remote sensing Article
Better Estimated IEM Input Parameters Using Random Fractal Geometry Applied on Multi-Frequency SAR Data Ali Ghafouri 1 , Jalal Amini 1, *, Mojtaba Dehmollaian 2 and Mohammad Ali Kavoosi 3 1 2 3
*
School of Surveying and Geospatial Engineering, Faculty of Engineering, University of Tehran, Tehran 1439957131, Iran;
[email protected] Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, Tehran 1439957131, Iran;
[email protected] Department of Geology, Exploration Directorate of National Iranian Oil Company, Tehran 1994814695, Iran;
[email protected] Correspondence:
[email protected]; Tel.: +98-912-459-4685; Fax: +98-21-8860-4534
Academic Editors: Timo Balz, Uwe Soergel, Mattia Crespi, Batuhan Osmanoglu, Nicolas Baghdadi and Prasad Thenkabail Received: 26 February 2017; Accepted: 4 May 2017; Published: 5 May 2017
Abstract: Microwave remote sensing can measure surface geometry. Via the processing of the Synthetic Aperture Radar (SAR) data, the earth surface geometric parameters can be provided for geoscientific studies, especially in geological mapping. For this purpose, it is necessary to model the surface roughness against microwave signal backscattering. Of the available models, the Integral Equation Model (IEM) for co-polarized data has been the most frequently used model. Therefore, by the processing of the SAR data using this model, the surface geometry can be studied. In the IEM, the surface roughness geometry is calculable via the height statistical parameter, the rms-height. However, this parameter is not capable enough to represent surface morphology, since it only measures the surface roughness in the vertical direction, while the roughness dispersion on the surface is not included. In this paper, using the random fractal geometry capability, via the implementation of the power-law roughness spectrum, the precision and correctness of the surface roughness estimation has been improved by up to 10%. Therefore, the random fractal geometry is implemented through the calculation of the input geometric parameters of the IEM using the power-law surface spectrum and the spectral slope. In this paper, the in situ roughness measurement data, as well as SAR images at frequencies of L, C, and X, have been used to implement and evaluate the proposed method. Surface roughness, according to the operational frequencies, exhibits a fractal or a diffractal behavior. Keywords: Synthetic Aperture Radar (SAR); Integral Equation Model (IEM); random fractal geometry
1. Introduction Studying the surface roughness geometry, especially studying the natural surfaces, requires appropriate satellite data and processing methodologies [1–3]. The Synthetic Aperture Radar (SAR) data acquired by the airborne and space-borne sensors has made it possible to examine the surface roughness, which provides useful information for geoscientists and geologists. The backscattered signal in all polarizations is affected by surface roughness, and contains the surface geometry information [4–6]. To measure the surface geometry using the SAR data, the surface geometric parameter(s) must be modeled against the backscattering coefficient on each polarization. Generally speaking, to describe and differentiate the patterns and the geometric surface texture, it is indispensable to model the interaction between the backscattered signals and the surface properties. The radar signal
Remote Sens. 2017, 9, 445; doi:10.3390/rs9050445
www.mdpi.com/journal/remotesensing
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is sensitive to certain surface roughness measures, which is determined by the operating radar signal The radar[7–9]. signal is sensitive to certain surface roughness measures, which is determined by the frequency operating radar signal frequency [7–9]. Of the available models, the standard theoretical models of backscattering include: (1) the Of the available models, the standard theoretical models of backscattering include: (1) the Kirchhoff Approximation (KA), which encompasses Geometric Optics (GO) and Physical Optics (PO); Kirchhoff Approximation (KA), which encompasses Geometric Optics (GO) and Physical Optics and (2) the Small Perturbation Model (SPM) [7,10]. (PO); and (2) the Small Perturbation Model (SPM) [7,10]. The Geometric Optics model for very rough surfaces, the Physical Optics model for moderate The Geometric Optics model for very rough surfaces, the Physical Optics model for moderate roughness, and the Small Perturbation Model for approximately smooth surfaces, have been roughness, and the Small Perturbation Model for approximately smooth surfaces, have been used. used. al. (1992) developed Integral Equation Model (IEM) a physically-based Fung Fung et al.et(1992) havehave developed the the Integral Equation Model (IEM) as aas physically-based electromagnetic transfer model which combines the Kirchhoff models and the SPM, which electromagnetic transfer model which combines the Kirchhoff models and the SPM, andand which constructs a more applicable model that can theoretically tolerate a really wide range of roughness constructs a more applicable model that can theoretically tolerate a really wide range of roughness dimensions. the IEM IEM isisnot notrestricted restrictedtotoany anyspecial special frequency range or roughness dimensions. ItIt is is of of note note that that the frequency range or roughness measures measures[11]. [11]. The rms-height parameter toto characterize thethe surface geometry [12]. In this paper, TheIEM IEMexploits exploitsthe the rms-height parameter characterize surface geometry [12]. In this the statistical rms-height and the Gaussian function (ACF) have been paper, the statistical rms-height and theauto-correlation Gaussian auto-correlation function (ACF)implemented have been as implemented the conventional IEM. In the conventional method, the surface roughness geometry the conventionalasIEM. In the conventional method, the surface roughness geometry is only considered is only considered in the vertical direction, and only the Euclidean geometry is applied. in the vertical direction, and only the Euclidean geometry is applied. However,using using the the so so called called fractal in in thethe horizontal direction However, fractalIEM, IEM,which whichconsiders considersroughness roughness horizontal direction anditsitsdispersion dispersionon onthe thesurface, surface,ititcan canbebepossible possibletotoimprove improvethe thesurface surfacemorphology morphologyestimation estimationand and andthe soar the precision the microwave discrimination means power-law surfacespectrum spectrumand soar precision of theofmicrowave discrimination by by means of of thethe power-law surface and the spectral slope parameter [13]. the spectral slope parameter [13]. this paper, as depicted in Figure 1, to provide the surface geometric characteristics for the IEM InInthis paper, as depicted in Figure 1, to provide the surface geometric characteristics for the computation, the in situ surface roughness measurement has been performed by the field surveying IEM computation, the in situ surface roughness measurement has been performed by the field operation. The dielectric constants have been extracted from the tables produced by Martinez et al. surveying operation. The dielectric constants have been extracted from the tables produced by (2000) [14]. Furthermore, the SAR satellite data of ALOS PALSAR, Sentinel-C, and TerraSAR, Martinez et al. (2000) [14]. Furthermore, the SAR satellite data of ALOS PALSAR, Sentinel-C, and respectively, in the bands L, C, and X, have been used to compute the surface roughness, i.e., the TerraSAR, respectively, in the bands L, C, and X, have been used to compute the surface roughness, geological morphology. i.e., the geological morphology. Inputs Surface Roughness (Morphology)
in situ measurement
+ Dielectric Constants
Extraction from References
Evaluation
Conventional Inputs
Using statistical
−
and
Fractal Inputs Using spectral exponent for ACF and −
Backscattering Comparison ° ° .
with SAR measurement
(a)
Inputs
}
°
Backscattering Coefficient ( ) Calculation using SAR Satellite measurements
Evaluation L-band : ALOS PALSAR C-band : Sentinel-1A/SAR-C X-band : TerrsaSAR-X
Roughness Comparison
.
with Field measurement
(b)
Figure1.1.Flowchart Flowchart of of the the study study and model evaluation: computation of of Figure and results resultsevaluation. evaluation.(a)(a)Direct Direct model evaluation: computation backscattering using using the the in in situ situ measured evaluation: calculation of surface backscattering measureddata; data;(b) (b)inverse inversemodel model evaluation: calculation of surface roughness using the backscattering coefficient. roughness using the backscattering coefficient.
