Three-Dimensional Ice-Flow Recovery from Ascending–Descending DInSAR Pairs and Surface-Parallel Flow Hypothesis: A Simplified Implementation in SNAP Software

Abstract

By considering two differential interferometric SAR signals, recovered from synthetic aperture radar (SAR) images, it has been possible to estimate the glacier velocity vector, from a method proposed by the authors Joughin, Kwok, and Fahnestock (JKF) in 1998. Although the JKF method normally works well under certain SAR observation conditions, we found a reformulated version of the main equation of this technique that may improve this interesting methodology. Thus, we present a mathematical review of this method, and a validation of our result in terms of accuracy, with some computer simulations. The innovation proposed is a simplified way to implement JKF’s work in the Sentinel Application Platform (SNAP) software, exemplified with some images from the Canadian Arctic. Generally, a north–east–up displacement estimation is considered, by using reference orthogonal coordinates, independent of the SAR image coordinates. However, we propose a methodology to estimate this velocity vector in terms of ascending or descending image coordinates. Given the importance of the JKF work, we believe that this investigation could contribute to the improvement of this technique, beyond the existence of other modern and independent methodologies.

1. Introduction

The dynamics of a glacier is based on the flow of excess ice from the accumulation zone to the ablation’s one. This flow depends on factors such as ice temperature, thickness, basal water content, and terrain slope [1]. In general, a glacier is considered in equilibrium when the volume of ice in the ablation zone is at least equal to that of the accumulation zone, ensuring its permanence over time. However, if the loss of ice during the summer exceeds that of the accumulation zone, both the thickness and surface area will retreat [2]. Unlike mountain glaciers, which are relatively small and only have tens of meters in thickness, circumpolar glaciers are much thicker and have tongues that can reach hundreds of meters or even kilometers in length [3]. The latter offers favorable conditions to analyze the glacier flow velocity of circumpolar regions, as a function of changing climatological parameters and the amount of ice accumulated over the years, through the comparison of radar satellite images.

As is well known, interferometric SAR (InSAR) and/or differential InSAR (DInSAR) techniques are widely used for monitoring topographic changes or terrain displacements induced by different phenomena, like earthquakes or seismic activity, magma accumulation due to volcanic lava flow, and particularly, glacier or ice-flow dynamics [4,5]. In the case of estimating the 3D ice-flow velocity vector $\mathbf{v}={(v_p,v_q,v_z)}^{\top }$, an interesting approach was proposed in the work of Joughin, Kwok, and Fahnestock (1998) [6] by considering at least two DInSAR signals, in ascending and descending configuration, respectively. Thus, by using the data of these interferometric pairs, and a reference digital elevation model (DEM) $h=h(p,q)$ of the region of analysis, the 3D-velocity vector $\mathbf{v}=v_p\mathbf{p}+v_q\mathbf{q}+v_z\mathbf{z}$, in a general orthonormal basis $\{\mathbf{p},\mathbf{q},\mathbf{z}\}$, can be estimated from the surface-parallel flow (SPF) assumption, i.e.,

$$v_z=\nabla h(p,q)·\mathbf{w}=h_p{v}_p+h_q{v}_q,$$

(1)

where $h_p=\partial h/\partial p$, $h_q=\partial h/\partial q$, and $\mathbf{w}:={(v_p,v_q)}^{\top }$ is the 2D horizontal velocity vector, respectively. However, the coordinates for the ascending and descending satellite data are different from the original reference coordinates where the points $(p,q,z)$ are considered. Generally, $\mathbf{p},\mathbf{q},\mathbf{z}$ referred to east, north, and up directions, respectively. Thus, we need information on the angles induced by unitary vectors ${\mathbf{y}}_a$ (ascending ground range), ${\mathbf{y}}_d$ (descending ground range), and $\mathbf{p}$. Specifically, $\alpha :=\mathrm{angle}({\mathbf{y}}_a,{\mathbf{y}}_d)$ and $\beta :=\mathrm{angle}({\mathbf{y}}_a,\mathbf{p})$, respectively (see Figure 1a). Nevertheless, if the available reference DEM h is in terms of azimuth and ground range coordinates (ascending or descending SAR satellite coordinates), we may assume without loss of generality that $\beta =0$, as we see later.

The basic parameters of InSAR geometry (see Figure 2a,b), as well as the reference coordinates of an ascending or a descending SAR system (see Figure 2c), are explained in Appendix A of this research. The relation between the deformation phase $\varphi$ of a DInSAR signal, line of sight (LOS) displacement $|A{B}^{\prime }|$, ground range displacement $dy$, and vertical displacement $dz$ (see Figure 2d), in their native ascending or descending coordinates, is also explained in this material. This information, although widely known, may be useful to understand part of the notation used in this work and how the 3D information of DInSAR data can be exploded. Particular attention must be taken into account, in the description of the moving of a point $A_{\ast }=(x,y,z)$ to a point $B_{\ast }=(x+dx,y+dy,z+dz)$ due to ice flow, where their respective orthogonal projections A and B onto a common azimuthal plane are displayed in Figure 2d.

