IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 5, SEPTEMBER 2011 939
Autofocus Correction of Phase DistortionEffects on SHARAD Echoes
Bruce A. Campbell, Member, IEEE, Nathaniel E. Putzig, Lynn M. Carter, and Roger J. Phillips
Abstract—SHARAD is a frequency-modulated (15–25 MHz)radar sounder that probes the upper few kilometers of the Martiancrust and polar layered deposits. At solar zenith angles less thanabout 100◦, the ionosphere of Mars can induce phase distortion insurface and subsurface radar echoes that substantially degradesthe signal-to-noise ratio and vertical resolution of the range-compressed data. We present a range-compression autofocus ap-proach that estimates the phase distortion of SHARAD data alongground-track segments of about 100 km, using a power-law im-age-sharpness metric and an empirically derived scaling betweenthe phase correction and radar frequency. This method is rapid,yields a greatly improved subsurface image, and provides a meansto track regional and temporal changes in the Martian ionosphere.
Index Terms—Mars, radar signal processing, spaceborne radar.
I. INTRODUCTION
SHARAD, the Shallow Radar sounder on the Mars Recon-naissance Orbiter, employs a linear chirp signal, from 15 to
25 MHz, with a total duration of 85.05 μs and a sampling rate of0.0375 μs. The received signal is correlated with a similar chirpwaveform to recover fine time resolution. The resulting rangeresolution is 15 m in vacuum, or about 9 m in materials witha dielectric constant of 3 (ice or loose dust), and close to 5 min dense rock. Focused synthetic-aperture processing is used toincrease the coherent gain of the sounder and narrow the along-track resolution to 300–500 m [1]. Because SHARAD uses adipole antenna only the 135-μs delay window limits the cross-track footprint, and off-nadir regions that have favorably tiltedslopes produce “clutter” echoes. The focused data are typicallypresented as a radargram, a 2-D power image with time delayon the vertical axis and distance along the ground track on thehorizontal axis. Recent studies demonstrate the value of thesedata in probing the polar layered terrain [2], [3], mid-latitudeglacial deposits [4], [5], lava flows and volcanic ash [6], [7],and sedimentary remnants of ancient flooding [8].
The range-compressed SHARAD echoes can be degraded byphase distortion induced by the Mars ionosphere over the band-width of the chirped signal [Fig. 1(a)]. At the frequencies used
Manuscript received December 21, 2010; revised February 15, 2011 andApril 1, 2011; accepted April 2, 2011. Date of publication May 19, 2011; dateof current version August 26, 2011.
B. A. Campbell is with the Center for Earth and Planetary Studies,Smithsonian Institution, Washington, DC 20013-7012 USA (e-mail:[emailprotected]).
N. E. Putzig and R. J. Phillips are with the Southwest Research Insti-tute, Boulder, CO 80302 USA (e-mail: [emailprotected]; [emailprotected]).
L. M. Carter is with the NASA Goddard Space Flight Center, Greenbelt, MD20706 USA (e-mail: [emailprotected]).
Digital Object Identifier 10.1109/LGRS.2011.2143692
Fig. 1. Portion of SHARAD observation 846602 over the north polar layereddeposits. Top panel shows signals range compressed with an ideal chirpfunction prior to focused synthetic-aperture processing. Lower panel shows theresults with inclusion of autofocus-derived ionospheric compensation. Rangeof SZA (from left to right) is 58◦–71◦.
by the Mars Advanced Radar for Subsurface and IonosphereSounding (MARSIS) instrument on Mars Express (1–5.5 MHz,which is close to the plasma frequency), the ionosphere alsoinduces a substantial shift in the round-trip delay of signals,but this effect is modest at SHARAD frequencies. Severalmethods have been proposed to compensate for ionosphericeffects on the received echoes, such as the phase-gradientautofocus technique [9] and a parameterized model for phasevariations with peak electron content and neutral-atmospherescale height [10], [11]. We present an alternative autofocusmethod, based on an empirically derived representation of thephase-correction function for the SHARAD chirp, with a single,variable parameter chosen to optimize an image-quality metricfor blocks of range-compressed radar echo data. The method israpid, yields a greatly improved surface and subsurface image,and provides a means to track relative changes in electron-density-induced phase properties of the Martian ionospherewith time and location.
II. MODELING THE IONOSPHERIC PHASE DISTORTION
The phase offset on each frequency component f of a radarchirp, to third order in the electron density distribution N withaltitude z, is [12], [13]:
Δφ(f) =−2π
c
[8.982
f
∫N(z)dz +
8.984
3f3
∫N2(z)dz
+8.986
8f5
∫N3(z)dz
](1)
where c is the speed of light in vacuum. The instantaneousfrequency over the duration of the chirp is given by:
f(t) = fH − f ′St (2)
1545-598X/$26.00 © 2011 IEEE
940 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 5, SEPTEMBER 2011
where t is time, fH is the maximum chirp frequency, and f ′S is
the rate of change of frequency. For SHARAD, the latter twoterms are 25 MHz and about 1.18× 1011 Hz/s, respectively.The image-restoring function, Φ(t), is the integrated phaseoffset as a function of time over the chirp duration. In theSHARAD frequency range, the second and third integral termsof (1) contribute well below 10% to the magnitude of the phaseoffset [12], but may impact higher-order terms (in t) of theintegrated phase.
