Focused Synthetic Aperture Radar Processing of Ice ... - IEEE Xplore

4496

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 8, AUGUST 2015

Focused Synthetic Aperture Radar Processing of Ice-Sounding Data Collected Over the East Antarctic Ice Sheet via the Modified Range Migration Algorithm Using Curvelets Shinan Lang, Xiaojun Liu, Bo Zhao, Xiuwei Chen, and Guangyou Fang

Abstract—In this paper, we propose a new algorithm to address the speckle noise problem in imaging of ice sheets. It is a waveequation-based ice-sounding imaging method using curvelets as building blocks of ice-sounding data, which successfully images the topography of ice sheets. First, theory analysis has been carried on to the proposed algorithm. Then, we give the specific steps to implement this algorithm. Finally, we apply this algorithm to the simulation point targets and High-Resolution Ice-Sounding Radar data to prove its validity of imaging of ice sheets. Furthermore, compared with five previous methods in two major aspects—the power of clutter reduction and the equivalent number of looks— the proposed algorithm reduces the speckle noise during the imaging processing without degrading the ability in clutter reduction. Index Terms—Curvelets as building blocks of ice-sounding data, East Antarctic Ice Sheet (EAIS), High-Resolution Ice-Sounding Radar (HRISR), imaging of ice sheets, modified range migration algorithm using curvelets, speckle noise.

I. I NTRODUCTION

T

HE Antarctic ice sheet is the largest continental ice on Earth, and its mass budget and stability have an important influence on global climate change and sea level rise [1]. To estimate the current ice mass balance and to predict future changes in the motion of the Antarctic ice sheets, we need to know the ice sheet thickness and the physical conditions of the ice sheet surface and bed. We require this information at fine resolution and over extensive portions of the ice sheets [2]. Radar sounding is an established technique for probing ice masses [3] and remotely sensing the basal conditions with sufficient resolution over a large area in a comparatively short

Manuscript received May 26, 2014; revised July 23, 2014, October 9, 2014, and December 12, 2014; accepted February 2, 2015. This work was supported in part by the National High Technology Research and Development Program of China (863 Program) under Grant 2011AA040202 and in part by the National Natural Science Foundation of China under Grant 41306203 and Grant 41101071. (Corresponding author: S. Lang.) S. Lang is with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China, and also with the University of Chinese Academy of Sciences, Beijing 100049, China (e-mail: [email protected]). X. Liu, B. Zhao, X. Chen, and G. Fang are with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2400473

period [4]. The Key Laboratory of Electromagnetic Radiation and Sensing Technology of the Institute of Electronics, Chinese Academy of Sciences (IECAS) drove its newly developed 150-MHz High-Resolution Ice-Sounding Radar (HRISR) over the East Antarctic Ice Sheet (EAIS) from Zhongshan Station to Kunlun Station and collected approximately 1848 km of sounding data during the 29th Chinese Antarctic Research Expedition (CHINARE 29). The HRISR system employed both normal coherent and incoherent integration techniques to improve the signal-to-noise ratio for obtaining good estimates of the ice thickness over much of the ice sheet. However, improved radar performance beyond that obtainable with integration is required to obtain ice-thickness measurements in a few areas [5]. There are three main ways of obtaining this improved performance: 1) increased transmitter power; 2) increased receiver sensitivity; and 3) additional focused synthetic aperture radar (SAR) processing [5]. We chose the third option: additional focused SAR processing because of the clutter reduction. Focused SAR imaging techniques have been applied to icesounding data to improve the gain and resolution for many years. The earliest application was a matched filter approach that correlated reference point target echo responses with radar data from the Greenland ice sheet [5], [6]. This method has been recently applied to data from the West Antarctic Ice Sheet [7]. The f –k migration algorithm, which is based on reverse propagation of the radar echoes back to their sources in the ice, has been also applied to data both from Greenland and Antarctica (see [4], [8], and [9]). All the focused SAR imaging techniques aforementioned cannot reduce the speckle noise caused by the coherent sum of the scatters, which is an inherent and characteristic feature of radar images unless they applied multilook processing. However, the multilook processing makes the decline of resolution of imaging [10]. A number of more advanced speckle reduction algorithms are available, such as the Lee and Frost filters, the wavelet-based filters, and simulated annealing [10]. However, these methods fall under the category of postprocessing, that is, processing after a wellfocused image is generated, which adds complexity of result processing. In this paper, we propose a modified range migration algorithm using curvelets to reduce the speckle noise during the imaging processing without degrading the ability in clutter reduction. This algorithm is based on the fact that curvelets allow

0196-2892 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

LANG et al.: SAR PROCESSING OF ICE-SOUNDING DATA COLLECTED OVER THE EAST ANTARCTIC ICE SHEET

4497

TABLE I BASIC HRISR R ADAR PARAMETERS

a sparse representation of wave propagators and smooth functions away from singularities along smooth curves [11], [12]. This theory has been successfully used in seismic imaging processing (see [11] and [19]). Since the imaging processing of ice-sounding data and seismic data has many similarities (see [10] and [13]–[16]), it is convenient to apply this theory to the imaging processing of ice-sounding data. The theoretical analysis and applied procedures of our algorithm are separately outlined in Sections III and IV, respectively. In Section V, we validate our algorithm by comparing the imaging results of ice sheets obtained from our algorithm with the ones obtained from the f –k migration algorithm, the 2-D matched filter approach, the traditional modified range migration algorithm, the traditional modified range migration algorithm combined with the curvelet-based filter, and the traditional modified range migration algorithm combined with the Wiener filter. II. E AST A NTARCTICA DATA C OLLECTION A. HRISR The HRISR system is maintained and operated by the Key Laboratory of Electromagnetic Radiation and Sensing Technology of IECAS. HRISR is designed to operate as a chirped pulse system with 150-MHz center frequency and 100-MHz bandwidth. The radar is configured to transmit short chirped pulses of 2- and 4-μs duration and long chirped pulses of 8-μs duration for normal operation. The transmit pulserepetition frequency (PRF) is 8000 Hz. The peak transmit power is 500 W, and the transmit and receive antenna gain is 9 dBi. The return signal is digitized by one 12-bit A/D converter sampling at 500 MHz. The dynamic range of the receiver is over 110 dB. A summary of the radar parameters is given in Table I. Compared with the instrumentation used in [4], the HRISR system is a one transmitter and one receiver system, which is restricted to nadir-looking sounder operation without side-looking mode. The specific instrumentation introduction will be discussed in our following paper by Prof. Xiaojun Liu.