Now, the organization of this research is as follows: In Section 2, velocity variables considering ascending and descending directions are described, by following the methodology exposed in reference [6], which basically consists in solving a matrix linear system of equations of order two, per each image pixel. This reference has motivated many authors to generalize the estimation of vector $\mathbf{v}$, even by considering the decomposition of the LOS component in ground and azimuth directions and solving matrix systems of orders bigger than two [7]. However, we reviewed the work in [6] carefully and found a subtle detail in one of the main equations. Consequently, we propose a reformulated equation that can be applied to a general choice of possible angles between ascending and descending tracks. Apparently, this particular reformulation has not been reported in any of the many citations of Joughin, Kwok, and Fahnestock’s (JKF) paper [6], see refs [8,9,10,11,12,13,14,15,16,17], to mention a few. Thus, a modified matrix equation that matches velocity components $v_p$ and $v_q$, in terms of deformation phases ${\varphi }_a$, and ${\varphi }_d$, recovered from ascending and descending data, respectively, is deduced in Section 2. Then, in Section 3, we apply our proposal to real data conformed by SAR images of the Canadian Arctic, by simplifying our velocity vector estimation in terms of the orthogonal coordinates $(p,q,z)=(y_d,x_d,z_d)$, induced by the descending coordinate system with $\beta =0$ in Figure 1b. After this reconstruction, in Section 4, we analyze the accuracy achieved with our corrected equation, in comparison to the original formula exposed in [6], by considering a simulation of ice-flow displacement. The simulation is implemented with synthetic scalar fields as the velocity components given by $\mathbf{v}$, under SPF assumption, and synthetic wrapped phases (SAR interferograms) for ${\varphi }_a$ and ${\varphi }_d$ are used in order to validate our result. Finally, some points of discussion are mentioned in Section 5.

2. Horizontal Velocity Vector in Terms of Ascending and Descending Data

Since the z-components of the 3D vectors for ascending azimuth ${\mathbf{x}}_a$, ascending across track ${\mathbf{y}}_a$, descending azimuth ${\mathbf{x}}_d$, descending across track ${\mathbf{y}}_d$, $\mathbf{p}$, and $\mathbf{q}$ are zero, they (or their projections onto plane $z=0$) can be considered as 2D vectors similar to the horizontal velocity vector $\mathbf{w}$, as depicted in Figure 1a. Now, as demonstrated in Appendix A (see Equation (A4)), the deformation phase $\varphi$ of a DInSAR signal, recovered from a pair of images acquired at times $t_1<t_2$, is related to vertical displacement $dz$, and the across-track one, $dy$, according to the expression $\varphi /\left(2k\right)=dzcos\left(\theta \right)+dysin\left(\theta \right)$, where $\theta$ is an observation angle. This is similar to having

$$\sigma :={\displaystyle \frac{\varphi }{2k\Delta tsin\left(\theta \right)}}\approx v_y+v_{z}cot\left(\theta \right),$$

(2)

by considering $v_z:=dz/dt$, $v_y:=dy/dt$ and $\Delta t:=t_2-t_1$, respectively. Thus, Equation (2), rewritten as an equality, for ascending and descending deformation phases ${\varphi }_a$ and ${\varphi }_d$, would be

$$\begin{array}{c}{\displaystyle {\sigma }_a:=\frac{{\varphi }_a}{2k\Delta t_{a}sin\left({\theta }_a\right)}=v_{y_a}+v_z{c}_a,}\\ {\displaystyle {\sigma }_d:=\frac{{\varphi }_d}{2k\Delta t_{d}sin\left({\theta }_d\right)}=v_{y_d}+v_z{c}_d,}\end{array}$$

(3)

where $\Delta t_a=t_2-t_1>0$, $\Delta t_d=t_2^{\prime }-t_1^{\prime }>0$, $c_a:=cot\left({\theta }_a\right)$, $c_d:=cot\left({\theta }_d\right)$, and $v_z=v_{z_a}=v_{z_d}$, respectively. The latter is because $z=z_a=z_d$ and $t_1^{\prime },t_2^{\prime }\approx t_1,t_2$, where $\Delta t_a\approx \Delta t_d$ (6 o 12 days, for Sentinel-1). Now, if $(v_{y_a},v_{y_d})$ denotes the coordinates of $\mathbf{w}={(v_p,v_q)}^{\top }$ in the non-orthogonal basis $\{{\mathbf{y}}_a,{\mathbf{y}}_d\}$ (with unit Euclidean norms $\left|\right|{\mathbf{y}}_a\left|\right|=\left|\right|{\mathbf{y}}_d\left|\right|=1$), then $\mathbf{w}=v_{y_a}{\mathbf{y}}_a+v_{y_d}{\mathbf{y}}_d$. Therefore, by calculating the dot product $\mathbf{w}·{\mathbf{y}}_a=v_{y_a}\left|\right|{\mathbf{y}}_a{\left|\right|}^2+v_{y_d}\left|\right|{\mathbf{y}}_a\left|\right|\left|\right|{\mathbf{y}}_d\left|\right|cos\left(\alpha \right)$ and $\mathbf{w}·{\mathbf{y}}_d$, we find that

