### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Incoherent Scatter Radar Equation and Ambiguity Functions
- 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
- 4. Numerical Results
- 5. Summary and Conclusion
- Acknowledgments
- References

[1] We present a computationally efficient, near-optimal approach to the estimation of ionospheric parameters from incoherent scatter radar measurements. The method consists of removing the range smearing of ionospheric autocorrelation function via a set of 1-D deconvolutions and performing nonlinear least squares fitting on the deconvolved autocorrelation functions. To stabilize the solution in the presence of noise, we incorporate regularization techniques. The computational cost is reduced significantly by estimating the ionospheric parameters at individual altitudes, in comparison to full-profile-type analysis, which attempts to estimate ionospheric parameters at all altitudes simultaneously. The performance of the new technique is evaluated in a numerical example and is shown to give estimates of almost equal quality as the full-profile technique but at a 95% reduction in computation.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Incoherent Scatter Radar Equation and Ambiguity Functions
- 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
- 4. Numerical Results
- 5. Summary and Conclusion
- Acknowledgments
- References

[2] Incoherent scatter radar (ISR) is the most comprehensive ground-based technique for studying the Earth's ionosphere. The incoherent scatter echo is the result of the scattering of electromagnetic energy, radiated from the radar, by electron density fluctuations in the ionospheric plasma, which in the most common case are influenced by much slower, massive positive ions. The frequency spectrum of the received signal provides information about electron and ion temperatures, ion composition and velocity. The analytical relationship between the spectrum and these parameters has been well established in the literature [e.g., *Dougherty and Farely*, 1960, 1961; *Farley*, 1966; *Hagfors*, 1961, 1971]. With the estimation of these parameters from incoherent scatter (IS) measurements, one can deduce many further ionospheric parameters such as electric field strength, conductivity and current, neutral air temperature, and wind speed.

[3] Although the exact forward theory of incoherent scatter was established more than four decades ago, inversion, i.e., the extraction of parameters from incoherent scatter spectra, has remained an open problem primarily because of two major factors. The first complication stems from the fact that variation of different plasma parameters may give rise to similar changes in the IS spectrum [*Vallinkoski*, 1988]. For example, the distinction between changes of the spectra due to ion composition or temperature ratio is very difficult. The same is true for ion mass and ion temperature for fixed temperature ratio. The second factor is the range smearing of information from one altitude over a number of altitudes, which is due to the length of transmitted waveform. An example is shown in Figure 1 where different lags of the autocorrelation function (ACF) are plotted prior to and after imposing the range ambiguity for an uncoded modulation of length 280 *μ*s with a 430 MHz radar (thin and thick curves, respectively). The estimation of ionospheric parameters from the range-smeared ACF would result in a greater ion temperature than electron temperature, which is not physical.

[4] Traditionally, incoherent scatter analysis has been based on performing nonlinear least squares fitting at individual altitudes (height-by-height or gated analysis) [*Lehtinen et al.*, 1996], where in the most common case the spacing between the altitudes is determined by the pulse length. The important assumption in this technique is that plasma parameters are considered constant in all regions which have influence on an individual set of spectral or ACF measurements. The effects of the transmitted waveform are then accounted for in the form of various corrections to ACF lag estimates. This method, although simple and fast, is based on an unrealistic assumption (constant parameter profiles for each range-gate) and suffers from coarse resolution of the parameters, as well as a bias which is introduced into the profiles [*Holt et al.*, 1992; *Lehtinen et al.*, 1996].

[5] Another method is the full-profile analysis [*Holt et al.*, 1992; *Lehtinen et al.*, 1996], which attempts to determine the complete altitude profiles of parameters simultaneously. The basic estimation procedure is to first calculate theoretical spectra on the basis of an initial parameter grid (followed by compensating for the smearing of information across the range, if applicable). Subsequently, the difference between the measured and predicted ACFs is used to update the plasma parameter values iteratively. The technique is optimal in the sense that all information including the range smearing effect as well as the covariance between the lag-estimate errors, can be incorporated into the analysis. The main limitation of the method, on the other hand, is its significant computational cost. Attempts have been made to reduce the cost by exploiting interpolation techniques from a coarse parameter grid to a fine range grid [*Holt et al.*, 1992; *Lehtinen et al.*, 1996]. The accuracy of the method, however, is highly dependent on the resolution of the parameter grid. A very fine spatial resolution, which is required to approximate fine details of the ambiguity function or to capture sharp gradients of parameter profiles, increases the number of parameters of the optimization search space and thus makes the technique more computationally expensive. One basic reason is due to the requirements of any nonlinear optimization technique, one of which is the computation of the derivative of the minimization function with respect to the search variables (parameters in the coarse grid). Even though analytical expressions of the derivatives of the lags of the theoretical autocorrelation function with respect to the ionospheric parameters are available, they cannot be exploited in the optimization procedures of the full-profile techniques. Therefore, forward difference, which slows down the speed of computation, is the only method that can be used for derivative calculation.