Fig. 1.

Area of tracks during CHINARE 29.

Fig. 2. HRISR system installed on a sled, which is pulled by a snow vehicle, during CHINARE 29.

B. Survey From Zhongshan Station to Kunlun Station During CHINARE 29 The HRISR data acquired over EAIS from Zhongshan Station (69◦ 22 24 S, 76◦ 22 40 E) to Kunlun Station (80◦ 25 01 S, 77◦ 06 58 E) during CHINARE 29 included one main transect, as shown in Fig. 1. The HRISR system was installed on a sled, which is pulled by a snow vehicle at a speed of 12–15 km/h, as shown in Fig. 2. The antenna is mounted 2 m above the snow surface, which allows the first 2-m air equivalent of the firn to be measured free of surface multiple reflections. The system has sounded the deep ice (∼3500 m) from Lambert Glacier. III. T HEORETICAL A NALYSIS OF C URVELETS , WAVE P ROPAGATION , AND I CE -S OUNDING DATA A. Brief Introduction to Curvelets Curvelets are basically 2-D anisotropic extensions to wavelets that have a main direction associated with them. The curvelet transform allows an essentially optimal sparse representation of objects that are twice continuously differentiable (C 2 ) away from discontinuities along C 2 edges (see [11], [17],

4498


where (r = ξ12 + ξ22 , ω = arctan(ξ1 /ξ2 )) are the polar coordinates in the frequency domain. The pair of windows Wj (r) and Vl (ω) are the radial and angular windows, respectively. They satisfy the conditions [18] ∞

Vl2 (ω − l) = 1,

ω∈R

l=−∞ ∞

(7) Wj2 (2j r) = 1,

r > 0.

j=−∞

Fig. 3. Tilings of curvelets in the (a) spectral domain and the (b) spatial domain. In the spectral domain, curvelets are supported near a “parabolic” wedge, and the shaded areas represent such a generic wedge. The essential support of a curvelet in the spatial domain is indicated by an ellipse.

and [18]). Analogous to wavelets, curvelets can be translated and dilated [11]. The dilation is given by a scale index j that controls the frequency content, whereas the translation that controls the time content is indexed by m1 and m2 . A curvelet has a main associated direction that can be changed through a rotation. This rotation is indexed by an angular index l. The relation between these indices and the location of the curvelet in the spectral and spatial domains is shown in Fig. 3(a) and (b) (see [11] and [17]). The decomposition scale aj is given by [18] aj = 2−j

πl2− 2 j 2

(2)

where x denotes the smallest integer being greater than or equal to x, and l = 0, 1, . . . , 4 · 2j/2 − 1. The number of directions is 4 · 2j/2 . The positions bj,l m are given by −1 −j −j/2 ) bj,l m = Rθj,l (m1 2 , m2 2

where m1 , m2 ∈ Z, and Rθj,l =

ϕ−1,0,m (x) = ϕ−1 (x − m),

T

cos θj,l − sin θj,l sin θj,l cos θj,l

ϕˆ−1 (ξ) = W0 (|ξ|)

rotation matrix with angle θj,l . The family of curvelet functions is given by ϕj,l,m (x) = ϕj,0,0 Rθj,l (x − bj,l m)

is the

with

W0 (r)2 = 1 −

2

W (2−j r) . (9)

Now, the system of curvelets shown in Fig. 3 given by ϕ−1,0,m (x) : m ∈ Z2 ∪ {ϕj,l,m (x) : j ∈ N0 , j

l = 0, 1, . . . , 4 · 2[ 2 ] − 1, m = (m1 , m2 )T ∈ Z2

(10)

satisfies a tight frame property. Every function f ∈ L2 (R2 ) can be represented as a curvelet series. We have f, ϕj,l,m ϕj,l,m (11) f= j,l,m

and the Parseval identity | f, ϕj,l,m |2 = f 2L2 (R2 )

∀ f ∈ L2 (R2 )

(12)

j,l,m

(4)

where x = (x1 , x2 )T is the spatial variable, and ϕj,0,0 is the mother curvelet. This mother curvelet is defined in the frequency domain by ϕˆj,0,0 (ξ) = Uj (ξ)

(8)

j≥0

(3)

m ∈ Z2

where

(1)

where j ≥ 0. The maximum value of j + 1 indicates the number of decomposition level. The equidistant sequence of rotation angles θj,l is given by θj,l =

Because there are many different windows that satisfy the constraint, many variants of curvelets can be constructed using different windows in the angular or radial direction of the spectral domain. The decay properties of the employed windows in the frequency domain determine the redundancy of the frame, whereas their smoothness is closely related to the decay properties of curvelets in the spatial domain [11]. For low frequencies, the curvelet elements are given by