In this paper, the IEM geometric input parameters have been calculated using three different In thisfor paper, the IEM geometric input parameters have beengeometry calculated using three different methods two types of surface assumptions; two conventional methods (stationary methods for two types of surface assumptions; two conventional geometry methods (stationary surface surface assumption) and the random fractal geometry (for power-law surface assumption). After assumption) and the random geometry (for power-law surfaceatassumption). After running running the model with each offractal the inputs, the backscattering coefficient each point corresponding the withofeach of themeasurement inputs, the backscattering coefficient at each point corresponding to each to model each pixel the SAR data is computed (Figure 1a). In addition, through the IEM pixel of the SAR measurement data is computed (Figure 1a). In addition, through the IEM inversion inversion method, the surface parameters are calculated, corresponding to the in situ measurements method, parameters are calculated, the in situ measurements (Figure (Figure the 1b).surface The IEM computation results, corresponding as well as thetoinverse IEM compared with the 1b). measurements, can show the model’s level efficacy. The IEM computation results, as well as theofinverse IEM compared with the measurements, can show the model’s level of efficacy.
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The IEM calculated and the SAR measured backscattering coefficients are mutually compared to evaluate the methodology (Figure 1a). Additionally, comparing the in situ rms-height with the inverse IEM results, the level of efficacy when using the power-law inputs versus the conventional ones can be cleared [12,15] (Figure 1b). In Section 2, the IEM backscattering model with the conventional inputs will be addressed. The methodology of random fractal geometry will be developed in Section 3, in order to determine the power-law surface parameters as the input of the IEM. Finally, the implementation and validation of the fractal IEM will be discussed in Section 4 for the multi-frequency SAR data. Section 5 presents the conclusion. 2. The IEM Backscattering Model Backscattering is the amount of the scattered signal per surface unit in the scattering angle from 0 ◦ to 180 degrees [16]. The calculation of the backscattering coefficient (σ ) process depends on the satellite antenna specifications. As defined by Fung et al. (1992) and explained by themselves (1994), the IEM model describes the relation of the backscattering coefficient to the surface roughness parameters, as well as the dielectric constant and the incidence angle. The IEM is characterized for the co-polarized and cross-polarized backscattering calculations [8,17]. Yet, in this paper, just the co-polarized equation is implemented. The co-polarized backscattering coefficient equation according to Fung et al. (2004), which is termed I2 EM by Ulaby (2014), is as follows [7,8,10,11,16]: ◦
σpp = where:
k2 −2k2 s2 cos2 θ +∞ n 2 W (n) (2ksinθ, 0) e ∑ Ipp 4π n! n =1
n I pp = (2k s cosθ ) f pp exp −k2 s2 cos2 θ + (k s cosθ )n Fpp
(1)
(2)
and pp is either the hh or vv polarizations; k stands for the radar wavenumber (k = 2π λ , λ wavelength); ( n ) s is the rms-height; θ denotes the incidence angle; and W is the Fourier transform of nth power of the ACF. f hh , f vv , Fhh , and Fvv are approximated by the following equations [10,16]: f hh =
2Rv cosθ sin2 θ 1 2 Fhh = 2 4Rh − 1 − (1 + R h ) cosθ ε sin2 θ ε cos2 θ 1 2 2 =2 1− 1 − R 1 + R − 1 − ( ) ( ) v v cosθ ε ε − sin2 θ f vv =
Fvv
−2Rh cosθ
(3a) (3b) (3c) (3d)
where the horizontally and vertically polarized Fresnel reflection coefficients, Rh and Rv , are given by the following: √ cosθ − ε − sin2 θ √ Rh = (4a) cosθ + ε − sin2 θ √ ε cosθ − ε − sin2 θ √ Rv = (4b) ε cosθ + ε − sin2 θ where ε is the ground relative dielectric constant; a complex number concerns the storage and dissipation of electricity. In the IEM, s and the surface ACF are the two geometric elements whose calculations are the main issue considered in this paper.
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Due to the nonlinearity of (1), the model inversion, i.e., solving the IEM for its parameters analytically, is almost impossible. There are a few arithmetic methods which can be employed to calculate the surface parameters knowing the backscattering coefficient and the imaging parameters [12]. In spite of the development of intelligent computation methods such as neural networks and the Bayesian method, one of the best and most direct methods in this respect is the Look-up table (LUT) [18,19]. In the LUT method, the backscattering coefficient values for different values of the surface roughness parameters and dielectric constant are calculated using (1), and then, the surface parameters corresponding to the backscattering coefficients can be calculated by interpolation and reversed matching. Section 2.1 describes the validity range and Section 2.2 exhibits the conventional input parameters calculation of this model. 2.1. The IEM Validity Region The IEM is applicable on a wide range of surfaces, from smooth to rough. The IEM’s validity range, given by Fung et al. (1994), is: ks < 3 (5a) q (5b) (kl )(ks) < µ |ε| q (ks)2 cos2 θ √ exp − 2 × 0.46 kl (1 − sinθ ) 1 0.46kl
(5c)
where µ is a constant and its Gaussian value, and the exponential ACFs are 1.6 and 1.2, respectively [10,20]. Dierking (1999) has also estimated the validity range of the IEM for different frequencies, similar to the ones specified by Fung et al. [21]. Furthermore, different studies have concerned the real validity range of this model and it is mostly shown that the applicable validity domain of this model is more limited [18,22,23]. 2.2. Conventional Inputs for IEM The rms-height is calculated using the following formula (6): v " u u1 s=t N
!
N
∑
i =1
z2i
#
− Nz
2
∀z=
1 N
N
∑ zi
(6)
i =1
Moreover, W (n) is another geometric term in the IEM, i.e., the Fourier transform of the nth power of the surface ACF, A( x ), either its exponential, or Gaussian regression [24,25] (i.e., respectively (7)–(9)): N−j
A( x ) =
∑ i =1 z i z i + j ∑iN=1 z2i
(7)
A( x ) = e
−| x | l
(8)
A( x ) = e
− x2 l2
(9)
where l is the correlation length. Because of its dependence on the profile length, as well as the rms-height, the calculation of the correlation length has been considered as a difficult problem [9,26]. In this paper, both exponential and Gaussian ACFs are used for conventional implementation, which have presented a better estimation and have been frequently recommended [3,12,15]. In the calculation of the ACFs, the correlation length is considered as one third of the semivariogram range [27].
