /N o e 2 is the corresponding Debye length, and the k-ω spectrum of electron density fluctuations in the equilibrium plasma is given by (32), namely n e (k,ω) 2 jωɛ o + σ i 2 n te (k,ω) 2 = jωɛ o + σ e + σ i 2 σ e 2 n ti (k,ω) 2 + jωɛ o + σ e + σ i 2 (41) or its multi-ion generalizations. The general framework of ISR spectral models represented by (38) (41) takes care of the macrophysics of the incoherent scatter process due to collective effects, while microphysics details of the process remain to be addressed in the specification of single-particle ACFs e jk Δr s. Examples of the usage of the framework in practical ionospheric incoherent scatter settings are provided in Section III. III. IONOSPHERIC APPLICATIONS A. Single-Particle ACFs e jk Δr According to the general framework presented in the previous section, the incoherent scatter spectrum of ionospheric plasmas can be specified in terms of single-particle ACFs e jk Δr of ionospheric charge carriers in thermal equilibrium. In general, an ACF e jk Δr can be explicitly computed if the pdf f(δr), where Δr is the component of Δr along k, is known. Alternatively, e jk Δr can also be computed directly given a set of realizations of Δr for a given time delay τ. In either case, pdfs f(δr) or pertinent sets of Δr data will reflect the dynamics of random particle motions taking place in ionospheric plasmas. B. ACF in Nonmagnetized and Collisionless Plasmas In a nonmagnetized and collisionless plasma, charge carriers will move along straight-line trajectories with random velocities v. In that case, the displacement vectors will be Δr = vτ (42) over intervals τ. Assuming Maxwellian-distributed velocity components v along wave vector k, we then have Gaussiandistributed displacements Δr = vτ with a pdf f(δr) = e Δr 2 2Δr 2 2πΔr2 (43) Fig. 6. (a) Cartoon depicting particle displacements Δr in a plasma with straight-line charge-carrier trajectories and (b) a sample ISR spectrum for a nonmagnetized and collisionless plasma in thermal equilibrium. and a variance (mean-squared displacement defined in the radar probing direction k) Δr 2 = v 2 τ 2 = C 2 τ 2 (44) where C = KT/m is the thermal speed of the charge carrier. Since the single-particle ACF of a Gaussian-distributed Δr is, in general, e jk Δr = e jkδr f(δr)d(δr) =e 1 2 k2 Δr 2 (45) we obtain e jk Δr = e 1 2 k2 C 2 τ 2 (46) for thermal equilibrium particles in a nonmagnetized and collisionless plasma, which, in turn, leads (via the general framework equations) to the most basic incoherent scatter spectral model exhibiting double-humped shapes, as shown in Fig. 6(b) when (46) is applied to both electrons and ions. C. Effect of Collisions The ACF (46) should also be applicable in collisional nonmagnetized plasmas so long as the relevant collision frequency ν is small compared to the product kc, i.e., ν kc, so that an average particle moves a distance of many wavelengths 2π/k in between successive collisions. More generally, the expression (44) for a mean-squared particle displacement remains valid in a nonmagnetized collisional plasma until the first collisions take place at τ ν 1. At larger τ, the meansquared displacement Δr 2 as well as the pdf f(δr) will, in general, depend on the details of the dominant collision process. Long-range Coulomb collisions between charged particles (e.g., electrons and ions) are frequently modeled as a Brownian motion process, a procedure which leads to a Gaussian f(δr) with a variance Δr 2 = 2C2 ν 2 (ντ 1+e ντ ) (47) as shown in Appendix III and further discussed in [1] the Brownian motion process is, by definition, one in which position and velocity increments are Gaussian random variables