(5)

where the notation “∧” denotes the variable in the frequency domain in this paper. ξ = (ξ1 , ξ2 )T is the variable in the frequency domain. Uj (ξ) is the scaled windows in the frequency domain and in polar coordinates, which is defined by ω Uj (r, ω) = 2−3j/4 Wj (2−j r)Vl (6) θj,1

holds. The terms f, ϕj,l,m are curvelet coefficients, which denote the projection of the function f on curvelet basis ϕj,l,m . B. Theoretical Analysis of Wave Propagation on Curvelets Waves with a given dominant wavelength are sensitive to variations in the medium with certain lengths scales only because the first Fresnel zone of a wave is proportional to the square root of the wavelength. Analogous to wavelets, curvelets have a bandlimited character and thus have an associated dominant frequency. Because of this bandlimited nature, curvelets with different dominant frequencies are sensitive to variations in the medium at different scales. This allows the possibility to smooth the background velocity with filters related to the dominant wavelength of the curvelets and propagate curvelets


of different scales through different smoothed versions of the medium [11]. The wave group maps each center of the curvelet onto a sum of curvelet-like waveforms whose locations and orientations are obtained by following the different Hamiltonian flow (see [12] and [20]). This statement means that the action of the solution operator of the wave equation on a curvelet of a particular scale can be approximated by moving the curvelet along the ray, associated with the center of the curvelet, through the medium smoothed for that particular scale (see [11] and [20]). The physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but, at the same time, with enough spatial localization so that they simultaneously behave like particles [12]. In fact, the procedure just outlined yields a leading-order contribution to the solution of the wave equation [11]. Hence, this procedure admits wave-equationbased ice-sounding imaging with curvelets. The wave propagates through the ice medium after the received wavefield being migrated to the air–ice interface in the wave-equation-based ice-sounding imaging with curvelets. The ice medium is the approximate homogeneous medium, which is a complex reflectivity structure with small propagation velocity perturbations [21]. This feature makes the ice medium not strictly the aforementioned smooth medium. However, Chauris [16], [19] has shown that wave-equation-based imaging with curvelets is still applicable in these kinds of medium. Hence, we ignore such smooth in this paper altogether. In addition, the wave-equation-based ice-sounding imaging operator belongs to the class of operators that is sparsfied by curvelets. However, the level of sparsity that can be achieved with a curvelet representation naturally will be somewhat less than the sparsity achieved for the smooth medium [12]. C. Curvelets as Building Blocks of Ice-Sounding Data The curvelet transform that allows an essentially optimal sparse representation of objects that are twice continuously differentiable (C 2 ) away from discontinuities along C 2 edges has been mentioned in Section III-A. Due to the wave character of ice-sounding data, the reflections recorded in ice-sounding data lie predominantly along smooth surfaces (or curves in 2-D) [4], just as geologic interfaces in the subsurface lie primarily along smooth surfaces. Hence, the ice-sounding data and their images can be sparsely represented using curvelets. At points where the recorded wavefronts or the subsurface contain pointlike discontinuities (e.g., at the edges of a fault in the subsurface), however, the level of sparsity that can be achieved with a curvelet representation naturally will be somewhat less than the sparsity achieved for the smooth parts of wavefronts or geologic interfaces [11]. In our work, we focus on the sparse representation of the wave-equation-based ice-sounding imaging operator using curvelets, rather than data compression with curvelets. The advantage of curvelets over the single samples (or spikes) that are currently used for building blocks of ice-sounding data is in accordance with the statement in [11]. By performing an intersubband statistical analysis of curvelet coefficients of the wave-equation-based ice-sounding imaging operator, one can

Fig. 4.

4499

Geometry of ice sounding.

distinguish between two classes of coefficients: those that represent useful signal and those dominated by speckle noise [18]. An intelligent thresholding procedure [22] is applied on the coefficients to eliminate the speckle noise. IV. M ODIFIED R ANGE M IGRATION A LGORITHM U SING C URVELETS The modified range migration algorithm using curvelets is a wave-equation-based ice-sounding imaging method, which is originally derived using the wavefield extrapolation approach and is implemented in the frequency and wavenumber domain. The complete of description of the wavefield extrapolation theory is given by Cafforio et al. [13] and Gazdag and Sguazzero [14]. As illustrated in Fig. 4, we assume that the snow vehicle is moving at constant velocity v0 and the HRISR is transmitting signal downward with fixed PRF. The wavefield of the radar electromagnetic signal s(x, z, t) in air or ice is governed by the 2-D wave equation given by ∂ 2 s(x, z, t) 4 ∂ 2 s(x, z, t) ∂ 2 s(x, z, t) = + 2 2 ∂t v ∂x2 ∂z 2

(13)

where x is the coordinate along the horizontal axis pointing to the moving direction, which is referred to as slow time or azimuth direction, z is the coordinate along the vertical axis pointing to nadir, which is referred to as fast time, t is the time variable, and v is the wave propagation velocity either in air or ice. The factor 4 over v 2 in (13) takes into account the signal’s two-way propagation from the transmitter to the target and then from the target to the receiver. The transmitter and the receiver are colocated for monostatic HRISR system. The coordinate z is zero along the snow vehicle moving path. Particularly, s(x, z = 0, t) is the signal received at (x, t), and s(x, z, t = 0) is the reflectivity from the target located at (x, z) according to the exploding reflectors model [13]. Therefore, the waveequation-based ice-sounding imaging turns out to be downward migration from s(x, z = 0, t) to s(x, z, t = 0). No motion compensation is performed before the modified range migration algorithm using curvelets in our situation. The reason for this is that cross-track motion has virtually no effect on the focusing of nadir targets [4]. Meanwhile, since the surface slope is very small and nearly constant at the location the measurements were made, the signal loss can be ignored [4]. Since the modified range migration algorithm using curvelets is applied in the frequency–wavenumber domain, let

4500


Fig. 5. Examples of the (a) angular window Vl (ω) and the (b) radial window Wj (r) used to design the curvelets. Because the decomposition level used in Section V is four, the maximum value of j is three. (a) illustrates Vl (ω) while j = 3 and l = 0, and (b) illustrates Wj (r) while j = 3.