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3. Power-Law Inputs for the IEM Functions, like y, proportional to some power of the input x (i.e., y = x p ), are power-law functions. Power-law surfaces have the spectrum in the form of S( f ) = c/ f α (α: spectral slope). In natural surface modeling, unlike the conventional geometry, the random fractal geometry provides better results. The term “fractal”, in many cases, is considered as the same as “self-similar”. Natural phenomena are statistically self-similar; i.e., every part of their structure has statistical properties (the mean and the standard deviation) similar to those of the whole structure [28]. The surfaces with a power-law roughness spectrum over the interval 0 ≤ f ≤ ∞ are ideal random fractals. For such surfaces, the rms-height s, the correlation length l, and the ACF do not exist [21,29] Thus, the surface modeling must be realized within the limits of the spatial frequencies (i.e., sampling rates) f min ≤ f ≤ f max . This is a semi-self-affine surface [13,30]. In other words, using the fractal geometry, natural surfaces can be modeled through the power-law form within f min and f max . Yordanov et al. (2002) present a general expression for the ACF valid for the arbitrary topological dimension [31]. Since after the roughness measurement of the study sites, the geometric parameters are considered to be calculated on some arbitrary linear profiles, the ACF for a linear profile of the samples can be realised in the form of [32]: A( x ) = 2π D−1
Z k max k min
k D−1−α cos(kx )dk
(10)
where k is the wavenumber and α has a limited amount; D < α < D + 2, D is the topological dimension, and for the linear profile, it is considered one (D = 1) [29,33], representing the slope of the linear best-fit of the power spectral density (PSD) on a logarithmic scale [29,32]. It is noted that this linear best-fit must be applied to the trendless profile. In other words, the spectral slope (α) might not include the trend part of the PSD, the early part of the PSD [29]. According to Dierking (1999), in addition to the ACF, two other regular geometric surface parameters, the power-law rms-height and the correlation length for a linear profile, can also be calculated using the spectral slope parameter [21]: s=
q
cLα−1 /(α − 1)
l=
( α − 1)2 L 2(2α − 1)
(11) (12)
where L is the profile length, and c is the spectral constant or the spectral offset [21]. Hence, in order to evaluate the role of the random fractals for improving the IEM results, having ascertained the surface morphology, using (8) and (9), the ACF for conventional-1 and conventional-2 methods, respectively, is achieved. The rms-height (s) can be calculated using (6) for these two conventional methods. For the power-law calculation of the input parameters, the welch method is applied to the in situ measured data to calculate the spectral slope, α. Having obtained α and using (11), the rms-height (s) is computed. Using (10), the ACF is computed. 4. Implementation and Evaluation Three different methods for the calculation of the input parameters of the IEM are implemented for the in situ field measurement to apply the methodology of the flowchart (Figure 1a). Afterward, following the second flowchart (Figure 1b), the LUT can be formed to determine the surface roughness of the terrestrial points corresponding to each measured SAR backscattering (on L, C, and X-bands) pixels. The case study is located in western Iran, between the cities of Ilam and Dehloran. From the view point of geology, this region is known as the “Anaran” anticline, located in the Zagros fold-thrust belt. Figure 2 depicts the geographic location of the case study and the 10 measurement sites. Figure 2b
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depicts the main geological formations of the region; however, the morphology of two of them (Gurpi Fm and Quaternary Sediments) is not considered to have been studied in this paper. The field measurement is planned in the form of a mesh of points; thus, the geometric parameters for any arbitrary profiles on the surface can be calculated. The in situ surface roughness measurement is performed on ten different sites with different morphologies using the programmable surveying Total Station (Trimble™5600). This instrument is used to measure a 25 × 25 mesh with a 50 cm interval on each measurement site. The measurement instrument has a better than one centimeter precision. Being measured as a mesh of points allows for the calculation of roughness parameters along any Remote Sens. 2017, 9, 445 6 of 16 arbitrary profile and certainly increases the precision of the parameters estimation. Generally, there are three formations belonging to different geological periods. Table 1 shows the details of these are three formations belonging to different geological periods. Table 1 shows the details of these geological formations. geological formations.
Figure 2. (a) The case study location on a map of Iran; (b) Surface roughness measurement sites Figure 2. (a) The case study location on a map of Iran; (b) Surface roughness measurement sites position on the geological map (As, Asmari Fm; Gs, Gachsaran Fm; Pd, Pubdeh Fm; Gu, Gurpi Fm; position on the geological map (As, Asmari Fm; Gs, Gachsaran Fm; Pd, Pubdeh Fm; Gu, Gurpi Fm; Qu, Qu, Quaternary Sediments); (c) Measurement sites position on the X-Band SAR backscattering image; Quaternary Sediments); (c) Measurement sites position on the X-Band SAR backscattering image; the the color-bar of the values of backscattering is beside the image in db. color-bar of the values of backscattering is beside the image in db. Table 1. Geological age and general specification of the morphology of the study area formations. Table 1. Geological age and general specification of the morphology of the study area formations. Geological Formation Geological Period General Morphology Appearance Asmari (As) Oligocene to Lower Miocene Rough Geological Formation Geological Period General Morphology Appearance Gachsaran (Gs) early Miocene Moderate Asmari (As) Oligocene Miocene Rough Pabdeh (Pd) Paleoceneto toLower early Miocene Smooth Gachsaran (Gs) early Miocene Moderate Pabdeh (Pd) Paleocene to early Miocene Smooth
Furthermore, the SAR measured data are described in Table 2. Three SAR measurement data are used in three microwave bands. The acquisition modes are also specified. Furthermore, the SAR measured data are described in Table 2. Three SAR measurement data are Tablemicrowave 2. Specifications of the SARacquisition measurement data inare three different frequencies for the study. used in three bands. The modes also specified. Satellite/Antenna Acquisition Mode Band Freq. (GHz) Incidence Angle/Pass/Look Table 2. Specifications of the SAR measurement data in three different frequencies for the study. ALOS–Palsar-2 Spot-light L 1.200 32.3°/Ascending/Left Sentinel-1A Strip Map Mode C 5.405 38.1°/Ascending/Right Incidence TerraSAR-X Staring Spot-Light XBand 9.650 Satellite/Antenna Acquisition Mode Freq. (GHz)22.7°/Descending/Left Angle/Pass/Look
ALOS–Palsar-2 Spot-light 1.200 32.3◦ /Ascending/Left The earth surface roughness of a single studyLarea in each of the radar bands is seen differently, ◦ /Ascending/Right Sentinel-1A Strip Map Mode C 5.405 38.1 due to the difference in the operational frequency (or wavelength). Figure 3 depicts the face of the TerraSAR-X X they are affected 9.650 22.7◦ /Descending/Left formations surface andStaring showsSpot-Light the extent to which by weathering and erosion.