[6] In this paper, we develop the theory of a new hybrid inversion technique which aims at obtaining estimates that are close to optimal at a fraction of computational cost. The technique is based on a correction to the effect of the transmitted waveform on the ACF lag profiles through a deconvolution process, and subsequent estimation of parameters from the plasma ACF at individual altitudes.

[7] Note that some coding schemes such as alternating codes [*Lehtinen and Haggstrom*, 1987] can be utilized to eliminate the range smearing from the non-zero lags of ACF measurements analytically (provided that the medium coherence is maintained over code transmission cycles). These modulation techniques, however, increase the noise in the form of uncorrelated clutter from unwanted ranges and as such are not suitable for high-gain radars such as Arecibo radar. In such cases, one can use the coding technique developed by *Sulzer* [1986] which essentially divides the radar power into several independent uncoded long-pulse transmissions at different frequencies. In this paper we focus on such situations and investigate the performance of the proposed approach on simulated data using the long-pulse modulation. The performance comparison of different analysis techniques on ISR experiments is the subject of a future publication.

[8] The paper is organized as follows: section 2 describes background information on the incoherent scatter radar equation as well as the concepts of range smearing and ambiguity. In section 3, we present a detailed description of our proposed method and its justification using the radar equation. Section 4 investigates the performance of the new method on simulated data and compares the corresponding estimation results with those of common methods currently used. The last section provides a summary and conclusion.

### 2. Incoherent Scatter Radar Equation and Ambiguity Functions

- Top of page
- Abstract
- 1. Introduction
- 2. Incoherent Scatter Radar Equation and Ambiguity Functions
- 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
- 4. Numerical Results
- 5. Summary and Conclusion
- Acknowledgments
- References

[9] In any plasma correlation measurement the received voltage from an altitude in the ionosphere contains information from not only that particular altitude but also a number of altitudes below. According to the incoherent scatter radar equation, the transmitted waveform plays an important role in how this information is combined over the range. In what follows, the radar equation is derived and ambiguity functions are described.

[11] The received scattered signal from the electron density fluctuations in the ionosphere can be represented by [*Kudeki*, 2003]:

where *n*(**r**, *t*) represents the fluctuating component of the plasma electron density at range **r** = *r* along the direction (the proportionality factor depends on the classic electron radius, and antenna-related factors such as radiation resistance of the antenna, antenna gain along the direction of transmission, transmitted power, etc.). Note that the received signal spectrum is centered at the carrier frequency, *ω*_{0}, instead of zero, since the antenna output is modulated by the carrier signal. At the detector, the received signal is converted to the baseband. The exponential term in (1) represents this conversion.

[12] Since there are many scattering electrons in the ionosphere, the scattered signals (or received voltages) can be treated as Gaussian random variables to a very good approximation, according to Central Limit Theorem [*Papoulis*, 1986]. Therefore, instead of working directly with the received voltage values, we model their joint statistics in the form of the autocorrelation function (ACF), 〈*v*(*t*)〉, where *τ* represents the time lag, and 〈.〉 and denote the expected value and conjugation operations, respectively.