S(kx , z, ω) be the wavefield of s(x, z, t) in the frequency and wavenumber domain and in polar coordinates, where kx is the wavenumber in the x dimension, and ω is the center angular frequency as defined in Section III-A. Because the snow vehicle has difficulty holding constant velocity, that is, the received data along slow time are not uniformly spaced, the 2-D fast Fourier transform (FFT) cannot be directly used to s(x, z, t). The azimuth nonuniform FFT (NUFFT) is implemented here to address this issue. NUFFT addresses the nonuniform sampling problem while performing the Fourier transform. A range FFT is performed first, followed by a range matched filter multiply to complete the pulse compression. Then, an azimuth NUFFT is performed to transform the data in the frequency– spatial domain into S(kx , z, ω) in the frequency–wavenumber domain. The algorithm we apply to realize NUFFT is proposed in [23]. The 2-D inverse FFT (IFFT) is available, which is given by (14) s(x, z, t) = S(kx , z, ω)ei(ωt+kx x) dωdkx .

where h is the antenna height, and kz_air is the wavenumber in air, which is obtained by substituting the wave propagation velocity in air, that is, the speed of light c, into (16), i.e., 2 kx c 2ω . (19) 1− kz_air = c 2ω According to the theoretical analysis in Section III, the sparse representation of the wave-equation-based ice-sounding imaging operator using curvelets is available now. A 2-D curvelet decomposition is performed to project S(kx , z = h, ω) onto curvelet frame to obtain curvelet coefficients. According to the curvelet decomposition algorithm described in [18] and [19], S(kx , z = h, ω) is multiplied by the scaled windows Uj (ξ) defined in (6) to obtain the curvelet coefficients Cˆμ in the frequency domain, which is given by Cˆμ = S(kx , z = h, ω)Uj (r, ω) −3j/4

=2 Substituting (14) into (13), the wave extrapolation equation is given by ∂S(kx , z, ω) = kz S(kx , z, ω) ∂z

(15)

where 2 kx v ω 2 2ω 2 kz = 4 − kx = 1− v v 2ω

(16)

is the wavenumber in the z dimension. The solution of (15) can be expressed as S(kx , z, ω)=S(kx , z = 0, ω) exp(ikz z).

(17)

The measured field is considered to be produced by sources exploding at a time t = 0. Since the wave propagation velocity is different in air and ice, we first propagate the received wavefield S(kx , z = 0, ω) to the air–ice interface. The expression of this propagation is given by S(kx , z = h, ω)=S(kx , z = 0, ω) exp(ikz_air h)

(18)

−j

S(kx , z = h, ω)Wj (2 r)Vl

ω θj,1

,

(20)

μ = (j, l, m). Since the Stolt interpolation is used in the following imaging steps and the interpolation is only applicable inside a circle of each curvelet [19], the selection principle of the radial window Wj (r) and the angular window Vl (ω) of curvelet decomposition and reconstruction is described in [16] and [19]. Based on this principle, the angular window Vl (ω) we choose is given by [16] j 2 π 3 (21) Vl (ω) = √ cos +1 ω− l 2 4 5 if ω ∈ [−π/(2+2j/2)+πl/(4 + 4j/2), π/(2 + 2j/2) + πl/(4 + 4j/2)] and zero otherwise, where ω, j and l are defined in Section III-A. The radial window Wj (r) we choose is given by a recursion constraint described in [24], with a change of variable from r to log r [16]. The examples of Vl (ω) and Wj (r) are displayed in Fig. 5. A 2-D IFFT is performed to obtain the curvelet coefficients Cμ in time domain. Soft thresholding is then applied on the


4501

Let sμ (x, z, t = 0) be the image of one input curvelet basis cμ (x, z = h, t). Substituting (23) into (14), sμ (x, z, t = 0) can be obtained, i.e., sμ (x, z, t = 0) = cˆμ (kx , z, ω)ei(ωt+kx x) dωdkx cˆμ (kx , z=h, ω)eikz_ice (z−h)ei(ωt+kx x) dωdkx

= =

Fig. 6. Coefficient histogram after soft thresholding procedure. (a) Wavelet coefficients of the imaging operator used in Fig. 11(a). (b) Curvelet coefficients of the imaging operator used in Fig. 11(a). (c) Wavelet coefficients of the imaging operator used in Fig. 11(b). (d) Curvelet coefficients of the imaging operator used in Fig. 11(b). The comparisons with wavelet coefficients qualitatively indicate the sparse representation of the wave-equation-based icesounding imaging operator using curvelets.

coefficients to eliminate speckle noise. The soft thresholding is defined by [22] x − sign(x)w, |x| ≥ w ηw (x) = (22) 0, |x| with w ≥ 0 a real-valued threshold, which is calculated by means of Candès et al. [25]. The coefficient histogram after soft thresholding procedure of practical data from Lambert Glacier and Kunlun Station is illustrated in Fig. 6. These figures qualitatively indicate the sparse representation of the wave-equation-based ice-sounding imaging operator using curvelets. Most of the curvelet coefficients are zero after the soft thresholding procedure. Then, each curvelet basis cμ (x, z = h, t) according to the nonzero coefficients is separately propagated to the source position at (x, z, t = 0). A 2-D FFT is performed to transform cμ (x, z = h, t) into cˆμ (kx , z = h, ω) in the frequency–wavenumber domain. According to (17), the propagation to depth z is given by cμ (kx , z = h, ω) exp (ikz_ice (z − h)) cˆμ (kx , z, ω)=ˆ

(23)

where kzi ce is the wavenumber in ice, which is obtained by substituting the wave propagation velocity in ice into (16), i.e.,

kz_ice

√ 2ω εr = c

1−

kx c √ 2ω εr

2

where εr is the relative dielectric constant of ice.

(24)

cˆμ (kx ,z=h, ω)e−ikz_ice hei(kz_ice z+kx x)dωdkx. (25)

The complex exponential term e−ikzi ceh is derived from the set of the origin of coordinate z. Equation (25) does not have the expression of a 2-D IFFT, and the computation of the double integral would take too much computer time. The so-called Stolt interpolation is used here to solve this problem. In order to take advantage of IFFT, we want to integrate (25) with respect to kzi ce instead of ω. Using (24), we can obtain cˆμ (kx , z = h, kzi ce ) as ckz_ice cˆμ (kx , z = h, kz_ice ) = cˆμ (kx , z = h, ω) √ . 2 2 εr kx2 + kz_ice (26) Substituting (24) and (26) into (25), the IFFT expression is given by sμ (x, z, t = 0) = cˆμ (kx , z = h, kz_ice )e−ikz_ice hei(kz_ice z+kx x)dkz_ice dkx . (27) The total image s(x, z, t = 0), which is obtained by a 2-D curvelet reconstruction using the reconstruction formula (11), is given by s(x, z, t = 0) =

Cμ sμ (x, z, t = 0)

(28)

˘ μ∈R

˘ is the index set that holds the multi-indices of where R curvelets that survive a certain threshold. Consequently, the modified range migration algorithm using curvelets contains the following sequence. 1) A range FFT is performed to transform the received data into the frequency–spatial domain, followed by a range matched filter multiply to complete the pulse compression. 2) An azimuth NUFFT is performed to transform the data in the frequency–spatial domain into S(kx , z = 0, ω) in the frequency–wavenumber domain. 3) The wavefield S(kx , z = 0, ω) in the frequency– wavenumber domain is propagated to the air–ice interface.

