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Incoherent Scatter Analysis Techniques EISCAT Radar School 24 August, 2003 John Holt Incoherent Scatter Analysis Techniques I. Introduction II. Data and Model Parameters III. Inverse Theory IV. Incoherent Scatter Spectrum as a Function of Basic Ionospheric Parameters V. Incoherent Scatter Measurements and Analysis VI. Future Trends 1 Introduction I. Data and Model Parameters A. Some History B. Voltage C. Power (Lag Profiles, Spectra, ACFs) D. Basic Derived Parameters E. Inferred Parameters 2 J. V. Evans, TR 274, 1962 J. V. Evans, TR 274, 1962 3 J. V. Evans, TR 274, 1962 J. V. Evans, TR 274, 1962 1. Above 300 km assume Ti is constant with height. The increase in the bandwidth of the signals is caused solely by an increase in Te/Ti. 2. Above 300 km assume Te/Ti is constant with height. The increase in the bandwidth of the signals is caused by an increase in Ti. Option 2 was preferred based on the large diurnal variation in temperature inferred from satellite drag measurements. 4 Perkins and Wand, 1964 I. Data and Model Parameters A. Some History B. Voltage C. Power (Lag Profiles, Spectra, ACFs) D. Basic Derived Parameters E. Inferred Parameters 5 A/D Data I. Data and Model Parameters A. Some History B. Voltage C. Power (Lag Profiles, Spectra, ACFs) D. Basic Derived Parameters E. Inferred Parameters 6 Power Measurements Range Delay Incoherent scatter power spectrum 2 2 4 λ 1 λ 2 ∑ D Fi (ω) N e0 (ω) + 4πD Fe (ω) ∑ Ni0 (ω) 2 2 1+ 2 4π i i e i W (ω) = 2 1 2 1 2 λ 2 1+ Fe (ω) + ∑ Fi (ω) 4π De i Di ω = angular frequency displacement from transmitter frequency λ = radar wavelength De= electron Debye length = (ε0KTe/4πNe2)1/2 Di= ion Debye length = (ε0KTi/4πNi2)1/2 ∞ 16π 2 KT ∞ 16 π2 KT Fe (ω) = 1 − ω∫ exp − 2 e τ 2 sin(ωτ)d τ − jω∫ exp − 2 e τ2 cos(ωτ)d τ 0 λ me 0 λ me ∞ 16π2 KT ∞ 16 π2 KT Fi (ω) = 1 − ω∫ exp − 2 i τ2 sin(ωτ)d τ − jω∫ exp − 2 i τ2 cos(ωτ)d τ 0 λ mi 0 λ mi ∞ 16 π2 KT N e0 (ω) = 2N e ∫ exp − 2 e τ2 cos(ωτ)d τ 2 0 λ me 7 Basic Derived Parameters • Electron density • Electron temperature • Relative ion concentrations (N2+, NO+, N2+, O+, He+, H+) • Ion temperatures (N2+, NO+, N2+, O+, He+, H+) • Ion velocities (N2+, NO+, N2+, O+, He+, H+) Errors in incoherent scatter power measurements The fundamental result is ∆Ps 1 Pn = 1 + Ps Nsamp Ps Where Ps is the error in the signal power, Ps is the signal power, Pn is the noise power and Nsamp is the number of samples. The variances and covariances of the autocorrelation function have the same dependence on the signal to noise ratio and number of samples. Later, we will look at how this dependence translates to the errors in the ionospheric parameters deduced from the ACF. 8 V. Inferred Parameters A. Ion production and loss (ion continuity equation) B. Neutral wind (ion momentum equation) C. Neutral temperature (ion energy equation) D. Electric field (Maxwells equations) II. Inverse Theory A. Forward Problem B. Inverse Problem C. Bayes’ Theorem D. Least Squares 9 Inverse problem theory Humans were naked worms; yet they had an internal model of the world. In the course of time up to the present, this model has been updated many times, following the development of new experimental possibilities (i.e., the development of their senses) or the development of their intellect. Sometimes the updating has been only quantitative, sometimes it has been qualitative. Inverse problem theory tries to describe the rules human beings should use for quantitative updatings. Albert Tarantola Inverse Problem Theory Definition of the Forward and Inverse Problems • Let S represent a physical system (e.g. the ionosphere) • Assume that we are able to define a set of model parameters which completely describe S (e.g. ion composition, temperatures, drifts, density, electron temperature…). • Operationally define some observable parameters whose actual values depend on the values of the observable parameters, given arbitrary values of the model parameters (e.g. I. S. radar spectra or ACFs). Forward problem: predict the values of the observable parameters, given arbitrary values of the model parameters. Inverse Problem: Infer the values of the model parameters from given values of the observable parameters. 