8 322 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 at all time delays τ and happen to be stochastic solutions of a first-order Langevin update equation with constant coefficients, as discussed in Appendix III. Given the variance (47), the corresponding single-particle ACF is e jk Δr = e k2 C 2 ν 2 (ντ 1+e ντ ) (48) having the asymptotic limits (46) as well as e jk Δr = e k2 C 2 ν τ (49) for ν kc and ν kc (or for ντ 1 and ντ 1), respectively. Note that when (49) is applicable, with ν kc, an average particle moves across only a small fraction of a wavelength 2π/k in between successive collisions. In Coulomb interactions, the time ν 1 between effective collisions (an accumulated effect of interactions with many collision partners) can be interpreted as the time interval over which the particle velocity vector rotates by about 90. Binary collisions of charge carriers with neutral atoms and molecules dominant in the lower ionosphere can be modeled as a Poisson process [10] and treated kinetically using the Bhatnagar Gross Krook collision operator (e.g., [11]). As shown in [10], in binary collisions with neutrals, the meansquared displacement of charge carriers is still given by (47), but the relevant pdf f(δr) is a Gaussian only for short and long delays τ, satisfying ντ 1 and ντ 1, respectively. At intermediate τ s, the ACF of a collisional plasma dominated by binary collisions will then deviate from (48), and as a result, collisional spectra will, in general, exhibit minor differences between binary and Coulomb collisions except in ν kc and ν kc limits (e.g., [10] and [12]). As the aforementioned discussion implies, the single-particle ACF in the high collision limit (ν kc) is insensitive to the distinctions between Coulomb and binary collisions and obeys a simple relation (49). In that limit, it is fairly straightforward to evaluate the corresponding Gordeyev integrals analytically and obtain (via the general framework equations) a Lorentzianshaped electron density spectrum (mainly the ion line ) n e (k,ω) 2 2k 2 D i ω 2 +(2k 2 D i ) 2 (50) N o valid for kh s 1 (wavelength larger than the Debye length), where D i C 2 i /ν i = KT i /m i ν i denotes the ion diffusion coefficient in the collisional plasma. This result is pertinent to D-region incoherent scatter observations (see Fig. 7) neglecting possible complications due to the presence of negative ions (e.g., [13]). D. Generalizations Using the high collision result (50), it can be easily shown that n e (k) 2 dω n e (k,ω) 2 = N o (51) 2π 2 Fig. 7. Collisional D-region spectrograms from Jicamarca Radio Observatory (from [14]). (a) ISR spectrogram (in decibels). (b) I-ion fit spectrogram (in decibels). Fig. 8. Backscattering geometry in a magnetized ionosphere parameterized by wave-vector components k and k and aspect angle α. which is, in fact, true in general i.e., for all types of plasmas with or without collisions and/or dc magnetic field so long as T e = T i and kh s 1. This result, in turn, leads to a wellknown volumetric radar cross-section formula 4πre 2 n e (k) 2 =2πreN 2 o (52) for ISRs, which is valid under the same conditions. E. Effect of an Ambient Magnetic Field B In a magnetized ionosphere with an ambient magnetic field B, it is convenient to express the scattered wave vector as k = ˆbk +ˆpk, where ˆb and ˆp are orthogonal unit vectors on the k-b plane which are parallel and perpendicular to B,

9 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 323 respectively, as shown in Fig. 8. We can then express the singleparticle ACF pertinent to a magnetized plasma in the form e jk Δr = e j(k Δl+k Δp) = e jk Δl e jk Δp (53) where Δl and Δp are particle displacements along unit vectors ˆb and ˆp. Assuming independent Gaussian random variables Δl and Δp, we can then write e jk Δr = e 1 2 k2 Δl2 e 1 2 k2 Δp2 (54) in analogy with the nonmagnetized case. The assumptions are clearly justified in the case of a collisionless ionosphere (or for intervals τ such that τν 1), in which case and, as shown in Appendix IV Δl 2 = C 2 τ 2 (55) Δp 2 = 4C2 Ω 2 sin2 (Ωτ/2), (56) where Ω qb/m is the particle gyrofrequency. The meansquared displacement (56) which is periodic in