Obviously, this variety of roughness must not be seen as the same in different wavelengths. The smoother surfaces are more susceptible to study erosion, whereas rougher areismore resistant The earth surface roughness of a single area in eachthe of the radarones bands seen differently, against level to which afrequency lithology (or getswavelength). eroded is dependent its material and of due to theweathering. difference The in the operational Figure on 3 depicts the face its age. the formations surface and shows the extent to which they are affected by weathering and erosion.
Obviously, this variety of roughness must not be seen as the same in different wavelengths. The smoother surfaces are more susceptible to erosion, whereas the rougher ones are more resistant against weathering. The level to which a lithology gets eroded is dependent on its material and its age.
(a)
(b)
(c)
due to the difference in the operational frequency (or wavelength). Figure 3 depicts the face of the formations surface and shows the extent to which they are affected by weathering and erosion. Obviously, this variety of roughness must not be seen as the same in different wavelengths. The smoother surfaces are more susceptible to erosion, whereas the rougher ones are more resistant against The level to which a lithology gets eroded is dependent on its material and Remote Sens. weathering. 2017, 9, 445 7 of 16 its age.
(a)
(b)
(c)
Figure Thegeological geologicalformations formations surface. surface. (a) Miocene; Figure 3. 3.The (a) Asmari Asmari(As) (As)fm., fm.,Oligocene Oligoceneto toLower Lower Miocene; Gachsaran (Gs) fm.,early earlyMiocene; Miocene;(c) (c)Pabdeh Pabdeh (Pd) (Pd) fm., The extent to to (b)(b) Gachsaran (Gs) fm., fm., Paleocene Paleocenetotoearly earlyMiocene. Miocene. The extent which they are affected by weathering and erosion can be seen. The photos are taken from ~1 m above which they are affected by weathering and erosion can be seen. The photos are taken from ~1 m above the surface. the surface. Remote Sens. 2017, 9, 445 7 of 16
mustbebeconsidered consideredthat thatprior prior to to any any backscattering backscattering processing, asas results assessment, It It must processing,asaswell well results assessment, the SAR imagery data introduced in Table 2, which are contaminated by the speckle noise, were the SAR imagery data introduced in Table 2, which are contaminated by the speckle noise, were subjected to a pre-processing stage, i.e., despeckling using the wavelet transform method [34]. subjected to a pre-processing stage, i.e., despeckling using the wavelet transform method [34]. Figure 4 illustrates how the roughness of the formations appears in the SAR images of different Figure 4 illustrates how the roughness of the formations appears in the SAR images of different frequencies. The backscattering in the images is shown in the equalized black and white images, but frequencies. The backscattering in the images is shown in the equalized black and white images, but the side color bars show the pixel values in db. The whiter the images, the rougher the face because the side color bars show the pixel values in db. The whiter the images, the rougher the face because of double, triple, and in general, multi-scattering signals. The blacker pixel values are because of the of double, triple, and in general, multi-scattering signals. The blacker pixel values are because of the smoothness of the surface. smoothness of the surface. 20
-10 0
0
-20
-20 -20
-30
-40
-40
-60
-40
-80
-50 -60
-100
-60
-120
-80
(a)
(b)
(c)
Figure Backscattering SAR measurement, as equalized black white images; (a)L-Band the Figure 4. 4. Backscattering onon thethe SAR measurement, as equalized black and and white images; (a) the L-Band SAR data, the ALOS-PALSAR satellite; (b) the C-Band SAR data, the Sentinel-1A satellite; (c) SAR data, the ALOS-PALSAR satellite; (b) the C-Band SAR data, the Sentinel-1A satellite; (c) the X-Band thedata, X-Band data, thesatellite; TerraSAR-X satellite; the value of backscattering is depicted SAR theSAR TerraSAR-X the value of backscattering is depicted beside eachbeside imageeach on the image on in db. i.e., Lower values, i.e.,show darksmoothness pixels, showand smoothness large values of color-bar in the db.color-bar Lower values, dark pixels, large dband values aredbbecause because of the surface roughness. theare surface roughness.
The numerical geometric characteristics on each site surface are tabulated in Table 3. The The numerical characteristics on each are tabulated 3.theThe rms-height and the geometric correlation length are calculated usingsite the surface conventional geometry;in theTable and rms-height and the correlation length are calculated using the conventional geometry; the α and fractal dimension using the power-law geometry. the fractal dimension the power-law geometry. Since there was using a considerably dry climatic condition when the satellite data was being acquired, Since there was a considerably dry climatic the satellite data it was being acquired, and the field measurement was performed incondition the samewhen seasonal conditions, was obviously and the fieldthat measurement was performed in the seasonal conditions, was perceived perceived the moisture until 5 cm below the same surface was absolutely 0%; it that is obviously why the dielectric that the moisture until used 5 cm from belowthe thereferences surface was is why dielectric constant values were whoabsolutely presented 0%; thesethat values for the dry climateconstant [14]. The values for the three geological formations of the study i.e.,for Asmari, Gachsaran, and The Pabdeh, values were used from the references who presented thesearea, values the dry climate [14]. values are 3.6, 4.0, and 4.1, respectively. for the three geological formations of the study area, i.e., Asmari, Gachsaran, and Pabdeh, are 3.6, 4.0, and 4.1, respectively. Table 53. illustrates Morphological andthe surface geometric of the study on the site visit Figure how spectral slopecharacteristics (α) is calculated usingsites thebased in situ measurements of and the in situ measurement. The roughness parameters are calculated using the equations and the PSD of the 10 selected study sites. The PSDs are estimated using the Welch’s power spectral methods presented in Section 3. Figure 5 shows how is estimated for each site surface. The Rmsdensity [35,36]. According to the previous section, the data trend or the result of large topography height and the correlation length are the conventional ones and the average of the multiple arbitrary must not be considered in the measurement of the spectral slope (α). For this aim, the red line profiles on each of the site surfaces. (i.e., a linear regression) in the plots of Figure 3 is drawn on a polynomial regression of the PSD, Site No. the Dominant Geological Formation (cm) Corr. Lengthobviously (cm) notwithstanding general trend part of the PSD.rms-Height The experimental analysis shows that Site 1 Site 2 Site 3 Site 4 Site 5 Site 6 Site 7
As Gs As-Gs Pd Pd Gs As
6.02 2.2 4.64 1.11 1.21 1.98 4.97
81.07 45.3 54.07 14.9 18.03 40.71 74.38
1.5345 1.7112 1.6865 1.7923 1.8825 1.8817 1.5349
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the regression of the PSD for frequencies between 0.85 and 3.15 (Hz) results in the best values for the spectral slope. This empirically decided frequency range is somehow equal for different sites. Table 3. Morphological and surface geometric characteristics of the study sites based on the site visit and the in situ measurement. The roughness parameters are calculated using the equations and methods presented in Section 3. Figure 5 shows how α is estimated for each site surface. The Rms-height and the correlation length are the conventional ones and the average of the multiple arbitrary profiles on each of the site surfaces. Site No.