The key aspect of a robust compensation algorithm for iono-spheric phase effects is flexibility in the relative strengths of, ata minimum, the quadratic and cubic components of Φ(t). Theconstant and linear components play no part in the sharpnessof the radar pulse compression. If we use only the first integralover N(z) in (1), we obtain an expression linearly related to thetotal electron content (TEC):
Δφ(f) ≈ −2π
c
[8.982
fTEC
]= 1.69× 10−6TEC
f. (3)
Substituting (2) for the frequency term in (3), integratingover time, expanding the resulting expression to third order, anddropping the linear term, yields:
Φ(t) ≈ 2.87× 10−18TEC
(f ′S
fH
)2 [t2 − 2
3
f ′S
fHt3]. (4)
This derivation shows that the integrated phase has a fixedrelationship between the coefficients on the quadratic andcubic terms, which may not provide adequate flexibility inmodeling ionospheric distortions. More range in this relativescaling could be obtained with various non-uniform verticaldistributions of electron content in (1), but only through theintroduction of additional parameters.
In order to provide that greater flexibility in a readily opti-mized format, we introduce an integrated phase function withscalar coefficient E and a power-law variation with frequency:
Φ(f) =E
f b(5)
where b is a parameter to be estimated. Substituting (2) for thefrequency and using a binomial series expansion (dropping theconstant and linear terms) yields:
Φ(t) =E(b2 + b)
2f bH
(f ′S
fH
)2 [t2 − (b+ 2)
3
f ′S
fHt3]. (6)
Note that the choice of b defines the relationship between co-efficients on the quadratic and cubic phase error terms. For b >0, this expression will have a more significant cubic componentthan present in (4). A simple proportionality between E andTEC cannot be validated, but we suggest that their correlationis of low order based on the narrow range of b demonstratedbelow and the general behaviors noted in Section III.
We optimize the E and b parameters by reference to animage-quality metric applied to the range-compressed dataprior to Doppler processing, refining the echo-amplitude met-rics proposed in [12], [14], [15]. Sharpness metrics are oftenused [e.g., [16]] to gauge the success of radar image restorationtechniques, where the quality factor is defined as the sum
over a sample region of the image pixel magnitudes (in ourcase, the echo power after range compression) raised to somepower k. Fienup and Miller [17] show that optimizing themetric for values of k < 2 gives greater weight to reducing theimage intensity in dark areas, while increasing values of k > 2give progressively greater weight to maximizing the brightestpoints in a scene. The two parameters dictating the stabilityof this method for SHARAD application are the along-tracksample size (in number of pulse records) and the power-lawexponent of the image-quality metric. For the SHARAD data,we wish to optimize the first surface return and the brightestsubsurface reflections, so a relatively high value of k is expectedto yield good results.
We employ a gradient search algorithm to maximize thesharpness metric for a particular block of echo data. To avoid in-ducing delay shifts due to linear phase variations with time, wefit the time-series of integrated phase terms with a polynomialrepresentation and subtract the constant and linear components.Tests for hundreds of SHARAD observations show that stableresults are obtained using an along-track sample of 6144 echorecords at the typical 4-fold onboard pre-summing, correspond-ing to about 35 s of data and about 105 km of ground-trackdistance. The solar zenith angle (SZA) changes by roughly twodegrees over each such window. The optimum exponent for thesharpness metric appears to be k = 5.
In principle, b could vary with time and location on Mars, butour tests using data collected over a period of two years showonly modest fluctuations in the optimum exponent value, withb = 1.93± 0.04. We thus hold the exponent fixed at this meanvalue, and optimize the focus solution for the value of E. Theradargram in Fig. 1(b) shows the significant improvement [fromFig. 1(a)] obtained for a typical dayside SHARAD observationof the north polar layered deposits.
III. MAPPING CHANGES IN IONOSPHERIC PROPERTIES
The total electron content of the Martian ionosphere varieswith the SZA, the Mars-Sun distance, and with the 26-day rota-tion period of the Sun [18]. The dependence of our initial valuefor E on SZA, prior to any other corrections, is shown in Fig. 2.This functional dependence on SZA is qualitatively consistentwith predictions of TEC variation from Chapman theory [13].Even without correcting for solar distance and rotation, valuesof E for SZA < 90◦ span a range considerably less than a factorof two about the fifth-order polynomial behavior shown by thesolid curve. At higher SZA, the values for E are more widelydistributed about the trend (by up to about a factor of 3–4).