τ is derived in Appendix IV by invoking circular particle orbits with periods 2π/Ω and mean radii 2C/Ω on the plane perpendicular to B. As a consequence of (55) and (56), ISR spectra in a magnetized but collisionless ionosphere can be derived from the singleparticle ACFs e jk Δr = e 1 2 k2 C2 τ 2 e 2k 2 C2 Ω 2 sin 2 (Ωτ/2) (57) for electrons and ions. Note that the ACF (57) becomes periodic and the associated Gordeyev integrals and spectra become singular in k 0 limit (expressed in terms of delta functions). Spectral singularities are, of course, not observed in practice, since collisions in a real ionosphere end up limiting the width of single-particle ACFs in τ in the limit of small aspect angles α. Despite the singularities in (57), it turns out that, for finite aspect angles α larger than a few degrees, the collisionless result (57) leads us to the most frequently used ISR spectral model at F-region heights. This is true because, given a finite k,thetermexp( (1/2)k 2 C2 τ 2 ) in (57) restricts the width of the ACF to a finite value of (k C) 1 even in the absence of collisions (or when collision frequencies are smaller than k C). It can then be shown that, for τ (k C) 1,aswellasΩτ 2π (easily satisfied by massive ions), the ACF (57) for ions recombines to a simplified form exp( (1/2)k 2 C 2 τ 2 ) as if the plasma were nonmagnetized. Also, with finite k, the ACF (57) for electrons simplifies to exp( (1/2)k 2 sin 2 αc 2 τ 2 ), since, for the light electrons, a condition k C Ω can be easily invoked to ignore the rightmost exponential in (57) [or even more accurately, replace it with its average value over τ, namely, 1 (k 2 C2 /Ω 2 )]. These ion and electron ACFs exhibit similar τ dependences and lead to similarly shaped Gordeyev integrals. F. Gordeyev Integral Calculations GivenanACFe jk Δr, the computation of the Gordeyev integral (38), namely J(ω) 0 dτ e jωτ e jk Δr (58) will, in general, require numerical integration. The application of a general-purpose chirp-z algorithm [16] which can be used with arbitrary shaped ACFs e jk Δr has been described in [4]. For F-region computations of the form J(ω) = 0 dτ e jωτ e 1 2 k2 C 2 τ 2 (59) at finite aspect angles α (see the discussion in the previous section), we can make use of the identity jz(θ) 0 dt e jθt e t2 4 = πe θ 2 j2 e θ2 θ 0 e t2 dt. (60) This identity can be verified by showing that Z(θ) satisfies Z(0) = j π and the differential equation dz dθ = 2 2θZ(θ) (61) where the right side of (61) is obtained by applying integration by parts to dz/dθ derived from the leftmost integral in (60). The right side of (60) can, in turn, be obtained from the solution of the differential equation (61) subject to initial condition Z(0) = j π after using an integrating factor e θ2. To compute Z(θ), we can make use of Dawson s integral function e θ2 θ 0 et2 dt defined in MATLAB use syntax mfun( dawson, θ) and Mathematica. In addition, Mathematica can calculate Z(θ) in terms of the error function with complex argument Erfi. Finally, Z (θ) for real θ, together with its analytic continuation into complex θ plane, is known as plasma dispersion function and has been tabulated by Fried and Compte [17]. Fig. 9 shows sample F-region ISR spectra computed for the 50-MHz Jicamarca ISR [15] using the collisionless ACF models discussed previously and the chirp-z transform. The panel on the left depicts model spectra for large aspect angles α, which are known to be very accurate despite the neglect of Coulomb collisions in the calculations. The small aspect angle results depicted on the right panel have also been computed using the collisionless model, but in computing the ion Gordeyev integral for small k, truncated versions of (57) in τ were utilized to avoid the secondary resonance peaks of the integrand. The accuracy of these small α spectra deteriorates with decreasing α, as Coulomb collision effects start becoming important in the same limit.