Dominant Geological Formation
rms-Height (cm)
Corr. Length (cm)
α
Site 1 Site 2 Site 3 Site 4 Site 5 Site 6 Site 7 Site 8 Site 9 Site 10
As Gs As-Gs Pd Pd Gs As Gs Gs Qu
6.02 2.2 4.64 1.11 1.21 1.98 4.97 1.66 2.86 1.76
81.07 45.3 54.07 14.9 18.03 40.71 74.38 19.31 50.9 31.31
1.5345 1.7112 1.6865 1.7923 1.8825 1.8817 1.5349 1.7501 1.7709 1.7472
Table 4 shows the capability of each radar frequency for the discrimination of each formation. The conformity in this table is determined according to the contents of Tables 2 and 3. Table 4. The status of the validity of the IEM based on (5a,b,c) for each geological formation on each of the study bands, based on Fung et al. (1994) [10]. Check marks show the conformity of the formation roughness parameters (on Table 3) to the IEM constraints at each microwave band and the X marks show the non-conformity.
Geological Formation Asmari Fm. Gachsaran Fm. Pabdeh Fm.
L-Band (5a) √ √ √
(5b) √
× ×
C-Band (5c) √ √ √
(5a) √ √ √
(5b) √ √ √
X-Band (5c) √ √ √
(5a)
(5b)
(5c)
× √ √
× × √
× √ √
In this table, it becomes clear which of the three constraints determined by Fung et al. (1994) via (5a,b,c) are approved or rejected in order to morphologically differentiate the formations. In this table, √ the approbation ( ) of an equation indicates the validity of the constraint for the relevant band and the rejection (×) of the equation means the invalidity of the constraint. 4.1. Geometric Parameters Analysis The microwave signal, depending on the radar frequency, is sensitive to a limited range of the roughness spectrum; thus, a surface seen as being rough in one frequency measurement may be seen as completely smooth in another. Figure 6 represents the backscattering of the incident radar signal in the radar bands L, C, and X on hh, vv, and hv polarizations for the earth surface of the main geological formations present in the field measured sites. As the frequency or the local incident angle increases, the smaller roughness scale becomes significant. The graphs confirm the different capability of the operational frequencies for the surfaces discrimination, presented in Table 4.
must not be considered in the measurement of the spectral slope ( ). For this aim, the red line (i.e., a linear regression) in the plots of Figure 3 is drawn on a polynomial regression of the PSD, notwithstanding the general trend part of the PSD. The experimental analysis obviously shows that theSens. regression of the PSD for frequencies between 0.85 and 3.15 (Hz) results in the best values for 9the Remote 2017, 9, 445 of 16 spectral slope. This empirically decided frequency range is somehow equal for different sites. Site1
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Figure 5. Calculation spectral slope ) via linear regression a polynomial regressionofofthe Figure 5. Calculation of of thethe spectral slope (α)( via thethe linear regression ofof a polynomial regression the PSD, calculated from the in situ measurements of the 10 selected study sites using the Welch’s PSD, calculated from the in situ measurements of the 10 selected study sites using the Welch’s method. method. Experimentally, the trend of the data is between 0 and 0.85 Hz, which has not been Experimentally, the trend of the data is between 0 and 0.85 Hz, which has not been considered in the considered calculation calculation of in thethe spectral slope.of the spectral slope.
Table 4 shows the capability of each radar frequency for the discrimination of each formation. The conformity in this table is determined according to the contents of Tables 2 and 3. Table 4. The status of the validity of the IEM based on (5a,b,c) for each geological formation on each of the study bands, based on Fung et al. (1994) [10]. Check marks show the conformity of the formation
for each site. Moreover, using the IEM inversion, by the values of ° on a 25 × 25 pixels SAR image, a 25 × 25 matrix of the rms-height can be calculated for each of the sites. The statistical rms-height, as well as the ACF, are calculated beside the extracted dielectric constant of each site from the published measured values to compute the conventional IEM backscattering coefficient, i.e., the values of ° and ° . The power-law IEM is computed using the Remote Sens. 2017, 9, 445 10 of 16 ACF and the rms-height is calculated using the spectral slope parameter. L-Band
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Figure Backscattering(σ( )° )for forthree three bands bands L, C, forfor thethe main Figure 6. 6. Backscattering C, and and XX on onhh, hh,vv, vv,and andhvhvpolarizations polarizations main geological formationsofofthe thestudy studyarea. area. geological formations ◦
Simulated IEM (σ°) [dB]
Simulated IEM (σ°) [dB]
Simulated IEM (σ°) [dB]
Figures 7–9, respectively, for the bands L, C, and X, present the comparisons between the 4.2. Evaluation of Backscattering and Surface Roughness Simulation backscattering coefficients calculated through the three methods of calculation of the IEM input What is needed discriminate geological formations according their morphology surface parameters, namelytotwo “Conventional” methods (exponential and to Gaussian ACFs) and is a the “Powerroughness [25]. In section, the of the IEM implementation to simulatereferred the backscattering law” method, in this comparison withresults the measured SAR backscattering coefficient, to as an ◦ “SAR Measurement”. This comparison is performed on 30 with randomly selected pixels for 10 measured coefficient (σ ) using the surface data and a comparison the SAR measured ones in the radar sites.L, Precisely point on theroughness diagonal line of each graph indicates that the IEM value bands C, and being X for athe surface computation is described (Figure 1a).simulated Via inverse IEM on the pixel is exactly equalmeasurements, to the measuredthe backscattering onthe that pixel. Therefore, using thecorresponding LUT, having ascertained the SAR rms-height of surface is simulated. these graphs, thesimulated farness of the diagonal line shows theperformed simulation using error [25,37]. Aninevaluation of the surface roughness can be the rms-height of the field measurements (Figure 1b). Conventional Inputs-1 Conventional Inputs-2 Power-Law Inputs Having obtained the surface geometric characteristics for a 25 × -15 -15 -15 25 points mesh and then, taking Asmari fm. ◦ Gachsaran fm. arbitrary profiles according to geological-17 maps and the site visit, a -17matrix 25 × 25 of σ is computed -17 Pabdeh fm. ◦ for each site. Moreover, using the IEM inversion, by the values of σ on a 25 × 25 pixels SAR image, a -19 -19 -19 25 × 25 matrix of the rms-height can be calculated for each of the sites. -21 -21 The statistical rms-height, as well as -21 the ACF, are calculated beside the extracted dielectric constant -23 -23 -23 of each site from the published measured values to compute the conventional IEM backscattering ◦ ◦ coefficient, i.e., the values of σhh and σvv-25. The power-law IEM is computed using the ACF and the -25 -25 -25 -23 -21 -19 -17 -15 -25 -23 -21 -19 -17 -15 -25 -23 -21 -19 -17 -15 SAR Measurement (σ°) [dB] SAR Measurement (σ°) [dB] SAR Measurement (σ°) [dB] rms-height is calculated using the spectral slope parameter. (a) (b) (c) Figures 7–9, respectively, for the bands L, C, and X, present the comparisons between the backscattering coefficients calculated through three of calculation IEM input Figure 7. Backscattering simulation accuracy viathe the IEM formethods the geological formations onof thethe L-band, parameters, namely two “Conventional” methods (exponential andusing Gaussian ACFs) andgeometry a “Power-law” (a) using the conventional geometry and the exponential ACF; (b) the conventional method, in the comparison with the measured SAR backscattering coefficient, referred to as an “SAR and Gaussian ACF; (c) using the Power-Law geometry for the calculation of the IEM input parameters. Measurement”. This comparison is performed on 30 randomly selected pixels for 10 measured sites. Precisely being a point on the diagonal line of each graph indicates that the IEM simulated value on the corresponding pixel is exactly equal to the measured backscattering on that pixel. Therefore, in these graphs, the farness of the diagonal line shows the simulation error [25,37].
