#### 2.1. Direct Model

In this section, the rationale for expressing the link between SAR image intensity and the local slopes of the surface is presented. The proposed direct model is divided in three parts: first, the natural surface under study is properly modeled using fractal geometry; in particular, a fBm is used. Actually, to cope with the non-differentiability of the fBm process and the limited range of fractalness of natural surfaces, a smoothed version of the original fBm process must be introduced (physical fBm).

The second step consists in adopting a scattering model; i.e., a method for the evaluation of the field scattered from the previously modeled natural surface. In this step, a relationship between the backscattering coefficient and the geometrical (in particular, local slopes) and electromagnetic parameters of the surface is derived.

The last step consists in the determination of a model for the SAR image, linking the measured intensity map to the backscattering coefficient and to the local slopes of the surface. This complete model-based (surface, scattering and imaging models) approach is a key feature to properly devise an inversion procedure allowing to somehow circumvent the ill-posedness of the problem and physically justify information extraction from even a single SAR image. The proposed model is in sharp contrast with other existing techniques in which only the imaging models, mostly derived from optics, are used [

10,

16] and only sparse considerations on electromagnetic scattering are made [

33].

#### 2.1.1. Surface Model

Several models describing natural surfaces’ roughness have been presented. Furthermore, there is increasing experimental evidence that the fractal geometry represents the most appropriate mathematical environment to describe the shape of natural surfaces [

34,

35,

36]; in addition, extensive literature deals with fractal scaling of natural surface roughness [

37,

38,

39]; furthermore, fractal approaches provide a description of the surface with a minimum number of independent parameters [

40]. One of the reasons for this success is the ability of fractal models to properly account for the statistical scale-invariance properties (in particular, self-affinity) of natural surfaces. Analytic models reported here are able to handle a variety of natural surfaces: bare and moderately vegetated soils, as well as ocean surfaces. In particular, we adopt here a (topological) 2-D fBm stochastic process

z(

x,

y) defined as follows [

40]:

where Pr{} stands for “probability”,

$\overline{\zeta}$ is the considered height increment,

z(

x,y) is the surface elevation and

is the distance between the two considered points of coordinates (

x,

y) and (

x′,

y′) and

H: Hurst coefficient (0 < H < 1) related to the fractal dimension D = 3 − H;

T: topothesy [m], i.e., the distance over which chords joining points on the surface have a surface-slope mean-square deviation equal to unity.

Fractal geometry exhibits self-affinity properties at any scale and does not allow the derivative operation at any point. Therefore, mathematical fractals cannot be directly applied in scattering problems. However, natural surfaces are observed, sensed, measured, and represented via instruments that are, for their intrinsic nature, band-limited. In other words, no actual natural surface holds property (1) at every scale, and some properties of fBm mathematical surfaces may be relaxed [

40].

#### 2.1.2. Scattering Model

The second step is the selection of a scattering model. Here we consider the SPM since it provides analytical closed-form equations, showing a range of validity adequate to SAR applications. Indeed, this method provides a very simple relation between fractals parameters and the backscattered field.

Considering a monostatic configuration and assuming that the surface can be described as a physical fBm, the backscattering coefficient under SPM can be expressed as [

41]

wherein

m and

n stand for the transmitted and received polarization (horizontal or vertical), respectively,

k is the electromagnetic wavenumber of the incident field.

S_{0} characterizing the spectral behavior of the physical fBm surface, is expressed in [m

^{(−2−2H)}] and related to

T and

H [

40];

${\beta}_{mn}$, accounting for the incident and reflected fields polarization, is a function of both the complex dielectric constant

ε_{r} of the surface and the local incidence angle

θ [

40]. Note that, with this model, we are able to deal only with the co-polarized case; in this case, it is possible to consider the term |

${\beta}_{mn}$|

^{2} constant with

θ in the angular interval of interest.

Figure 1 shows a graphical comparison between the proposed fractal scattering model and the Lambertian one. The strong non-linearity of the proposed fractal model can be appreciated; unlike Lambertian law, it is also not upper-bounded, thus forecasting an infinite energy scattered in normal incidence conditions. This behavior, clearly in contrast with the physics of scattering phenomena, is due to the approximation used to obtain the SPM solution, which is applicable only in a small slope regime. However, both the scattering models go to zero under grazing incidence conditions.