10 324 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 Fig. 9. Sample ISR spectra computed for the 50-MHz Jicamarca ISR (reprinted from [15]). The panel on the left depicts large aspect angles of α =30, 60, and 90. The right panel depicts the spectra for small aspect angles α. Note that the spectra of the smallest aspect angles are once again double humped in the right panel when the effective mass of the electrons m e/ sin 2 α is much larger than the ion mass for small α. G. Coulomb Collisions in a Magnetized Ionosphere The inclusion of Coulomb collisions in ISR spectral theories pertinent for the magnetized F-region ionosphere is not straightforward. This level of realism was not available in the early incoherent scatter models. In fact, the early ISR theories predicted ion resonance peaks at small aspect angles for precisely the same reasons that we have seen during our discussions of the collisionless models previously. Jicamarca Radio Observatory was built near the magnetic equator to make overhead measurements of the theoretically predicted ion resonance peaks in the incoherent scatter spectra. The failure to detect the expected peaks at Jicamarca led to the realization that ion ion Coulomb interactions cannot be left out of the ISR spectral models [18]. The first comprehensive spectral model including the ion ion Coulomb collisions was developed by Woodman [19], [20], using a simplified Fokker Planck kinetic equation with velocity-independent collision rates. In retrospect, Woodman s model amounts to assuming Brownian motion dynamics for the charge carriers in a magnetized plasma setting. The model leads to independent and Gaussian distributed displacements Δl and Δp with variances Δl 2 = 2C2 ν 2 (ντ 1+e ντ ) (62) Δp 2 = 2C2 ( cos(2γ)+ντ e ντ ν 2 +Ω 2 cos(ωτ 2γ) ) (63) where γ tan 1 ν/ω (e.g., [21]). These variances can be employed in the ACF model (54) in the usual way. This leads to an ACF [ e jk Δr exp k2 C 2 Ω 2 + ν 2 (cos(2γ)+ντ ] e ντ cos(ωτ 2γ)) (64) in the zero aspect angle limit, a resonance-free result with a finite width so long as the collision frequency ν>0. Plots of ACF (64) for ions and electrons having the variances (62) and (63) are shown in Fig. 10(a) and (b), respectively, assuming zero and nonzero collision rates ν. The accuracy and limitations of the Brownian-motion model for Coulomb collisions in a magnetized plasma are addressed in detail in our companion paper [1]. The paper contrasts the Brownian motion results (ACFs and Gordeyev integrals) with the results obtained from stochastic solutions of a generalized Langevin equation with velocity-dependent friction and diffusion coefficients appropriate for F-region plasmas. Fig. 11 shows some examples of electron Gordeyev integrals obtained with the generalized Langevin method for ISRs with 50- and 500-MHz operation frequencies. IV. DISCUSSION AND CONCLUSION We have presented in this paper a unified treatment and a review of incoherent scatter spectral theories pertinent to ionospheric applications. Our presentation is comprehensive and complete in the sense that we have provided nearly full derivations of a number of different spectral models that have appeared in the literature in a diverse set of publications over several decades. It is our hope that newcomers to the field may get a quick start by reading our presentation (including the appendices) before spending additional time with the original papers dating back from the 1960s when incoherent scatter theories were first developed, in addition to reading our companion paper that includes some of the most recent developments in ISR science. It should be pointed out that incoherent scatter spectral models have been independently derived by a relatively large number of authors using a number of distinct approaches, e.g., Woodman [19], Dougherty and Farley [22], Fejer [23], Salpeter [24], and Hagfors [25], all leading to identical results. This multiplicity of successful approaches utilized in incoherent scatter