parameters, namely two “Conventional” methods (exponential and Gaussian ACFs) and a “Powerlaw” method, in comparison with the measured SAR backscattering coefficient, referred to as an “SAR Measurement”. This comparison is performed on 30 randomly selected pixels for 10 measured sites. Precisely being a point on the diagonal line of each graph indicates that the IEM simulated value on the corresponding pixel is exactly equal to the measured backscattering on that pixel. Therefore, Remote Sens. 2017, 9, 445 11 of 16 in these graphs, the farness of the diagonal line shows the simulation error [25,37]. Conventional Inputs-1
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Figure 7. 7.Backscattering via the theIEM IEMfor forthe thegeological geological formations L-band, Figure Backscatteringsimulation simulation accuracy accuracy via formations onon thethe L-band, Remote Sens. 2017, 9, 445 11 of 16 (a)(a) using the conventional geometry and the exponential ACF; (b) using the conventional geometry using the conventional geometry and the exponential ACF; (b) using the conventional geometryand Remote Sens. 2017, 9, 445 11 of 16 the Gaussian ACF; (c) ACF; using the geometry forgeometry the calculation the IEM input and the Gaussian (c) Power-Law using the Power-Law for theof calculation of parameters. the IEM Conventional Inputs-1
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Gachsaran fm. Conventional Inputs-1 Pabdeh fm. Asmari fm. Gachsaran fm. Pabdeh fm.
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Figure 8. Backscattering simulation accuracy via the IEM for the geological formations on the C-band, (a) using the conventional geometryaccuracy and the exponential ACF; (b) using the conventional geometry Figure 8. 8. Backscattering simulation via the IEM IEMfor forthe the geological formations the C-band, Figure Backscattering simulation accuracy via the geological formations onon the C-band, and the Gaussian ACF; (c) using the Power-Law geometry for the calculation of the IEMand (a)(a) using thethe conventional geometry and thethe exponential ACF; (b)(b) using thethe conventional geometry using conventional geometry and exponential ACF; using conventional geometry input parameters. theand Gaussian ACF; (c) ACF; using the geometry forgeometry the calculation of calculation the IEM input the Gaussian (c) Power-Law using the Power-Law for the of parameters. the IEM input parameters. Conventional Inputs-1
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Figure accuracyvia viathe theIEM IEMfor forthe the geological formations X-band, Figure9.9.Backscattering Backscattering simulation simulation accuracy geological formations on on the the X-band, (a)(a) using the conventional geometry ACF; (b) using conventional geometry and using conventionalsimulation geometryand andthe theexponential exponential ACF; usingthe the conventional geometry Figure 9.the Backscattering accuracy via the IEM for the(b) geological formations on the X-band, the Gaussian ACF; (c)ACF; using(c) the using Power-Law geometry for the calculation ofcalculation the IEM input parameters. and the Gaussian the Power-Law geometry for the of the IEM (a) using the conventional geometry and the exponential ACF; (b) using the conventional geometry input and parameters. the Gaussian ACF; (c) using the Power-Law geometry for the calculation of the IEM
For an parameters. assessment of the aforementioned calculation methods, it is necessary to compare the input For an assessment of the aforementioned calculation methods, it is necessary to compare the simulated surface roughness parameter (rms-height) resulting from the inverse IEM (i.e., extracted simulated surface roughness parameter (rms-height) resulting from the IEM (i.e., extracted For an assessment the calculation methods, isinverse necessary compare thethe from Look-up table) withofthe inaforementioned situ measured parameter for the tenitsites. For thistocomparison, from Look-up table) with the in situ measured parameter for the ten sites. For this comparison, the simulated surface roughness parameter (rms-height) resulting from the inverse IEM (i.e., extracted point graphs of Figures 10–12 depict the comparison of the computed rms-height of 30 selected pixels point graphs of table) Figures 10–12 depict comparison of the computed of 30comparison, selected pixels Look-up with themeasured in situthe measured parameter for the tenrms-height sites. this the onfrom bands L, C, and X, and the values on their corresponding fieldFor points. on bands L, C, and X, and the measured values on their corresponding field points. point graphs of Figures 10–12 depict the comparison of the computed rms-height of 30 selected pixels Precisely being aapoint on diagonalline line ofeach eachgraph graphindicates indicates that the simulated rms-height Precisely on the the diagonal the simulated rms-height on bands L, C,being and X,point and the measured valuesof on their correspondingthat field points. using the inverse IEM on the corresponding pixel is exactly equal to the field measured rms-height. usingPrecisely the inverse IEM on theon corresponding pixel exactly equal to thethat field being a point the diagonal line of is each graph indicates themeasured simulatedrms-height. rms-height Evidently, the proximity of the points to the diagonal diagonalline lineannounces announcesthe thesimulation simulation correctness [37]. Evidently, the proximity of the points to the correctness [37]. using the inverse IEM on the corresponding pixel is exactly equal to the field measured rms-height. Evidently, the proximity of the points to the diagonal line announces the simulation correctness [37]. Conventional Inputs-1
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from Look-up table) with the in situ measured parameter for the ten sites. For this comparison, the point graphs of Figures 10–12 depict the comparison of the computed rms-height of 30 selected pixels on bands L, C, and X, and the measured values on their corresponding field points. Precisely being a point on the diagonal line of each graph indicates that the simulated rms-height using the inverse IEM on the corresponding pixel is exactly equal to the field measured rms-height. Remote Sens. 2017, 9, 445 12 of 16 Evidently, the proximity of the points to the diagonal line announces the simulation correctness [37]. Conventional Inputs-1
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Figure simulationaccuracy accuracyvia viathe theinverse inverse IEM geological formations Figure10. 10.The TheRms-height Rms-height simulation IEM forfor thethe geological formations on on the conventionalgeometry geometryand andthe theexponential exponential ACF; using conventional theL-band, L-band,(a) (a)Using Using the conventional ACF; (b)(b) using thethe conventional geometry ACF; (c) (c) using usingthe thePower-Law Power-Lawgeometry geometry calculation of the geometryand and the the Gaussian Gaussian ACF; forfor thethe calculation of the IEMIEM Remote Sens. 2017, 9, 445 12 of 16 Remote Sens. 2017, 9, 445 12 of 16 input inputparameters. parameters.