#### 2.1.3. Imaging Model

In this section, we derive a relationship between the SAR intensity image and the slopes of the observed surface in order to apply the SfS algorithm. The following model for the intensity SAR image is used [

42]:

where

I is the SAR intensity image and

G is a calibration constant.

Substituting (3) in (4), we obtain the desired relationship between the intensity map

I and the range and azimuth slopes

In order to express the intensity image in terms of the local surface slopes, let us introduce the partial derivatives of the surface z(x,y) along ground-range (p) and azimuth (q) directions

Hence, it is necessary to link the local incidence angle

θ to the local slopes

p and

q; it can be easily noted that

θ is the angle between the propagation unit vector and the surface normal unit vector, so that it can be expressed as:

Expressing also

$\mathrm{sin}\theta $ in terms of local slopes, the backscattering coefficient

${\sigma}_{mn}^{0}$ can be rewritten in the following form:

Finally, the SAR image intensity can be linked to the local slopes as follows:

Incidentally we note that the direct model presented in (9) clarifies the ill-posedness of the SfS problem (two unknowns,

p and

q, to be retrieved from (9)) and identifies the numerous parameters (linked to both the sensor and the surface) on which SAR imagery depends. The use of analytical models for both scattering phenomena and surface shape makes us more aware on how these parameters affect SAR image formation, emphasizing the ill-posedness of the SAR SfS problem, and more in general, of any inversion procedure from a single SAR image. At the same time, a proper modeling of the problem gives the chance of circumventing the ill-posedness, making possible the estimation of the parameters of interest. Indeed, the sensitivity analysis shown in [

30] demonstrates the major contribution to the SAR image formation due to macroscopic roughness (i.e., to the local incidence angle), whereas the other parameters provide a minor contribution.

#### 2.2. Model Inversion

The retrieval of information from remote sensing data of the atmosphere, ocean, and land is basically an inverse problem. It is frequently an ill-posed problem, since many geophysical parameters concur to determine the quantities detected by the sensor (the observations), where different combinations of these parameters can give rise to similar observed quantities. Moreover, errors are always present in the measurements as well as in the knowledge of the causal model (the so-called direct or forward model) that relates the parameters to the observations.

Due to the usual lack of a priori knowledge of the characteristics of the observed surface, the proposed inversion method assumes spatial invariance of the dielectric properties of the sensed scene, more precisely the spatial homogeneity of the dielectric constant of the imaged topography. This hypothesis is embedded in the stricter assumption of invariance of the Fresnel reflection coefficient β_{mn}, which is made for the sake of simplicity in order to obtain a less involved inversion technique. Spatial changes of dielectric characteristics (i.e., in general of the complex relative permittivity ε_{r}) in SAR images can be mainly related to changes in the material that covers surface (for example water, snow, soil, rock, ice, oil, etc.) or to changes in soil moisture content (dry or wet soil).

As described previously in the paper, natural landscapes exhibit a fractal behavior varying both in space and in scale. This means that both

H and

T may be modeled as a function of spatial and scale variables. However, in order to deal with a simplified forward model and to exploit existing algorithms for fractal parameter estimation, e.g., [

28,

43], the proposed model assumes a scale-invariance of both the Hurst coefficient and topothesy; i.e., the multifractal behavior of natural surfaces is neglected.

While these problems are relevant to every SAR image and inversion technique, SfS has to deal with another important issue: being available only one intensity image, the inversion procedure has to retrieve two unknowns—

p and

q—from a single intensity measurement. From these considerations we can understand the successful choice of fractal geometry: it is required the knowledge of only two independent parameters, that, in turn, are independent from radar viewing geometry and sensor parameters (frequency, resolution, look angle) [

43].

In this section, the proposed inversion technique is analytically derived, and its limits are analyzed. In order to simplify the inversion procedure, and then retrieve the range slope map, we assume a small-slope regime for the surface. Thanks to this hypothesis, a first-order approximation of the intensity map

I, through a McLaurin series expansion with respect to

p and

q can be considered:

where

o(∙) stands for small-o notation.

It can be proved that

${\frac{\partial h}{\partial q}|}_{\begin{array}{c}p=0\\ q=0\end{array}}=0$ [

28], so that we obtain a linear function of the partial derivative

p only. The intensity function

I is, in the first-order approximation, linearly linked only to the range slope of the surface. As a consequence,

The coefficients of the McLaurin series expansion, a_{0} and a_{1}, depend on the specific adopted direct scattering model. In particular, these coefficients are a function of the sensor look angle, which is hence an important parameter for the determination of the validity limits of the proposed linear model. Finally, we note that the obtained result highlights a key property of SAR—and, more in general, of side-looking radars-imaging behavior, showing a clear mathematical definition of a preferential imaging direction due to their particular acquisition geometry.