10 The Bayesian approach to inverse problems The proper formulation of inverse problems is obtained by using the language of probability calculus, and by using a Bayesian interpretation of probability. Inverse problem theory has to be developed from the consideration of uncertainties (either experimental or in physical laws), and the right (well- posed) question to set is: given a certain amount of a priori information on some model parameters, and given an uncertain physical law relating some observable parameters to the model parameters, in what sense should I modify the a priori information, given the uncertain results of some experiments. Tarantola BUT 11 There is another point of view Courts have consistently held that academic license does not extend to shouting “Bayesian” in a crowded lecture hall. Press et al. Numerical Recipes A narrow definition of inverse theory Consider two positive functionals, A and B. A measures something like the agreement of a model to the data (e.g. χ2) and B measures something like the “smoothness” of the solution. Then the central idea in inverse theory is the prescription: Minimize: A + λB For various values of 0 < λ < ∞ and then settle on a “best” value of λ by one or another criterion. 12 Bayesian solution to inverse problems • Let f(d,m) be a joint probability density on the parameters (d,m) • Define the marginal probability densities f M (m ) = ∫ d d f ( d , m ) f D (d ) = ∫ d m f ( d , m ) M M • Define the conditional probability densities f (d 0, m) f ( d , m 0) f M |D = f D| M = ∫ M dm f (d 0, m) ∫ M dd f ( d , m 0 ) • Then f D| M ( d 0 | m ) f M ( m ) f M | D ( m | d 0) = Bayes Theorem ∫ dm f D|M ( d 0 | m ) f M (m) Monte Carlo Simulation 13 Least squares If the measurements and a priori information have Gaussian uncertainties, then the maximum likelihood point in the probability density representing the posterior information in the model space is the minimum of the objective function: 1 S(m) = (g(m) - dobs )t C-1 (g(m) - dobs ) + (m - m prior ) t C-1 (m - m prior ) 2 D M Where g(m) is the linear model, d is the data vector, m is the model vector, d=g(m), The Gaussian uncertainties in the observed data are described by the covariance operator CD and the Gaussian uncertainties in the a priori information are described by Cm. For uncorrelated errors: 1 nd (g i (m) - d iobs ) 2 n m (m - mprior ) α α 2 S(m ) = ∑ +∑ α 2 2 i=1 (σ D ) i 2 α =1 (σ M ) Solution of least squares problem • Normal Equations - possible roundoff errors – Gauss Jordan elimination - most common – Cholesky decomposition - efficient – LU decomposition - no covariance matrix • QR decomposition - fewer roundoff problems than normal equations – Householder transforms - reflections – Givens transforms - rotation. Good for sequential addition of new data • Singular value decomposition - Essential if fit parameters are not independent 14 Nonlinear least squares • Iterative linear least squares. Start with trial values for parameters and move to successively better solutions • Requires partial derivatives of objective function with respect to fit parameters. May be computed numerically, or use analytic partial derivatives. • If the fit is not linear enough, successive linear least squares solutions may not converge. Levenberg-Marquardt method alleviates this problem by including a term in the steepest descent direction. • Multiple local minima may still be a problem. This is not usually a problem when fitting the incoherent scatter ion line. III. Incoherent Scatter Spectrum as a Function of Basic Ionospheric Parameters A. Forward Problem B. Monte Carlo Simulations C. Inverse Problem 15 III. Incoherent Scatter Spectrum as a Function of Basic Ionospheric Parameters A. Forward Problem B. Monte Carlo Simulations C. Inverse Problem Incoherent scatter spectrum 16 17 18 19 III. Incoherent Scatter Spectrum as a Function of Basic Ionospheric Parameters A. Forward Problem B. Monte Carlo Simulations C. Inverse Problem 20 21 22 23 24 25 IV. Incoherent Scatter Measurements A. Lag Profiles B. Traditional Single-Altitude Analysis C. Full-Profile Analysis D. Other Methods Basic features of monostatic radar signal return • Return includes signals from a volume which extends cT/2 in the radar los direction, where T is the length of the transmitted waveform. • To achieve good height resolution, must use as short a pulse as possible. If the pulse is too long, the measurements are smeared over a range extent larger than the spatial scale of the model parameters. • But pulse must be long enough to provide sufficient lag extent (spectral resolution) to allow model parameters to be extracted with sufficient accuracy. • A further complication arises from the finite impulse response of the receiver, which causes the measured signal to be a time average of the signal returning to the radar. The impulse response can be shortened by increasing the bandwidth of the receiver, but this introduces additional noise into the measurement. • Modulated waveform, e.g. alternating codes, can simultaneously achieve good spatial and frequency resolution, but only at the expense of noise in the form of uncorrelated clutter from unwanted ranges. 