## 1 Introduction

[2] The Super Dual Auroral Radar Network (SuperDARN) is a chain of HF radars which monitor ionospheric plasma convection in the northern and southern hemispheres by detecting backscatter from ionospheric plasma irregularities [*Greenwald et al.*, 1985; *Chisham et al.*, 2007]. A typical SuperDARN radar has 16 look directions (“beams”) separated by 3.24° in azimuth, with 75–100 range gates along each beam separated by 45 km. The dwell time on any particular beam is typically 3–7 s (integration period) which results in a 1–2 min azimuthal scan. Examples of field of view plots of a single scan are shown in Figure 1. Figure 1a shows signal-to-noise ratio (SNR) (“backscattered power”), Figure 1b shows Doppler velocity, and Figure 1c shows Doppler spectral width. These data are fairly representative in displaying a range of echo types including (1) an extended region of low-velocity ground scatter at greater ranges on the more westward beams, (2) meteor wind scatter at the very near ranges, (3) a high-velocity ionospheric scatter feature on the middle beams, and (4) spotty noise/interference elsewhere but especially on the more northward beams.

[3] The nature of the primary targets detected by SuperDARN radars introduces certain complications. The principal dilemma arises because the radar was designed to detect targets with Doppler velocities of up to 2 km/s out to a range of 4500 km. These conditions impose mutually exclusive requirements on the nominal pulse repetition frequency (PRF). To avoid ambiguities in range, we require a long interpulse period (PRF≤33.3 Hz), while to avoid ambiguities in Doppler velocity, we must have a shirt interpulse period (PRF ≥320 Hz). Some techniques which have been used to solve this problem are complementary codes, alternating codes [*Lehtinen*, 1986], and aperiodic sequences [*Uppala and Sahr*, 1994]. In order to resolve this dilemma, the radars employ multipulse sequences to simultaneously determine the range and Doppler velocity of targets [*Farley*, 1972; *Greenwald et al.*, 1985; *Hanuise et al.*, 1993; *Baker et al.*, 1995; *Barthes et al.*, 1998; *Ponomarenko and Waters*, 2006]. This means that instead of transmitting solitary pulses that are separated by a fixed time determined by the PRF, the radars periodically emit sequences of pulses that are separated unevenly in time by integer multipliers of an “elementary lag time” *τ*_{0}=1.5–2.4 ms. By sampling the returns from a fixed range for each pulse of the sequence using a coherent receiver, all products of the complex autocorrelation function (ACF), *R*_{k}=*V*(*t*)*V*^{∗}(*t*+*k**τ*_{0}), where *V* is the receiver voltage sample and *k* is the lag number, can be calculated from 0 to *n**τ*_{0}, where *n* is the number of lags, with occasional misses at certain lags. An ACF is calculated for each range gate from the returns from each multipulse sequence. Averaging the returns over multiple sequence transmissions partially suppresses the contributions from pulses that encounter other scattering regions at the same sampling times (cross-range interference, CRI, a type of clutter) [*Baker et al.*, 1995]. This averaging occurs within what is called an integration period. An example of a standard SuperDARN multipulse sequence is shown in Figure 2. One observes that with this eight pulse sequence, all but two of the lags can be computed up to a lag of 24.

[4] An integration period is typically 3–7 s in length. The total number of multipulse sequences transmitted during an integration varies between about 15 and 60. The ACFs calculated from all the sequences are then integrated in order to minimize interference and increase gain. The integrated ACFs are fit to model functions in order to resolve Doppler velocity (*v*), spectral width (*w*), and backscatter power (signal-to-noise ratio, SNR) as functions of range. Figure 3a shows an ACF from the Fort Hays West radar taken from the period of Figure 1. The ACF consists of a real part (red curve), *R**e*{*R*}, and an imaginary part (blue curve), *I**m*{*R*}, in quadrature. Note that the real part has a maximum at lag zero, and the imaginary part has a value of zero at lag zero. The Doppler shift imposed on the frequency of the returned signal is manifested as a systematic variation of phase with lag. The phase *φ* at lag *τ* is calculated as *φ*(*τ*)= arctan(*I**m*{*R*(*τ*)}/*R**e*{*R*(*τ*)}). Figure 3b illustrates the variation of phase with lag for the ACF of Figure 3a. The maximum Doppler frequency shift, *f*_{dmax}, that can be resolved is related to the basic lag time, *τ*_{0}, by *f*_{dmax}=1/(2*τ*_{0}). Typically, this value is ≈ 300 Hz, corresponding to a maximum Doppler velocity of ≈ 4000 m/s. The lag power *P* at lag *τ*is calculated as *P*(*τ*)=|*R*(*τ*)|. The SNR is found using the fitted signal level at lag zero, *R*_{0}, of the ACF. The spectral width is obtained as a decay of the amplitude of the ACF with lag, i.e., a decrease in *P*(*τ*) with *τ*. Figure 3c shows the lag powers for the ACF of Figure 3a. A detailed discussion of the physical significance of spectral width in terms of signal composition is given by *Ponomarenko and Waters* [2006]. In order to actually calculate *v*, *w*, and SNR from the radar data, fits must be performed to the lag phases and powers of the ACFs.

[5] FITACF is the name of the traditional routine used to process SuperDARN ACFs. While it has performed reasonably well since the inception of SuperDARN, its performance has rarely been tested, mainly because of the absence of a realistic data simulator accounting for both regular and random components of the backscatter echoes. Some other algorithms have been developed over the last few years that attempt to improve data quality but to compare their performance to that of FITACF objectively; again, one needs to have a controlled set of inputs such as can be provided by a comprehensive data simulator. An appropriate simulator has recently been developed by *Ribeiro et al.* [2013] based on the collective scatter model initially conceived by *Ponomarenko et al.* [2008].

[6] In this paper, we examine three different ways of extracting Doppler velocity, spectral width, and SNR from SuperDARN ACFs. We first analyze the conventional FITACF package, in use for almost 30 years. The second method is FITEX2, which is an iteration on a routine called FITEX, which was developed in order to fit a specific multipulse sequence. Finally, we test so-called LMFIT, which uses the Levenberg-Marquardt algorithm [*Levenberg*, 1944; *Marquardt*, 1963] to fit the complex ACF in a single procedure. The aim of this analysis is to compare the performances of the three routines and determine which is the most reliable at extracting Doppler velocity and spectral width from SuperDARN ACFs.