4502


Fig. 8. Position of targets used in the simulation. (a) Arrangement of five point targets. (b) Arrangement of three point targets.

TABLE II S IMULATION PARAMETERS W ITH P OINT TARGETS

Fig. 7. Flowchart of the modified range migration algorithm using curvelets for ice sounding.

4) The wavefield S(kx , z = h, ω) at air–ice interface is multiplied by the scaled windows Uj (ξ) to obtain the curvelet coefficients Cˆμ in the frequency domain. 5) A 2-D IFFT is performed to obtain the curvelet coefficients Cμ in the time domain. Soft thresholding is applied on these coefficients to eliminate speckle noise. 6) A 2-D FFT is performed to transform one curvelet basis cμ (x, z = h, t) according to the nonzero coefficients into cˆμ (kx , z = h, ω) in the frequency–wavenumber domain. 7) The Stolt interpolation is performed to change variables from ω to kz_ice . 8) A 2-D IFFT is performed to obtain one image sμ (x, z, t = 0). 9) The 2-D curvelet reconstruction is performed to obtain the total image s(x, z, t = 0), unless all the images sμ (x, z, t = 0) according to the nonzero coefficients are acquired. The flowchart of the modified range migration algorithm using curvelets for ice sounding is shown in Fig. 7. Compared with the 2-D matched filter approach and the traditional modified range migration algorithm, the modified range migration algorithm using curvelets reduces the speckle noise during the imaging processing without degrading the ability in clutter reduction. Compared with the f –k migration algorithm used in [4], [9], and [15], the modified range migration algorithm using curvelets not only reduces the speckle noise during the imaging processing without degrading the ability in clutter reduction but also replaces the summation with respect to ω with Stolt

interpolation to take advantage of the 2-D IFFT to improve the computational efficiency. V. R ESULTS AND C OMPARISONS A. Simulation With Point Targets This simulation demonstrates the validity of the modified range migration algorithm using curvelets. The simulation contains two parts. In the first part, five point targets are simulated. Three of them have the same two-way propagation time that one point target is arranged in nadir ice and the other two point targets are placed on ice surface along azimuth direction. The rest of the point targets are arranged equally spaced around the nadir target, as shown in Fig. 8(a). The nadir target coordinates are located at (0, 2000 m), and the equal interval length is 500 m. In the second part, three point targets are simulated. All the three point targets are consistent with the first part with the two point targets placed on ice surface along azimuth direction being removed, as shown in Fig. 8(b). The simulation parameters are given in Table II. The decomposition level indicated by the maximum value of j + 1 used here is four. To check on the focusing in more detail, the two nadir targets in Fig. 8(a) and (b) are analyzed, as shown in Fig. 9. Fig. 9(a) demonstrates that the power of the two nadir targets are equal, which indicates that the two point targets on the ice surface have been suppressed. Fig. 9(b)–(d) demonstrates the impulse response of the nadir target in Fig. 8(b). The measured range resolution [10] is 0.75 m, the range peak sidelobe ratio (PSLR) is −12.04 dB,


Fig. 9.

4503

Analysis of the nadir targets. (a) Power comparison. (b) Expanded target contours. (c) Range profile. (d) Azimuth profile.

Referring to the simulation parameters given in Table II, the measured azimuth resolution given by (29) is 2.17 m. The azimuth PSLR is −15.47 dB, and the azimuth ISLR is −10.53 dB. These values agree with the theoretical values. The two other targets are similarly well compressed, as shown in Fig. 10.

B. HRISR Data Processing

Fig. 10.

Compressed targets.

and the range integrated sidelobe ratio (ISLR) is −9.86 dB. The measured azimuth resolution is calculated by [3] σ= where λ Ls h d nice

λ 2Ls

h+

d nice

radar wavelength; synthetic aperture length; aircraft height; target depth; refractive index of the ice layer.

(29)

The decomposition level indicated by the maximum value of j + 1 used here is four. Fig. 11(a) and (b) shows the sounderderived profiles after the modified range migration algorithm using curvelets processing for two parts of the main transect, as shown in Fig. 1. Fig. 11(a) illustrates the data collected along the traverse of Lambert Glacier, whereas Fig. 11(b) illustrates the data collected along the traverse near Kunlun Station. The traverse in Fig. 11(a) is shorter than that in Fig. 11(b) because the maximum measurable depth during CHINARE 29 is about 3500 m, derived from the parameters setting. The initial 2 km in Fig. 11(b) indicates that the snow vehicle halts and starts up again. The first, about 500 m below the ice surface, is the blind area caused by the receiver blanking switch, which was necessary to avoid receiver saturation. Well-known internal reflecting horizons (IRHs) are seen between 500 m to near the echo-free zone. IRHs in Fig. 11(a) are occasionally blurred, which we deduce is due to the inner deformation of the ice

4504


Fig. 11. HRISR data results after the modified range migration algorithm using curvelets processing. (a) Data collected along the traverse of Lambert Glacier. (b) Data collected along the traverse near Kunlun Station.