11 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 325 Fig. 10. (a) Ion (O + ) and (b) electron ACFs at 50 MHz and aspect angle α =0with and without Coulomb collisions modeled as a Brownian motion process (for T e = T i = 1000 KandB o =25μT). theory work is an interesting fact in its own right, deserving careful study and reflection, and perhaps the underlying reason why we could identify a general framework that seems to work with all known ionospheric variations of thermal equilibrium incoherent scatter theory. We believe that at the heart of the multiplicity of successful approaches is the existence of redundancies in a handful of physical principles commonly used when one approaches the incoherent scatter problem different approaches are distinguished by how these redundancies are handled, ignored, or even taken advantage of. More specifically, it seems possible to construct a fully developed incoherent scatter theory by employing, in addition to Maxwell s equations, any two of the following three sets of elements: 1) generalized Nyquist theorem and Kramers Kronig relations; 2) plasma kinetic equations; 3) single-particle statistics. Fig. 11. Electron Gordeyev integrals (real part) computed for Bragg wavelengths of (a) 3 and (b) 0.3 m using the stochastic solution method described in [1]. For instance, Farley et al. use (1, 2), Fejer and Hagfors use (2, 3) even though they describe their inclusion of collective effects in different ways and our present approach relies on (3, 1). In Farley s use of (1, 2), the emphasis is on the Nyquist theorem and plasma kinetic theory, because once the admittances (normalized conductivities) are obtained from the kinetic equations, the Nyquist theorem directly provides the spectral results. The method is straightforward and elegant, but it requires a fair amount of experience in handling the plasma kinetic equations and performing some nontrivial contour integrations in the complex plane. In our own approach based on (3, 1), because of our work in formulating the relevant particle statistics (3), we were able to bypass an explicit use of the plasma kinetic equations (i.e., 2). Also, it can be added that Woodman s as well as Hagfors avoidance of (1) comes at the

12 326 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 expense of some intermediate steps in their derivations which are effectively unidentified proofs of the fluctuation dissipation theorem for a given plasma setting. It is interesting to note that our approach not only gives us all the relevant aspects of ISR spectral models for thermal equilibrium ionospheres but also provides us with a kinetictheory-free derivation of Landau damping. Collisionless damping of longitudinal modes in a warm plasma, the well-known Landau damping effect, is normally derived in elementary plasma physics books starting with Vlasov s equation. The derivations rely on the use of the Landau contour in selecting the proper expression for the medium conductivity σ that is consistent with causality. The damping of longitudinal modes is then observed from the imaginary part of the characteristic roots ω of 1+σ(k,ω)/(jωɛ o ), representing the denominator of longitudinal wave response (31), after a pole-zero cancellation takes place at ω =0. The causality constraint is built in to our general framework relations (38) and (39) via the use of Kramers Kronig relations, and thus, the use of (38) and (39) to determine σ in terms of the Gordeyev integral of e jk Δr also leads to Landau-damped roots of 1+σ(k,ω)/(jωɛ o ).The damping of longitudinal modes is, in fact, manifest in finite lifetimes of single-particle ACFs e jk Δr found for all cases examined in this paper, except for collisionless and magnetized plasmas in the α 0 limit it is the lack of any damping mechanism at all that causes the problems in the theory in that case (fluctuation dissipation