26 Lag Profiles • Cross products are computed from a set of N receiver samples z(i) from one IPP • LPM = z(m)⋅z(n) formed from set z(1),z(2),…,z(N) up to maximum lag lmax = m-n • First diagonal = lag 0, second diagonal = lag 1, etc. • Products are accumulated element by element for a specified number of IPPs. • The integrated LPM is analyzed to determine ionospheric parameters – The products are combined to form ACFs or spectra which are analyzed individually • Summation rule • Deconvolution techniques – The entire matrix is analyzed simultaneously to produce profiles of ionospheric parameters vs range Formation of single-pulse ACFs from lag profile matrices Triangular Triangular summation rule. summation rule. Gated samples Best range resolution Trapezoidal Lag products summation rule treated as integrals of plasma lag- profile matrix 27 Another approach – full-profile analysis Another approach is to leave the lag-profile matrix matrix unmodified and use our theoretical knowledge of the incoherent scatter correlation function with lag to represent the lag variation of the lag-profile matrix. This brings our (very good) a priori knowledge about the lag-profile matrix into the analysis as soon as possible. Further, it facilitates including any additional a priori information about the ionospheric parameter profiles to contribute to the analysis. OASIS full-profile analysis • We will discuss the MIT OASIS program here. GUISDAP also has a full-profile capability. • We assume that: – The data are Gaussian random variables – A priori constraints can be expressed as Gaussian random variables – The model is linear in the vicinity of the solution • Then, the problem reduces to minimizing the L2 norm of a vector – least squares. 28 Components of OASIS model 1. Model for height variation of ionospheric parameters 2. Model for the incoherent scatter power spectrum 3. Model for the radar measurement process, which is primarily a function of the transmitted waveform and receiver impulse response 4. A priori constraints on the model parameters 1. Parameter profiles B-splines are a representation of piecewise polynomials pp( x) = p ( x ) i x ∈ [ti , ti+1 ], (i = 1, 2,..., m) Splines are piecewise polynomials which have n-1 continuous derivatives at the knots., e.g. step-functions (first-order splines), broken lines (second- order splines) and cubic-splines (fourth-order splines). 1. Nonzero only on k intervals B k ( x) = 0 j outside x = [t j , t j +k ] 2. Always positive within their range B k ( x) > 0 j inside x = [t j , t j + k ] 3. Well-defined normalization. 29 Collision Frequency Profile In the E-region, where ion-neutral collisions are important we can do better. The collision frequency will have a diffusive equilibrium height variation determined by the masses of the dominant molecular species. In OASIS, we actually take the height variation from the MSIS model and scale that. The collision frequency determination is then reduced to finding a single scalar parameter instead of a collision frequency at each altitude. OASIS ionospheric parameter profile model n Si = ∑ aij B k ( r ) j j =1 where S = N (r ), T (r ),T ( r ), V (r ), P (r ) ij e i e los O a = B-spline coefficients ij k B (r ) j r = range N (r ) = electron density e T (r ) = ion temperature i T (r ) = electron temperature e V = ion line-of-sight drift los P (r ) = %O O + 30 Components of OASIS model 1. Model for height variation of ionospheric parameters 2. Model for the incoherent scatter power spectrum 3. Model for the radar measurement process, which is primarily a function of the transmitted waveform and receiver impulse response 4. A priori constraints on the model parameters 2. Plasma lag-profile matrix • Given parameter profiles, we can calculate the corresponding incoherent scatter ACF at any point within the range of the profiles • Doing this at a specified set of ranges yields a matrix whose rows are ACFs • The ACF matrix is then normalized to include the effects of the single-electron cross section, electron density and range from the radar. • This matrix will be denoted by σ(τ,r) in the following 31 Components of OASIS model 1. Model for height variation of ionospheric parameters 2. Model for the incoherent scatter power spectrum 3. Model for the radar measurement process, which is primarily a function of the transmitted waveform and receiver impulse response 4. A priori constraints on the model parameters Incoherent scatter