sheet [1]. IRHs in Fig. 11(b) are continuously imaged, which is one of the options for ice core drilling. Bottom topography in Fig. 11(a) is very flat. Combined with IRHs in Fig. 11(a), we deduce that there may be a small lake. Bottom topography in Fig. 11(b) becomes increasingly rough along the transverse with peak-to-peak deviations of about 1000 m. We compared the imaging results of ice sheets obtained from our algorithm with the ones obtained from the f –k migration algorithm, the 2-D matched filter approach, the traditional modified range migration algorithm, the traditional modified range migration algorithm combined with the curvelet-based filter, and the traditional modified range migration algorithm combined with the Wiener filter to demonstrate the advantage in speckle noise elimination of our algorithm. Figs. 12 and 13 illustrate all results, which give a more detailed view of Fig. 11. We compared these results in two major aspects: one is the power of clutter reduction and the other is equivalent number of looks (ENL). First, we compared our algorithm with the f –k migration algorithm [4]. The red curves in Fig. 14(a) and (b) denote our algorithm, whereas the yellow curves denote the f –k migration algorithm. These curves have basically uniform trend. This means that the abilities in clutter reduction of the f –k migration algorithm and our algorithm are almost the same. The ENL is defined by [26]

ENL =

(E[P ])2 VAR[P ]

(30)

where P is equal to the intensity of a pixel; and E and VAR are the mean value and variance of P , respectively. The mean value E and the variance VAR are separately calculated by [27] Q M 1 E[P ] = D(Pi + k, Pj + l) (2M + 1)(2Q + 1) k=−M l=−Q

(31)

and VAR[P ] =

1 (2M + 1)(2Q + 1) Q M [D(Pi + k, Pj + l) − E[P ]]2

(32)

k=−M l=−Q

where (Pi , Pj ) is the pixel location, M and Q are the size of calculation window whose values often select M = Q = 2 (window size 5×5), and D is the calculation area of Figs. 12(a) and (b) and 13(a) and (b). We choose the area (3250 m, 3500 m) × (400 m, 500 m) of Fig. 12(a) and (b) and the area (2250 m, 2500 m) × (400 m, 500 m) of Fig. 13(a) and (b) to calculate ENL, as shown in Fig. 15(a) and (b). From (30)–(32), the ENL of Fig. 12(a) and (b) are 15.6591 and 9.7733, respectively; whereas the ENL of Fig. 13(a) and (b) are 9.6163 and 4.1684, respectively. This ENL comparison shows that our algorithm has stronger ability in eliminating speckle noise than the f –k migration algorithm. The calculating times of Fig. 12(a) and (b) are 5352.2707 and 7544.6923 s, respectively; whereas the calculating times of Fig. 13(a) and (b) are 5110.2333 and 7203.6059 s, respectively. These calculating time comparisons show that our algorithm is more efficient than the f –k migration algorithm. Second, we compared our algorithm with the 2-D matched filter approach [5]. The green curves in Fig. 14(a) and (b) denote the 2-D matched filter approach. The comparisons between the green and red curves mean that the abilities in clutter reduction of the 2-D matched filter approach and our algorithm are almost the same. We choose the same area shown in Fig. 15(a) and (b) of Figs. 12(a) and (c) and 13(a) and (c) to calculate ENL. The ENL of Fig. 12(a) and (c) are 15.6591 and 10.1716, respectively; whereas the ENL of Fig. 13(a) and (c) are 9.6163 and 3.6903, respectively. This ENL comparison shows that our algorithm has stronger ability in eliminating speckle noise than the 2-D matched filter approach. Third, we compared our algorithm with the traditional modified range migration algorithm. The difference between our


4505

Fig. 12. Results of HRISR data collected along the traverse of Lambert Glacier. (a) Modified range migration algorithm using curvelets result. (b) f –k migration algorithm result. (c) Two-dimensional matched filter approach result. (d) Traditional modified range migration algorithm result. (e) Traditional modified range migration algorithm combined with the curvelet-based filter result. (f) Traditional modified range migration algorithm combined with the Wiener filter result.

algorithm and the traditional modified range migration algorithm is that the traditional algorithm utilizes the single samples for imaging. The imaging approach of the traditional algorithm is the same with our algorithm, except Steps 4, 5, and 6. The blue curves in both Fig. 14(a) and (b) denote the traditional modified range migration algorithm. The comparisons between the blue and red curves illustrate that the abilities in clutter reduction of the traditional modified range migration algorithm and our algorithm are almost the same. We choose the same area shown in Fig. 15(a) and (b) of Figs. 12(a) and (d) and 13(a) and (d) to calculate ENL. The ENL of Fig. 12(a) and (d) are 15.6591

and 9.6917, respectively; whereas the ENL of Fig. 13(a) and (d) are 9.6163 and 6.8556, respectively. This ENL comparison shows that our algorithm has stronger ability in eliminating speckle noise than the traditional modified range migration algorithm. Fourth, we compared the results obtained from our algorithm with the ones obtained from the traditional modified range migration algorithm combined with the curvelet-based filter. The curvelet-based filter falls under the category of postprocessing. The result obtained from the traditional modified range migration algorithm is then processed by the

4506


Fig. 13. Results of HRISR data collected along the traverse near Kunlun Station. (a) Modified range migration algorithm using curvelets result. (b) f –k migration algorithm result. (c) Two-dimensional matched filter approach result. (d) Traditional modified range migration algorithm result. (e) Traditional modified range migration algorithm combined with the curvelet-based filter result. (f) Traditional modified range migration algorithm combined with the Wiener filter result.

curvelet-based filter to achieve the final result without speckle noise. We choose the area (1250m, 1500 m) × (400 m, 500 m) of Fig. 12(a) and (e) and the area (1750 m, 2000 m) × (400 m, 500 m) of Fig. 13(a) and (e) to calculate ENL, as shown in Fig. 16(a) and (b). The ENL of Fig. 12(a) and (e) are 36.3961 and 36.3962, respectively; whereas the ENL of Fig. 13(a) and (e) are 12.9522 and 11.4420, respectively. This ENL comparison shows that our algorithm has almost the same capability in eliminating speckle noise with the curvelet-based filter. However, the curvelet-based filter is an image processing approach that should be arranged after a well-focused image is generated.