It can be shown that for the expression in (11)

Linearization greatly simplifies the inversion procedure, allowing the estimation of the local range slopes in a very straightforward and fast way.

In order to get a valid slope map, it is necessary to devise a procedure aimed at evaluating the calibration constant G that appears in (11). In this work, we propose a very simple way to estimate G from the data; from (11) we have

Since a_{0} is known, (13) implies that it is possible to estimate the value of G simply by measuring from data the intensity of a flat homogeneous region. Speckle and roughness could affect the estimation, so it is necessary to consider a sufficiently wide flat area or, alternatively, a despeckling algorithm should be applied to the noisy image to obtain a good estimate of G. Obviously, the main disadvantage of this procedure is that it requires a basic knowledge of the sensed scene in order to identify a flat region in the SAR image.

If such a priori knowledge is not available, assuming the validity of the linear model in (11), the calibration constant

G can be calculated as follows

where μ

_{I} is the estimated mean value of the intensity map

I. This procedure, based on the assumption that, according to the fractal model assumed, the global mean slope observed in a SAR image of a natural surface is zero, greatly simplifies the inversion technique, since the knowledge of a flat region present in the intensity image is no longer necessary. Furthermore, this hypothesis drastically reduces the amount of a priori knowledge required: if only the shape of the surface is of interest, no knowledge is required, apart from sensor parameters. Additionally,

H can be estimated from data using a linear regression of the range cut of the image power spectra (for more details see [

28]). It is noteworthy that the retrieval algorithm proposed in [

28] estimates a large-scale Hurst coefficient (i.e., the macroscopic roughness), while, at least in principle, we need a microscopic characterization of the resolution cell. However, thanks to the scale-invariance hypothesis,

H is assumed to be constant at any observation scale. In addition, it can be easily shown that both

T and

ε_{r} do not affect topography estimation if (14) is used, since they simplify in (13). Thus, using (14):

This means that, if

G is evaluated according to (14), the estimated range slope depends only on the ratio between the coefficients

a_{0} and

a_{1}. According to (12), in this ratio both |

β_{mn}|

^{2} (in which ε

_{r} is present) and

S_{0} (in which

T is present) cancel out. From now on, in order to simplify the notation, we perform a formal substitution, incorporating the calibration constant within the coefficients

a_{0} and

a_{1}, so that, hereafter, the linear imaging model becomes

#### Error Source Evaluation

In this section, some considerations about range slope estimation are emphasized and the main error sources are analyzed and discussed. Starting from (11), it is clear that errors on range slope estimation are caused by errors on the linear model coefficients and the calibration constant. Errors on coefficients are linked to model error; i.e., the linear approximation of the intensity image and to parameter errors; i.e., errors on geometric and electromagnetic parameters of both surface and sensor; errors on G are due to model error concerning G evaluation (see Equation (14)) and, as we will also show, to errors committed on a_{0} and a_{1}.

In the following, we note by

$\widehat{x}$ an estimator of x. Considering an additive error Δ

G on

G estimation we obtain:

Equation (17) means that an error ΔG cause a rescaling and a translation of the estimated range slope and, as a consequence, the estimated topography suffers from a rescaling and a tilt depending on the relative error and the coefficients ratio.

The same effects are caused by an error on model coefficients:

where Δ

a_{0} and Δ

a_{1} the error on coefficients.

In a more realistic scenario, errors on both calibration constant and coefficients are present; in this case, more severe distortion effects will be experienced, thus:

Indeed, (11) suggests that an error on model coefficients causes an error on calibration constant estimation, thus:

where

µ_{p} is the range-slope expected value and

G_{truth} is the true calibration constant. Equation (20) establishes that an additive error on

a_{0} and

a_{1} causes a multiplicative error on

G, and then an additive error given by

The error Δ

G takes into account both calibration constant model error

$\mathsf{\Delta}{G}_{m}$ and coefficients error

$\mathsf{\Delta}{G}_{c}$. It can be noted that the two error sources are factorizable, thus

Finally,

Figure 2 shows at a glance the error source on range slope estimation.