theorem requires some mode of dissipation in the system after all). APPENDIX I HILBERT TRANSFORM AND KRAMERS KRONIG RELATIONS Let h(t) be a real-valued causal and absolute integrable impulse response function of a linear time-invariant system such that h(t) h e (t)+h o (t) H(ω) H e (ω)+h o (ω) =H R (ω)+jh I (ω) (65) where h e (t) and h o (t) sgn(t)h e (t) are even and odd parts of h(t), indicates a Fourier transform pair, and H R (ω) and H I (ω) denote the real and imaginary parts of the Fourier transform H(ω). Since H e (ω) =H R (ω) and H o (ω) =jh I (ω) and h o (t) =sgn(t)h e (t) H o (ω) = 1 2 2π jω H e(ω) (66) where indicates convolution, it follows that H I (ω) = 1 πω H R(ω) = 1 π and, likewise H R (ω) = 1 πω H I(ω) = 1 π H R (ω ) ω ω dω (67) H I (ω ) ω ω dω. (68) Hilbert transformation formulas (67) and (68) are known as Kramers Kronig relations [9], and it can be shown that they are also valid when h(t) is complex valued. APPENDIX II CONFIRMATION OF THE GENERAL FRAMEWORK From (24), applied to any species s in its own reference frame, we have (not showing the subscripts s) n t (k,ω) 2 N o = =2Re dτe jωτ e jk Δr 0 dτe jωτ e jk Δr =2Re{J(ω)} (69) in view of (38) and the fact that e jk Δr is a conjugate symmetric function of τ, which is, in turn, a consequence of the spectrum n t (k,ω) 2 being real valued; this confirms the first expression in (39). Next, from (69) and (37), ω 2 k 2 e2 N o 2J R =2KTσ R σ R = ω2 N o e 2 k 2 KT J R = ɛ oω 2 k 2 h 2 J R, (70) where σ R and J R refer to real parts of conductivity σ = σ(k,ω) and the Gordeyev integral J = J(ω), respectively, and h ɛ o KT/N o e 2 is the Debye length. Applying the Hilbert transform formula (67) to expression (70), we find that σ I = 1 π = 1 π ɛ o k 2 h 2 ɛ o ω 2 k 2 h 2 ω 2 J R ω ω dω = ɛ oω 2 k 2 h 2 J I + 1 ɛ o ω π k 2 h 2 J R ω ω dω + 1 ɛ o π k 2 h 2 J R dω = (ω + ω)j R dω ɛ o [ ω 2 k 2 h 2 J I + ω ] (71) given that, first, J(ω) is the Fourier transform of a causal function, second, J R = J R (ω) is an even function, and third, J Rdω = π this is obtained by evaluating the inverse Fourier transform of both sides of (69) at τ =0 and using e jk Δr 1 as τ 0. Finally, in view of (70) and (71), we find, in confirmation of the conductivity expression given in (39) σ = σ R + jσ I = ɛ oω 2 k 2 h 2 J R + j ɛ o k 2 h 2 [ω2 J I + ω] = jωɛ o k 2 h 2 [1 jω(j R + jj I )] = jωɛ o k 2 [1 jωj]. (72) h2

13 KUDEKI AND MILLA: INCOHERENT SCATTER SPECTRAL THEORIES PART I 327 APPENDIX III CONFIRMATION OF MEAN-SQUARED DISPLACEMENT (47) FOR A BROWNIAN PROCESS The dynamics of particles of mass m undergoing Brownian motion (e.g., [26] and [27]) is described by a first-order linear differential equation m dv + mνv = δf (73) dt for the particle velocity v = dx/dt where δf is a Gaussian white noise with zero mean. Note that (73), known as the Langevin equation, has the form of Newton s second law including a zero-mean random force δf which is taken to be independent of x and v and a friction force proportional to v. Since δf is a Gaussian random variable and the coefficients of (73) are constants, the predicted solution v(t) of (73) and its integral x(t) are also Gaussian random variables. To determine the mean-squared displacement x 2 of a Brownian particle as a function of time t, we multiply (73) by x and take the expected value of the product over an ensemble of x and v = dx/dt realizations to obtain d dt { d dt x2 } { } d + ν dt x2 =2C 2 (74) where C 2 v 2. Integrating (74) twice with the use of initial conditions (d/dt)x 2 = x 2 =0at t =0, we find, in agreement with (47), that x 2 = 2C2 ν 2 (νt 1+e νt ). (75) Brownian motion velocity v = dx/dt is said to be a continuous Gauss Markov process, since v(t) is, first, a Gaussian random variable and, second, according to (73), v(t) is Markovian, which means that past values of v are of no help in predicting its future values if the present value is available. Coulomb interactions of charged particles