This feature makes the curvelet-based filter more complicated to use. Finally, we compared the results obtained from our algorithm with the ones obtained from the traditional modified range migration algorithm combined with the Wiener filter. The Wiener filter also falls under the category of postprocessing and should be arranged after a well-focused image is generated, that is, the result obtained from the traditional modified range migration algorithm is then processed by the Wiener filter to achieve the final result without speckle noise. We choose the same area shown in Fig. 16(a) and (b) of Figs. 12(a) and (f) and


4507

Fig. 14. Comparisons between A-scopes. (a) Comparisons between A-scopes from Fig. 12(a)–(d). The blue curve denotes the traditional modified range migration algorithm, the green curve denotes the 2-D matched filter approach, the yellow curve denotes the f –k migration algorithm, and the red curve denotes the modified range migration algorithm using curvelets. (b) Comparisons between A-scopes from Fig. 13(a)–(d). The blue curve denotes the traditional modified range migration algorithm, the green curve denotes the 2-D matched filter approach, the yellow curve denotes the f –k migration algorithm, and the red curve denotes the modified range migration algorithm using curvelets.

Fig. 15. Area for calculating ENL. (a) The red square denotes the area for calculating the ENL of Fig. 12(a)–(d). (b) The black square denotes the area for calculating the ENL of Fig. 13(a)–(d).

Fig. 16. Area for calculating ENL. (a) The red square denotes the area for calculating the ENL of Fig. 12(a), (e), and (f). (b) The black square denotes the area for calculating the ENL of Fig. 13(a), (e), and (f).

4508


13(a) and (f) to calculate ENL. The ENL of Fig. 12(a) and (f) are 36.3961 and 36.0412, respectively; whereas the ENL of Fig. 13(a) and (f) are 12.9522 and 11.1930, respectively. This ENL comparison shows that our algorithm has almost the same capability in eliminating speckle noise with the Wiener filter; however, it is easier to use. Through the preceding comparisons of two major aspects, we can derive the conclusion that the modified range migration algorithm using curvelets reduces the speckle noise during the imaging processing without degrading the ability in clutter reduction. In addition, the comparisons of ENL also point out that the modified range migration algorithm using curvelets enhances both IRHs and the bedrock interface.

VI. C ONCLUSION We have proposed a new algorithm named the modified range migration algorithm using curvelets to address the problem of speckle noise. This algorithm is a wave-equation-based icesounding imaging method using curvelets as building blocks of ice-sounding data, which successfully images the topography of ice sheets. Validation with point targets simulation indicates the ability of the modified range migration algorithm using curvelets for clutter reduction. The comparisons using HRISR data indicate that the proposed algorithm reduces the speckle noise during the imaging processing without degrading the ability in clutter reduction. The proposed algorithm is not only suitable to continuous wave linear frequency modulation (LFM) radar system but also appropriate to impulse radar system. When the proposed algorithm is applied to the data set acquired by the impulse radar system, the range pulse compression performance should be replaced by deconvolution process [28]. Additionally, this algorithm can be also applied to data collected from nadir-looking airborne HRISR systems to reduce the along-track surface clutter with appropriate motion compensation approach. The proposed algorithm will soon be applied to the large data set acquired during CHINARE 31 to improve data interpretation. Meanwhile, the real-time ice-sounding imaging algorithm is developing for the upcoming airborne radar system, which aims to better understand how ice sheets develop and evolve. ACKNOWLEDGMENT The authors would like to thank CHINARE 29 and the teammates for their generous help in the experiments in Antarctica. They would also like to thank Prof. B. Sun and Dr. J. Guo of the Polar Research Institute of China for their valuable assistance in data acquisition over Antarctica. They would also like to thank Dr. X. Cui of the Polar Research Institute of China for his help in understanding the physical and chemical characteristics of polar ice sheets and Prof. X. Hou of Xi’an Jiaotong University for understanding the characteristics of wavelets and curvelets and numerous helpful discussions. They would also like to thank N. Cicero for editing this paper and anonymous reviewers for their positive comments and suggestions for improving this paper.