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Figure 11. The Rms-heightsimulation simulationaccuracy accuracyvia viathe the inverse IEM for the geological formations onon Figure 11. The Rms-height accuracy via theinverse inverseIEM IEMfor for the geological formations Figure 11. The Rms-height simulation the geological formations on the C-band, (a) Using the conventional geometry and the exponential ACF; (b) using the conventional the C-band, geometryand andthe theexponential exponential ACF; using conventional the C-band,(a) (a)Using Usingthe the conventional conventional geometry ACF; (b)(b) using thethe conventional geometry and theGaussian GaussianACF; ACF;(c) (c)using usingthe thePower-Law Power-Lawgeometry geometry for the calculation ofthe thethe IEM geometry and the using the Power-Law geometry for the calculation of IEM geometry and the Gaussian ACF; for the calculation of IEM input parameters. input inputparameters. parameters.
Simulated RMS [cm] Simulated RMS [cm]
1.75 1.75
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Asmari fm. Asmari fm. fm. Gachsaran Gachsaran fm. Pabdeh fm. Pabdeh fm.
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Figure 12.12. The Rms-height accuracy via the inverse inverseIEM IEMfor forthe thegeological geologicalformations formations Figure The Rms-heightsimulation simulationaccuracy accuracyvia viathe the onon Figure 12. The Rms-height simulation inverse IEM for the geological formations on thethe X-band, (a) using the conventional geometry and the exponential ACF; (b) using the conventional X-band,(a) (a)using usingthe theconventional conventionalgeometry geometryand andthe theexponential exponentialACF; ACF;(b) (b)using usingthe theconventional conventional the X-band, geometry and the Gaussian ACF; (c) using the Power-Law geometryfor forthe thecalculation calculation of IEM geometry and the Gaussian ACF; (c) using the Power-Law geometry ofthe thethe IEM geometry and the Gaussian ACF; (c) using the Power-Law geometry for the calculation of IEM input parameters. inputparameters. parameters. input
InTable Table5,5,5,the thestandard standard deviation deviation of of the the calculated calculated rms-height rms-height ininthe the methods of of IEM InIn of the calculated rms-heightin themethods methods IEM Table the standard deviation of IEM implementation are tabulated. Therefore, the statistical dispersion of the calculated results on each implementationare aretabulated. tabulated. Therefore, Therefore, the the calculated results on on each implementation thestatistical statisticaldispersion dispersionofof the calculated results each sitefor foreach eachpolarization polarization versusthe the fieldmeasured measuredvalues valuesare arecomparable, comparable,in inorder orderto toevaluate evaluatethe the site site for each polarization versus versus thefield field measured values are comparable, in order to evaluate efficiency of the proposed power-law calculation of the inputs instead of the conventional one. efficiency of of thethe proposed power-law calculation of the inputs instead of the conventional one.one. the efficiency proposed power-law calculation of the inputs instead of the conventional Evidently, the smaller the standard deviation, the higher the accuracy of the simulated data. Evidently,the thesmaller smallerthe thestandard standarddeviation, deviation, the data. Evidently, the higher higherthe theaccuracy accuracyofofthe thesimulated simulated data. Table5.5.The Thestandard standarddeviation deviationof ofthe therms-height, rms-height,calculated calculatedusing usingthe theinverse inverseIEM IEMfor forthe the Table bands L, C, and X, using three methods of inputs calculation, two geological formations on geological formations on bands L, C, and X, using three methods of inputs calculation, two
conventional(1: (1:The Theexponential exponentialACF; ACF;2:2:The TheGaussian GaussianACF) ACF)and andthe thepower-law power-lawgeometry. geometry.The Thebold bold conventional values show a decrease of the standard deviation of using the power-law inputs versus both the values show a decrease of the standard deviation of using the power-law inputs versus both the conventional ones. conventional ones.
L-Band L-Band As. Gs. As. Gs. Inputs using the Conventional Geometry1 1.049 0.919 Inputs using the Conventional Geometry1 1.049 0.919 Inputs using the Conventional Geometry2 0.985 0.799 Inputs using the Conventional Geometry2 0.985 0.799 0.839 0.721 Inputs using the Power-law Geometry Geological Formation Geological Formation
C-Band C-Band Pd. As. Gs. Pd. As. Gs. 0.873 0.352 0.378 0.873 0.352 0.378 0.834 0.375 0.309 0.834 0.375 0.309 0.970 0.326 0.370
X-Band X-Band Pd. As. Gs. Pd. As. Gs. 0.505 0.207 0.239 0.505 0.207 0.239 0.506 0.232 0.283 0.506 0.232 0.283 0.512 0.224 0.240
Pd. Pd. 0.341 0.341 0.355 0.355 0.285
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Table 5. The standard deviation of the rms-height, calculated using the inverse IEM for the geological formations on bands L, C, and X, using three methods of inputs calculation, two conventional (1: The exponential ACF; 2: The Gaussian ACF) and the power-law geometry. The bold values show a decrease of the standard deviation of using the power-law inputs versus both the conventional ones. L-Band
Geological Formation Inputs using the Conventional Geometry1 Inputs using the Conventional Geometry2 Inputs using the Power-law Geometry
C-Band
X-Band
As.
Gs.
Pd.
As.
Gs.
Pd.
As.
Gs.
Pd.