lag-product data • The expectation value of the lagged product obtained by multiplying complex receiver samples z(t) measured by a monostatic radar at times t and t ′ is: ∞ ∞ z (t ) ( z (t ') ) = A∫ dr ∫ d τ Wt ,t ' ( τ, r ) σ(τ, t ) 0 −∞ • The plasma lag profile matrix σ(t,τ) includes the effect of the plasma correlation function, single-electron scattering cross section, electron density and range from the radar. • The radar ambiguity function Wt,t′(τ,r) includes the effects of transmitter modulation and receiver filtering. • The radar normalization factor A depends on transmitter power, antenna area, system losses, etc. 32 Radar ambiguity function ∞ Wt ,t ' (τ, r ) = ∫ d νWt A (ν, r )Wt 'A (ν − τ, r ), −∞ where Wt A is 2r Wt A (τ, r ) = h(t − τ) P (τ − ) c and where h is the receiver impulse response and P is the complex envelope of the transmitted waveform. The function WtA(τ,r) is known as the amplitude ambiguity function for sampling time t and indicates how the signal sampled at time t is composed of signals scattered from ranges r and received at times t. Discrete form of model prediction for measured lag products Replacing the integrals by an integration rule we get the model prediction for the measured lag products m2 lmax M ij = ∑ ∑w k = m1 l = 0 jkl σ i + k ,l (ri + k ) If we take this to be a two-dimensional Riemann sum, then Wjkl is a matrix representation of the ambiguity function for lag j. The range summation index k is relative to the range index i because the shape of the ambiguity function is not a function of range. The lag summation is over all lags for which the lag products are not zero, that is lags less than the length of the transmitted waveform. 33 Two-dimensional radar ambiguity functions for a four-pulse modulation and boxcar impulse response Two-dimensional radar ambiguity functions for a single-pulse modulation and wideband Butterworth filter Note triangular Weighting of single-pulse| lag products 34 Sample two-dimensional radar ambiguity function for a 100- µs single-pulse modulation Waveform and filter Used to collect data described below Components of OASIS model 1. Model for height variation of ionospheric parameters 2. Model for the incoherent scatter power spectrum 3. Model for the radar measurement process, which is primarily a function of the transmitted waveform and receiver impulse response 4. A priori constraints on the model parameters 35 Constraints and other data • Two types of constraint may be included, the radar calibration constant and a linear combination of splines and their derivative. Basically, a regularization term, ∂ j Pi C= ∑a j =0,3 ij ∂rj But perhaps easier to relate to physics. For example, a20=1, a30=-1, else aij=0 i =1,5 yields Ti=Te • In addition to incoherent scatter lag product measurements, measured values of foF2 can be included OASIS least squares fit ( M ij − mij ) 2 ( Di − di )2 (C − c ) 2 F =∑ +∑ + ∑ i 2i i, j ∆mij 2 i ∆di2 i ∆ci The first term is the weighted squared residual of the lagged products. The second term is the weighted squared residual of any other data. The third term is the weighted squared residual of the a priori constraits. 36 Fit (lines) to smooth simulated data (points) when the electron temperature contains a Gaussian feature narrower than cT/2 300 µs pulse → 45 km resolution 8.33 km wide Gaussian Experimental lag-profile matrix and Oasis fit to the matrix 300 km pulse 37 OASIS power profile derived from measurements made with 640-µs single pulse Deconvolved 640 µs measurement is good match to 100 µs measurement OASIS power profile derived from a simultaneous fit to 100-, 300- and 640-µs measurements 38 Ion and electron temperature profiles derived from fit to 100-, 300- and 640-µs data Ion and electron temperature a posteriori probability densities 39 Power profile derived from 100 and 300-µs observations of the ERIC-2 ionospheric depletion experiment VI. Future Trends A. Adaptive Integration B. Analysis on Demand C. Direct Determination of Inferred Parameters from Measurements D. Incorporation of Measurements Directly into Assimilative Models 40 41 42 43 VI. Future Trends A. Adaptive Integration B. Analysis on Demand C. Direct Determination of Inferred Parameters from Measurements D. Incorporation of Measurements Directly into Assimilative Models 44

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incoherent scatter, J. Geophys, Annales Geophysicae, incoherent scatter radar, J. Atmos, Cornell University, latitude ionosphere, Atmospheric Sciences, Radio Science, Journal of Atmospheric and Terrestrial Physics

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