[13] From (1), the ACF of the received voltage can be expressed as:

where **r**′ = *r*′′ denotes the range *r*′ along the direction ′. Within the integrand above 〈*n*(**r**, *t*)〉 is the space-time ACF of the density fluctuation *n*(**r**, *t*). With the assumption that the fluctuation ACF vanishes rapidly with an increasing magnitude of **x** ≡ **r**′ − **r**, we can proceed as:

[14] The inner integral represents a spatial Fourier transform, as follows:

where *R*(**k**, *r*, *τ*) denotes the ACF of electron density fluctuations at altitude *r*, time lag *τ* and the wave vector **k**(=−2**k**_{0}) and is directly related to ionospheric parameters [e.g., *Farley*, 1966]. Inserting (4) into (3) yields:

[15] Equation (5), which demonstrates the relationship between the received voltage and the target in a statistical sense, is called the incoherent scatter (or soft target) radar equation. Inspection of this equation brings us to the two following observations:

[16] 1. Incoherent scatter radar equation holds for each lag of ionospheric ACF independently from other time lags. Therefore, to evaluate the plasma ACF at time lag *τ*, we need to consider only the same lag of the received signal ACF.

[17] 2. Using an incoherent scatter radar, it is not possible to achieve point estimations of the plasma ACF. Instead, weighted averages of this quantity over a finite range interval are obtained. The weights are merely dependent on the modulated waveform and vary from lag to lag.

[18] The functions describing the averaging operation on the underlying plasma ACF are called soft target radar ambiguity functions, (**p**_{τ}(*t*) = *V*_{0}(*t*)). These functions essentially indicate that the signal coming from range *r* contains information from several altitudes, where the altitude interval is equal to the distance covered by the product of the waveform and its shifted version. Note that the 0th lag contains the most range smearing of information. As we move to farther lags, this altitude interval decays as the common part between the pulse and its shifted version diminishes.

[19] Note that in a more general case, the ambiguity functions are dependent on both range and time lag, as developed in [*Lehtinen and Huuskonen*, 1996]. This dependence on time lag is caused by a non-ideal receiver, whose impulse response contains the time average of previous samples. In this work, however, we assume the receiver has a sufficiently narrow impulse response, and as such the ambiguity in the lag direction is negligible.

[20] Also note that our definition of the soft target ambiguity function differs from that of hard target radar applications where it is defined as [see for example *Blahut*, 2004]:

In this equation *χ*(*τ*, *ν*) is the hard target ambiguity function in terms of a delay (range) resolution variable, *τ*, and a Doppler resolution variable, *ν*. The hard target radar equation is used as a measure of detectability of two hard targets with separation in range and velocity. The difference between the two definitions is due to different filters used in radar receivers. In a hard target application normally a matched filter is used, whereas in soft target radars the received signal is usually over-sampled. Through the rest of this paper we use the term “ambiguity function” to refer to soft target radar applications only.

### 4. Numerical Results

- Top of page
- Abstract
- 1. Introduction
- 2. Incoherent Scatter Radar Equation and Ambiguity Functions
- 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
- 4. Numerical Results
- 5. Summary and Conclusion
- Acknowledgments
- References

[50] In this section we investigate the performance of the hybrid technique on a numerical experiment and compare the corresponding estimation results with those of common methods currently used in terms of mean-squared error and computation time.

[51] In data simulation, we take into account the fluctuations of both the incoherent scatter signal and the receiver noise by simulating the received signals. The data simulation algorithm is summarized as follows:

[52] 1. Generate the altitude profiles of the ionospheric parameters. For simplicity, we restrict the ions in the ionosphere to oxygen only, which is a realistic assumption in the lower *F* region and around the height of peak electron density. Figure 3 shows an example of three parameter profiles (profiles of electron and ion temperatures and electron density) used in the simulations.

[53] 2. Generate noisy voltage samples on the basis of the parameters at each altitude and the transmitted pulse shape, which itself requires the following steps:

[54] a. Form the incoherent scatter spectrum at individual altitudes on the basis of parameter profiles.

[55] b. Generate two sets of random spectra based on of the square root of the spectrum at each altitude. With this configuration, the generated signal will have stronger variations whenever the spectrum itself is stronger (consideration of signal fluctuation).

[56] c. Form the complex spectra.

[57] d. Transform into the time domain.

[58] e. Multiply the complex ACF by the pulse shape.

[59] 3. Add voltage samples from different heights.

[60] 4. Add random noise to the voltage samples generated in (3) (This random noise represents sky and receiver noise and is independent from the signal).

[61] 5. Form the lag-profiles.

[62] 6. Accumulate over many pulse transmissions.

[63] Note that a radar frequency of 430 MHz and an uncoded long pulse of length 280 *μ*s are used in simulations. The lag spacing is set to 10 *μ*s, and the data is accumulated over 12000 pulse transmissions, which yields an error of about 4% in the lags of the measured ACF.