By normalizing E to the polynomial representation of theaverage behavior with SZA, we may examine the effect ofMars-Sun distance on the phase distortion (Fig. 3). As expected,there is a correlation between smaller Mars-Sun distance andenhanced daytime ionospheric phase effects. The variation inTEC with the solar rotation cycle is on the order of ±10%[20], but the irregular sampling in time and SZA provided bySHARAD tracks makes it difficult to distinguish any similar,relatively rapid fluctuations in E. While we can thus normalizethe effect of Mars-Sun distance on E, the level of uncertaintydue to the 26-day solar cycle sets a lower confidence bound
CAMPBELL et al.: AUTOFOCUS CORRECTION OF PHASE DISTORTION EFFECTS ON SHARAD ECHOES 941
Fig. 2. Logarithm of average phase-compensation factor E versus solar zenithangle based on about 24 000 independent, 105-km-long segments of SHARADground tracks. Solid curve corresponds to fifth-order polynomial fit.
Fig. 3. Deviations from the polynomial trend shown in Fig. 2 of phase-compensation factor E, for values of the solar zenith angle less than 75◦, versussun-Mars distance (in units of 106 km). Solid line is least squares fit showingthat required phase focusing correction decreases with increasing Mars-Sundistance (i.e., with lower TEC in the ionosphere).
for identifying ionospheric anomalies such as those due toremanent magnetic fields in the Martian crust [19], [21], [22].
Our coverage of the dayside of Mars shows very modestdeviations in E from the background trend (rms value of about10%, consistent with the uncertainties over the solar cycle), andin particular no evident correlation with the strong magneticanomalies in the southern highlands [21], [22]. The absenceof signatures similar to the dayside ionosphere “bulges” pos-tulated in [19] suggests that such changes affect primarily thevertical distribution of electrons, with minimal impact on theirintegrated density as sensed by SHARAD.
Because the electron density is smaller on the nightside ofMars, its modulation by remanent magnetic fields is expected tobe greater. The transition from dayside to nightside conditionsin the ionosphere occurs between solar zenith angles of 90◦
and 105◦, with higher altitudes sunlit to larger angles [18]. Inorder to avoid this transition region, the near-global nightsidemapping of TEC from MARSIS data was constrained to SZA >100◦ [12]. For SHARAD echoes, the distorting effects of the
TABLE ISTANDARD DEVIATION OF RATIO BETWEEN DERIVED VALUE OF E AND
THE NET CORRECTIONS FOR DEPENDENCE ON SZA AND SUN-MARS
DISTANCE. EACH DISTRIBUTION IS BASED ON AT LEAST 1200INDEPENDENT SHARAD OBSERVATIONS OF 100-km SEGMENTS
OF GROUND TRACK
ionosphere become negligible beyond about SZA = 100◦ so wehave a view only of the day-night transitional regime, and thenear-circular, fixed-local-time orbit of MRO limits geographiccoverage at SZA = 90◦–100◦ to regions at latitudes greater thanabout 60◦.
Deviations from the best-fit trend (Fig. 2) do increase intothe nightside (Table I), which we attribute to a combinationof effects. First, values of E less than about 2× 1014 haveno impact on the range-compression performance; i.e., theycontribute a cumulative phase difference of less than aboutπ/2 over the duration of the chirp. This basic uncertaintyfactor is increasingly significant to the solutions at higher SZAwhere E is smaller. Second, small changes in E, again moresignificant at higher SZA, are influenced by the nature/mixingof terrain within an autofocus footprint and appear as spuriousspatial variations. Finally, there is likely some dependence ofthe electron content on crustal magnetic fields that modulate therelease of charged particles during the night [12], [23]. Futurework with an increased number of autofocused SHARADtracks, and their integration with MARSIS results, may providegreater confidence in mapping local electron-content fluctua-tions caused by remanent crustal magnetic fields.
IV. CONCLUSION
We present a simplified representation of the phase-distortioncorrection function for SHARAD data, and implement it in anautofocus optimization approach that provides a scaling valuerelated to the electron content of the Martian ionosphere. Theseestimated values are qualitatively consistent with earlier modelsfor the dependence of TEC on solar zenith angle and Mars-Sun distance. Compensating for these two sources of variabilityleaves only about 10% residual fluctuations in the daysidebehavior at any chosen SZA, well below the factor-of-twovariations observed for the nightside ionosphere in our resultsand in those from MARSIS data [12]. SHARAD coverage atuseful solar zenith angles is as yet relatively sparse, but asadditional data are acquired, it may be possible to apply thistechnique to study regions of anomalous ionospheric propertiesassociated with crustal remanent magnetic fields.
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