so-called Coulomb collisions in a plasma are observed to give rise to random walk and diffusion effects which, in certain circumstances, exhibit a general agreement with the Brownian motion process of a suitable friction coefficient ν. More advanced Markovian models of Coulomb collisions based on a Langevin equation with v-dependent coefficients ν and δf are examined in our companion paper [1]. APPENDIX IV CONFIRMATION OF Δp 2 =(4C 2 /Ω 2 )sin 2 (Ωτ/2) In the absence of collisions, or in between collisions, charged particles move in circular orbits in the plane perpendicular to the geomagnetic field B, for example, the xy plane, with a gyrofrequency Ω=±eB/m i,e and an orbit radius of v 2 x + v 2 y ρ = v Ω =. (76) Ω Thus, the location of any particle projected to, for example, the x-axis is given by x o + ρ cos(ωt + φ), where x o and φ are random constants with uniform distributions. Hence, particle displacement along x over a time lag τ is Δx = ρ cos (Ω(t + τ)+φ) ρ cos(ωt + φ) = ρ [cos(ωt +Ωτ) cos(ωt )] = ρ [cos(ωt ) (cos(ωτ) 1) sin(ωt ) sin(ωτ)] Δp. (77) Squaring and taking the expected value over independent variables φ, v x, and v y, we find that [ 1 Δp 2 = ρ 2 2 (cos(ωτ) 1)2 + 1 ] 2 sin2 (Ωτ) [ 1 = ρ 2 ( 1 2 cos(ωτ) + cos 2 (Ωτ) ) + 1 ] 2 2 sin2 (Ωτ) = ρ 2 [1 cos(ωτ)] = 2C2 ( ) Ω 2 Ωτ sin2 2 = 4C2 Ω 2 [1 cos(ωτ)] (78) as claimed, since the expected values cos 2 (Ωt ) =sin 2 (Ωt ) = (1/2) and cos(ωt ) sin(ωt ) =0 over random t, while ρ 2 = (v 2 x + v 2 y)/ω 2 =2C 2 /Ω 2 with v 2 x = v 2 y C 2. ACKNOWLEDGMENT The first author is grateful for having had the good fortune of learning the theory and practice of ISRs firsthand from its pioneers D. Farley, J. Fejer, and R. Woodman, in specific and having had access to the wonderful radio instruments at Jicamarca and ALTAIR. REFERENCES [1] M. A. Milla and E. Kudeki, Incoherent scatter spectral theories Part II: Modeling the spectrum for modes propagating perpendicular to B, IEEE Trans. Geosci. Remote Sens., vol. 49, no. 2, Feb. 2011, to be published. [Online]. Available: [2] M. P. Sulzer and S. A. González, The effect of electron Coulomb collisions on the incoherent scatter spectrum in the F region at Jicamarca, J. Geophys. Res., vol. 104, no. A10, pp , Oct [3] N. N. Rao, Elements of Engineering Electromagnetics. Upper Saddle River, NJ: Prentice-Hall, [4] M. A. Milla and E. Kudeki, F -region electron density and T e/t i measurements using incoherent scatter power data collected at ALTAIR, Ann. Geophys., vol. 24, no. 5, pp , Jul [5] W. E. Gordon, Incoherent scattering of radio waves by free electrons with applications to space exploration by radar, Proc. IRE, vol. 46, no. 11, pp , Nov [6] K. L. Bowles, Observation of vertical-incidence scatter from the ionosphere at 41 Mc/sec, Phys. Rev. Lett., vol. 1, no. 12, pp , Dec [7] H. Nyquist, Thermal agitation of electric charge in conductors, Phys. Rev., vol. 32, no. 1, pp , Jul [8] H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev., vol. 83, no. 1, pp , Jul [9] K. C. Yeh and C. H. Liu, Theory of Ionospheric Waves. New York: Academic, [10] M. Milla and E. Kudeki, Particle dynamics description of BGK collisions as a Poisson process, J. Geophys. Res., vol. 114, no. A7, p. A07 302, Jul [11] J. P. Dougherty and D. T. Farley, A theory of incoherent scattering of radio waves by a plasma 3. Scattering in a partly ionized gas, J. Geophys. Res., vol. 68, no. 19, pp , [12] T. Hagfors and R. A. Brockelman, A theory of collision dominated electron density fluctuations in a plasma with applications to incoherent scattering, Phys. Fluids, vol. 14, no. 6, pp , Jun