[64] In the implementation of the hybrid technique, we apply Tikhonov regularization with the first-order gradient operator (refer to (21)) on the lag profiles in the first step. The deconvolved lag profiles are thus obtained by solving the set of equations denoted in (22) for each time lag (with integer multiple of lag spacing (Δ*τ*)) individually. From various approaches that exist to solve the equations in (22), we use conjugate gradient method, which is a fast converging iterative method [*Golub and Van Loan*, 1989]. In the second step of the method, the Levenberg-Marquardt algorithm [*Levenberg*, 1944; *Marquardt*, 1963] is used as the NLLS optimization procedure.

[65] The proper value of the regularization parameter, *λ*, is achieved by means of L-curve shown in Figure 4 along with the proper regularization parameter (*λ* ≃ 25.6), identified by the dashed lines. Figure 5 illustrates the estimated parameter profiles for this regularization parameter. While using smaller values of this parameter results in a much larger variation in the estimated parameters, using larger values introduces biases into the estimation results (see Figure 6). The ionospheric parameter estimation errors are also plotted in Figures 7a–7c as a function of regularization parameters, where the errors are measured as with *a*_{ij} and _{ij} denoting the *i*th ideal and estimated ionospheric parameter at altitude *j*, respectively, and *n* the number of altitudes. According to this figure, there is a range of regularization parameters over which the parameter estimation errors are minimum (or within 10% of the minimum) and the proper regularization parameters selected by L-curve fits well in this range. Figure 7d also illustrates the computation time required to complete the analysis on a 2.4 GHz workstation.

[66] The height-by-height estimation results, which are obtained by the application of triangular weighting prior to LS optimization, are illustrated in Figure 8. Note that the fitting is performed only at parameter grids specified by squares, and the estimated parameter profiles (dotted curves) are produced by linear interpolations on these grids. The coarse resolution achieved by this method is evident from the plots.

[67] The optimal parameter estimation results of the full-profile technique is illustrated in Figure 9. The resolution of the parameter grid in this optimal case is 26 km. The optimality is determined on the basis of the performance of the technique over a wide range of parameter grid resolutions. The plots in Figures 10a–10d illustrate the errors in estimated parameters as well as the timing required to complete the analysis (on a 2.4 GHz workstation) for parameter grids with spatial resolutions from 10 to 60 km (corresponding to 30- to 6-point parameter grids). These errors are minimum (or very close to minimum) over a limited range of spatial grid resolutions. This range of optimal grid densities varies slightly for different ionospheric parameters. However, in general we can consider the interval between 21 and 29 km to be within 10% from the optimal. For comparison, we also plot the estimated temperatures for the two spatial resolutions of 57 km and 12 km in Figure 11. The high variation of the estimated parameters from the true parameters in both cases, because of underfitting or overfitting of the solution to the noisy data, is evident from the plots.

[68] From Figures 7a–7c, 8 and 10a–10c, it is clear that the hybrid technique is able to provide faithful estimates of the ionospheric parameters that are nearly as good as the ones given by the full-profile technique (provided that the optimal regularization parameter and grid density are chosen) while it outperforms the height-by-height method in terms of estimation accuracy. Two main differences between the performances of the two methods, however, can be inferred from the plots in Figures 7d and 10d. First, the hybrid method requires nearly constant computational time for various regularization parameters, unlike the full-profile technique where the analysis time is linearly dependent on the parameter grid density. Second, the hybrid technique reduces the computation time to nearly 95%. The restriction of the optimization search space to the number of ionospheric parameters at each range and hence the ability to utilize analytical derivatives of the ACF lags with respect to the physical parameters serves as the main key to the efficiency of the method.

[69] Note that in our simulation setup, the errors in lag estimates become highly correlated. However, we include only the variances of the errors (diagonal elements of the error covariance matrix) in the above analysis as the computation of the full covariance matrix is not tractable, especially as the number of altitudes increases. The estimation errors noted in Figures 7a–7c and 10a–10c have been obtained numerically using Monte-Carlo-type simulations. Therefore no underestimation of errors occurs as is the case when errors are obtained analytically using the diagonal error covariance matrix [*Huuskonen and Lehtinen*, 1996]. Nevertheless, the high correlations in the parameter profiles estimated by the hybrid method are in part consequences of ignoring these error covariances.