R EFERENCES [1] X. Cui et al., “Progress and prospect of ice radar in investigating and researching Antarctic ice sheet,” Adv. Earth Sci., vol. 24, no. 4, pp. 392–402, 2009. [2] T. H. Jacka et al., “Recommendations for the collection and synthesis of Antarctic ice sheet mass balance data,” Global Planet. Change, vol. 42, no. 1–4, pp. 1–15, Jul. 2004. [3] M. E. Peters et al., “Along-track focusing of airborne radar sounding data from West Antarctica for improving basal reflection analysis and layer detection,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 9, pp. 2725–2736, Sep. 2007. [4] J. Paden, “Synthetic aperture radar for imaging the basal conditions of the polar ice sheets,” Ph.D. dissertation, Univ. Kansas, Lawrence, KS, USA, 2006. [5] J. J. Legarsky et al., “Focused synthetic aperture radar processing of icesounder data collected over the Greenland ice sheet,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 10, pp. 2109–2117, Oct. 2001. [6] J. J. Legarsky, “Synthetic Aperture Radar (SAR) processing of glacial ice depth sounding data, Ka-Band backscattering measurements and applications,” Ph.D. dissertation, Univ. Kansas, Lawrence, KS, USA, 1999. [7] F. Hélière et al., “Radio echo sounding of Pine Island Glacier, West Antarctica: Aperture synthesis processing and analysis of feasibility from space,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp. 2573–2582, Aug. 2007. [8] R. B. Alley et al., “Deformation of till beneath ice stream B, West Antarctica,” Nature, vol. 322, no. 6074, pp. 57–59, Jul. 3, 1986. [9] J. Li et al., “High-altitude radar measurements of ice thickness over the Antarctic and Greenland ice sheets as a part of operation icebridge,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 2, pp. 742–754, Feb. 2013. [10] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Boston, MA, USA: Artech House, 2005, ch. 6, p. 265, ch. 8, p. 329, ch.4, p. 131. [11] H. Douma, “A hybrid formulation of map migration and wave-equationbased migration using curvelets,” Ph.D. dissertation, Colorado Sch. Mines, Golden, CO, USA, 2006. [12] E. J. Candès et al., “The curvelet representation of wave propagators is optimally sparse,” Commun. Pure Appl. Math., vol. 58, no. 11, pp. 1472– 1528, Nov. 2005. [13] C. Cafforio, C. Prati, and F. Rocca, “SAR data focusing using seismic migration techniques,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 2, pp. 194–207, Mar. 1991. [14] J. Gazdag and P. Sguazzero, “Migration of seismic data,” Proc. IEEE, vol. 72, no. 10, pp. 1302–1315, Oct. 1984. [15] J. Li, “Mapping of ice sheet deep layers and fast outlet glaciers with multi-channel-high-sensitivity radar,” Ph.D. dissertation, Univ. Kansas, Lawrence, KS, USA, 2009. [16] H. Chauris and T. Nguyen, “Seismic demigration/migration in the curvelet domain,” Geophysics, vol. 73, no. 2, pp. S35–S46, Mar./Apr. 2008. [17] E. Candès et al., “Fast discrete curvelet transforms,” Multiscale Model. Simul., vol. 5, no. 3, pp. 861–899, Mar. 2006. [18] J. Ma and G. Plonka, “A review of curvelets and recent applications,” IEEE Signal Process. Mag., vol. 27, no. 2, pp. 118–133, Mar. 2010. [19] H. Chauris, “Seismic imaging in the curvelet domain and its implications for the curvelet design,” Proc. 76th Annu. Int. Meet., SEG, Expanded Abstracts, 2006, pp. 2406–2410. [20] H. Smith, “A parametrix construction for wave equations with C1,1 coefficients,” Ann. Inst. Fourier, vol. 4, no. 8, pp. 797–835, 1998. [21] S. Fujita, H. Maeno, and K. Matsuoka, “Radio-wave depolarization and scattering within ice sheets: A matrix-based model to link radar and icecore measurements and its application,” J. Glaciol., vol. 52, no. 178, pp. 407–424, Jul. 2006. [22] G. Hennenfent, “Sampling and reconstruction of seismic wavefields in the curvelet domain,” Ph.D. dissertation, Univ. British Columbia, Vancouver, B.C., Canada, 2008. [23] G. Fan and Q. Liu, “Fast Fourier transform for discontinuous functions,” IEEE Trans. Antennas. Propag., vol. 52, no. 2, pp. 461–465, Feb. 2004. [24] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, “Shiftable multi-scale transforms,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 587–607, Mar. 1992. [25] E. Candès et al., CurveLab, 2004. [Online]. Available: https://www. curvelet.org [26] A. Morcira, “Improved multilook techniques applied to SAR and SCANSAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 29, no. 4, pp. 529–534, Jul. 1991.


[27] Z. Mouyan, Deconvolution and Signal Recovery. Beijing, China: Nat. Defence Ind. Press, 2001, ch. 6, p. 187. [28] Y. Su et al., Ground Penetrating Radar Theory and Applications. Beijing, China: Sci. Press, 2006, ch. 7, pp. 192–194.

Shinan Lang received the B.E. degree from the University of Science and Technology Beijing, Beijing, China, in 2010. She is currently working toward the Ph.D. degree in the Institute of Electronics, Chinese Academy of Sciences, Beijing, China. She is also currently with the University of Chinese Academy of Sciences, Beijing. Her research interests are in developing advanced signal and array processing algorithms for processing and interpreting data of radio echo sounding of ice sheets and radar depth sounder image processing technology.

Xiaojun Liu received the B.S. and M.S. degrees in electronics science and technology from the North University of China, Taiyuan, China, in 1995 and 1998, respectively, and the Ph.D. degree in electrical engineering from the Chinese Academy of Sciences, Beijing, China, in 2001. Since 2001, he has been with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences (IECAS), Beijing, where he is currently a Professor and leads a group on the research studies of remote sensing of ice sheets with projects supported by the National Natural Science Foundation of China, the National High Technology Research and Development Projects (863 Projects) of China, etc. His research interests include application of radars to remote sensing of polar ice sheets, sea ice, atmosphere, and land. He developed several radar systems currently being used at IECAS for sounding and imaging of polar ice sheets and has also participated in field experiments during CHINARE 26, CHINARE 28, CHINARE 29, and CHINARE 30.

4509

Bo Zhao was born in Shanxi, China, in 1983. He received the B.E. degree in electronics science and technology from the Civil Aviation University of China, Tianjin, China, in 2004 and the M.S. and Ph.D. degrees in electrical engineering from the Chinese Academy of Sciences, Beijing, China, in 2006 and 2009, respectively. Since 2009, he has been with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing, where he became an Associate Professor in 2014. He participated in field experiments in Antarctica during the 26th Chinese Antarctic Research Expedition. His research interests include system development of radar sounding, digital system design, and signal processing.

Xiuwei Chen was born in Liaoning, China, in 1978. He received Ph.D. degree in computer application technology from the Chinese Academy of Sciences, Beijing, China, in 2011. Since 2011, he has been with the Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing. His research interests include radar system design and radar signal processing.

Guangyou Fang received the B.S. degree from Hunan University, Changsha, China, in 1984 and the M.S. and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1990 and 1996, respectively, all in electrical engineering. From 1990 to 1999, he was an Engineer, an Associate Professor, and a Professor with the China Research Institute of Radiowave Propagation, Xinxiang, China. From 2000 to 2001, he was a Visiting Scholar with the University of Trieste, Trieste, Italy, and with the International Centre for Science and High Technology—United Nations Industrial Development Organization, Trieste. From 2001 to 2003, he was a Special Foreign Research Fellow of the Japan Society for the Promotion of Science, working with Prof. M. Sato at Tohoku University, Sendai, Japan. Since 2004, he has been a Professor with the Institute of Electronics, Chinese Academy of Sciences, Beijing, China, where he is currently the Director of the Key Laboratory of Electromagnetic Radiation and Sensing Technology. He has authored over 100 publications. His research interests include ultrawideband radar, ground-penetrating radar signal processing and identification methods, terahertz imaging technology, and computational electromagnetics.