14 328 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 1, JANUARY 2011 [13] J. D. Mathews, The incoherent scatter radar as a tool for studying the ionospheric D-region, J. Atmos. Terr. Phys., vol. 46, no. 11, pp , Nov [14] J. L. Chau and E. Kudeki, First E- and D-region incoherent scatter spectra observed over Jicamarca, Ann. Geophys., vol. 24, no. 5, pp , Jul [15] E. Kudeki, S. Bhattacharyya, and R. F. Woodman, A new approach in incoherent scatter F region E B drift measurements at Jicamarca, J. Geophys. Res., vol. 104, no. A12, pp , Dec [16] Y. L. Li, C. H. Liu, and S. J. Franke, Adaptive evaluation of the Sommerfeld-type integral using the chirp z-transform, IEEE Trans. Antennas Propag., vol. 39, no. 12, pp , Dec [17] B. D. Fried and S. D. Compte, The Plasma Dispersion Function. New York: Academic, [18] D. T. Farley, The effect of Coulomb collisions on incoherent scattering of radio waves by a plasma, J. Geophys. Res., vol. 69, no. 1, pp , Jan [19] R. F. Woodman, Incoherent scattering of electromagnetic waves by a plasma, Ph.D. dissertation, Harvard Univ., Cambridge, MA, Mar [20] R. F. Woodman, On a proper electron collision frequency for a Fokker Planck collision model with Jicamarca applications, J. Atmos. Sol.-Terr. Phys., vol. 66, no. 17, pp , Nov [21] E. Kudeki and M. A. Milla, Incoherent scatter spectrum theory for modes propagating perpendicular to the geomagnetic field, J. Geophys. Res., vol. 111, no. A6, pp. A A , Jun [22] J. P. Dougherty and D. T. Farley, A theory of incoherent scattering of radio waves by a plasma, Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 259, no. 1296, pp , Nov [23] J. A. Fejer, Scattering of radio waves by an ionized gas in thermal equilibrium, Can. J. Phys., vol. 38, no. 8, pp , Aug [24] E. E. Salpeter, Electron density fluctuations in a plasma, Phys. Rev., vol. 120, no. 5, pp , Dec [25] T. Hagfors, Density fluctuations in a plasma in a magnetic field, with applications to the ionosphere, J. Geophys. Res., vol.66,no. 6,pp , Jun [26] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys., vol. 15, no. 1, pp. 1 89, Jan [27] D. T. Gillespie, The mathematics of Brownian motion and Johnson noise, Amer. J. Phys., vol. 64, no. 3, pp , Mar Erhan Kudeki (M 05) received the Ph.D. degree from the School of Electrical Engineering, Cornell University, Ithaca, NY, in His dissertation was on the plasma instabilities of the equatorial ionosphere. After spending approximately two years as a Postdoctoral Researcher with Cornell University and with the Jicamarca Radio Observatory near Lima, Peru, he joined the Electrical and Computer Engineering faculty at the University of Illinois at Urbana Champaign, Urbana, in 1985, where he currently teaches courses on circuits and linear systems, electromagnetics, and communication and remote sensing applications of radiowave propagation. His research is mainly focused on radar remote sensing of the upper atmosphere and ionosphere, with special emphasis on equatorial plasma electrodynamics and incoherent scatter radar techniques. He is a coauthor of an undergraduate textbook on circuits and linear systems entitled Analog Signals and Systems (Pearson/Prentice Hall, 2009) published in the Illinois ECE Series. Dr. Kudeki was the recipient of the National Science Foundation CEDAR Prize Lecture in 2006 for his work on incoherent scatter observations at small magnetic aspect angles. Marco A. Milla (M 07) received the B.S. degree in electrical engineering from Pontificia Universidad Católica del Perú, Lima, Peru, in 1997 and the M.S. and Ph.D. degrees from the University of Illinois, Urbana, in 2006 and 2010, respectively. He joined the Jicamarca Radio Observatory as a Signal-Processing Engineer in 1998 and spent about five years working on different research and engineering projects. Since 2004, he has been with the Remote Sensing and Space Sciences Group, Department of Electrical and Computer Engineering, University of Illinois. His doctoral research involves the development of incoherent scatter radar techniques for the estimation of ionospheric state parameters. In particular, he has studied Coulomb collisions and magnetoionic propagation effects on the incoherent scatter spectrum measured with antenna beams pointed perpendicular to the Earth s magnetic field.