1.049 0.985 0.839
0.919 0.799 0.721
0.873 0.834 0.970
0.352 0.375 0.326
0.378 0.309 0.370
0.505 0.506 0.512
0.207 0.232 0.224
0.239 0.283 0.240
0.341 0.355 0.285
Comparing the values of the standard deviation in Table 5 shows that the use of random fractals geometry has improved the results by more than 10% on average, i.e., dividing the largest standard deviation values of the two conventional methods by the standard deviation of the Power-law method canRemote give such a value. Sens. 2017, 9, 445The bold numbers are used to indicate that the Power-law geometry overcomes 13 of 16 both of the conventional methods. For both co-polarizations, the same improvement is achieved. Figure 13 standard deviation values on the to easily the different 13illustrates illustratesthethe standard deviation values onbar-charts the bar-charts tocompare easily compare the frequencies, as well asas the different different frequencies, well as the formations. different formations. L-Band
1.4 1.2 1
Pow er-law IEM
1.049 0.985 0.919 0.839
0.8
0.799 0.721
0.97
0.4
0.2
0.2
0.1
0
0
Gachsaran
(a)
0.505 0.506 0.512
0.5
0.3
Pabdeh
0.355 0.341
0.35
0.873 0.834
0.4
X-Band
0.4
0.6
0.6
Asmari
C-Band
0.7 Conventional IEM-1 Conventional IEM-2
0.3 0.25
0.375 0.352 0.326
0.378 0.369 0.37
0.2
0.283 0.232 0.224
0.239
0.285
0.24
0.207
0.15 0.1 0.05 Asmari
Gachsaran
(b)
Pabdeh
0
Asmari
Gachsaran
(c)
Pabdeh
Figure 13. 13. The The Standard Standard Deviation Deviation of of the the calculated calculated rms-height rms-height values values using using the the IEM IEM Look-up Look-up table table Figure calculated via three methods methods of of inputs inputs calculation, calculation, conventional-1: conventional-1: using the calculated via three using the the exponential exponential ACF, ACF, the conventional-2: using the Gaussian ACF, and the power-law geometry for the geological formations conventional-2: using the Gaussian ACF, and the power-law geometry for the geological formations of of the study sites, using backscattering coefficient L-band; C-band; X-band. the study sites, using thethe backscattering coefficient on on (a) (a) L-band; (b)(b) C-band; andand (c) (c) X-band.
5. Discussion 5. Discussion In this thispaper, paper,the theIntegral Integral Equation Model (IEM) backscattering model, as as well as its inverse In Equation Model (IEM) backscattering model, as well its inverse (using (using the Look-up table method), are implemented to measure the geometry of the earth surface the Look-up table method), are implemented to measure the geometry of the earth surface roughness roughness or morphology. For such implementation, the input parameters are calculated using three or morphology. For such implementation, the input parameters are calculated using three different different methods: two conventional geometries (the exponential and the Gaussian correlation methods: two conventional geometries (the exponential and the Gaussian correlation functions, the functions, ACFs) and onemethod. power-law The implementation of the fractal geometry in this ACFs) andthe one power-law Themethod. implementation of the fractal geometry in this paper is paper is performed through the spectral slope parameter (α), which shows the coincidence of the performed through the spectral slope parameter (α), which shows the coincidence of the power-law power-law ACF with the earth surface. The(i.e., IEM (i.e., °, coefficients) backscattering are ACF with the earth surface. The IEM results σ◦results , backscattering arecoefficients) compared with compared with the Synthetic Aperture Radar (SAR) measurement (i.e., TerraSAR-X) and the inverse the Synthetic Aperture Radar (SAR) measurement (i.e., TerraSAR-X) and the inverse IEM results (i.e., IEM results (i.e., werethe compared with the in situ measurement. rms-height) wererms-height) compared with in situ measurement. The fractal of the of the The fractal nature nature of the roughness roughness causes causes aa higher higher efficiency efficiency of the power-law power-law geometry geometry versus versus the conventional geometry when modeling the surface; Figures 7–12 show better results for the the conventional geometry when modeling the surface; Figures 7–12 show better results for the fractal fractal modeling and andeliminate eliminate uncertainties, affirming [21,38] Comparing studies. Comparing the two modeling moremore uncertainties, affirming [21,38] studies. the two conventional conventional geometry implementations, from aswavelength, the moderate the geometry implementations, apart from theapart C-band as the the C-band moderate thewavelength, Gaussian ACF Gaussian ACF causes less deviation for higher wavelengths, and the exponential ACF results are causes less deviation for higher wavelengths, and the exponential ACF results are better for lower better for lower supports the deductions presented in [12,37,39]. The values of the wavelengths; thiswavelengths; supports the this deductions presented in [12,37,39]. The values of the standard deviation standard deviation in Table 5 show a general improvement, but when considering the behavior of in Table 5 show a general improvement, but when considering the behavior of different geological different geological surfaces with various levels of roughness on each of the SAR frequency image, we deduce that for lower frequencies (e.g., L-band), the rougher surfaces are in the fractal regime, confirming the conclusions presented in [21,40]; and for the higher frequencies (e.g., X-band), the smoother ones are in accordance with [6]. Therefore, the rougher surfaces in high frequencies and the smoother ones in low frequencies display an obvious diffractal behavior. In general, having the in
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surfaces with various levels of roughness on each of the SAR frequency image, we deduce that for lower frequencies (e.g., L-band), the rougher surfaces are in the fractal regime, confirming the conclusions presented in [21,40]; and for the higher frequencies (e.g., X-band), the smoother ones are in accordance with [6]. Therefore, the rougher surfaces in high frequencies and the smoother ones in low frequencies display an obvious diffractal behavior. In general, having the in situ measurements, applying the most famous backscattering model, and completing an evaluation with the SAR measured data, the results of significant studies such as [17,38,41,42] are somehow endorsed with comparable numerical results, showing the efficiency of applying the power-law methods for IEM implementation for the SAR backscattering studies. 6. Conclusions Using the random fractal geometry via the implementation of the power-law model in the backscattering modeling offers results which are up to 10% better for the calculation of the surface geometry. This methodology can improve the results of the spectral processings in the morphological mapping of the formations. Additionally, using the achieved information of morphology, we improved the scale of the geological maps. For each radar frequency, the specific size of the surface roughness can be measured via signal backscattering. Using the three SAR bands L, C, and X for the earth surface of three known-geological surfaces shows the capability of this technology in geological morphology mapping. The selection of the sites in this paper was undertaken considering that no large-scale topography has been inside the sampling mesh. Moreover, for the calculation of the spectral slope parameter (α), the trend part of the Power Spectral Density (PSD) graph has been ignored, and the spectral slope line was drawn on a polynomial regression of the PSD, which experimentally offered more accurate results. A proper estimation of the surface roughness spatial frequency and consequently the spectral slope α, using the power-law ACF, offered appropriate IEM input parameters computation; therefore, after running the model, a more precise backscattering coefficient which exhibited a lower standard deviation was achieved. Also, using the random fractal geometry, the model inversion with the Look-up table method offered better roughness approximation. However, the geometric behavior of the surface roughness against the SAR frequencies is not constant, i.e., as the micro-topography decreases, the fractal regim exists in higher frequencies, and vice versa. Acknowledgments: The authors would like to thank the University of Tehran Vice Chancellor for the Research and also the Research and Development department of NIOC Exploration Directorate for the support of this paper. All the respectable reviewers are acknowledged for their fruitful comments and suggestions about the paper. Author Contributions: A.G., J.A., M.D., and M.A.K. contributed to conception. A.G. and M.A.K. carried out the acquisition, analysis, and interpretation of data. A.G. drafted the manuscript. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare that they have no competing interests.
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