$$\begin{array}{c}\mathbf{w}·{\mathbf{y}}_a=v_{y_a}+v_{y_d}cos\left(\alpha \right),\\ \mathbf{w}·{\mathbf{y}}_d=v_{y_a}cos\left(\alpha \right)+v_{y_d},\end{array}$$

(4)

or equivalently

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\mathbf{w}·{\mathbf{y}}_a\\ \mathbf{w}·{\mathbf{y}}_d\end{array}\right]& {\displaystyle =\left[\begin{array}{cc}1& cos\left(\alpha \right)\\ cos\left(\alpha \right)& 1\end{array}\right]\left[\begin{array}{c}{v}_{y_a}\\ v_{y_d}\end{array}\right]}\hfill \\ & =:{\mathbf{B}}^{-1}\left[\begin{array}{c}{v}_{y_a}\\ v_{y_d}\end{array}\right],\hfill \end{array}$$

(5)

where symbol $=:$ defines a matrix ${\mathbf{B}}^{-1}$ ($2\times 2$-dimensional). Nevertheless, the system in Equation (5) in terms of inverse matrix $\mathbf{B}:={\left({\mathbf{B}}^{-1}\right)}^{-1}$ is

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{v}_{y_a}\\ v_{y_d}\end{array}\right]& {\displaystyle =\frac{1}{{sin}^2\left(\alpha \right)}\left[\begin{array}{cc}1& -cos\left(\alpha \right)\\ -cos\left(\alpha \right)& 1\end{array}\right]\left[\begin{array}{c}\mathbf{w}·{\mathbf{y}}_a\\ \mathbf{w}·{\mathbf{y}}_d\end{array}\right]}\hfill \\ \hfill & \\ \hfill & =:\mathbf{B}\left[\begin{array}{c}\mathbf{w}·{\mathbf{y}}_a\\ \mathbf{w}·{\mathbf{y}}_d\end{array}\right].\hfill \end{array}$$

(6)

If we consider that $\mathbf{p}={(p,q)}^{\top }={(1,0)}^{\top }$ and $\mathbf{q}={(0,1)}^{\top }$, it is clear from Figure 1a that ${\mathbf{y}}_a=cos\left(\beta \right)\mathbf{p}+sin\left(\beta \right)\mathbf{q}$ and ${\mathbf{y}}_d=cos(\alpha +\beta )\mathbf{p}+sin(\alpha +\beta )\mathbf{q}$; then, $\mathbf{w}$ corresponds to

$$\begin{array}{cc}\hfill v_{y_a}{\mathbf{y}}_a+v_{y_d}{\mathbf{y}}_d& {\displaystyle =\left[\begin{array}{cc}cos\left(\beta \right)& cos(\alpha +\beta )\\ sin\left(\beta \right)& sin(\alpha +\beta )\end{array}\right]\left[\begin{array}{c}{v}_{y_a}\\ v_{y_d}\end{array}\right]}\hfill \\ \hfill & \\ \hfill & {\displaystyle =:\mathbf{A}\left[\begin{array}{c}{v}_{y_a}\\ v_{y_d}\end{array}\right],}\hfill \end{array}$$

(7)

which is the same as

$$\mathbf{w}=\left[\begin{array}{c}{v}_p\\ v_q\end{array}\right]=\mathbf{A}\left[\begin{array}{c}{\sigma }_a-v_z{c}_a\\ {\sigma }_d-v_z{c}_d\end{array}\right]$$

(8)

from Equation (3), in terms of another $2\times 2$-dimensional matrix $\mathbf{A}$. Now, from the SPF assumption, Equation (8) is equivalent to writing

$$\left[\begin{array}{c}{v}_p\\ v_q\end{array}\right]+(h_p{v}_p+h_q{v}_q)\mathbf{A}\left[\begin{array}{c}{c}_a\\ c_d\end{array}\right]=\mathbf{A}\left[\begin{array}{c}{\sigma }_a\\ {\sigma }_d\end{array}\right]$$

(9)

or

$$\left(\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\mathbf{A}\left[\begin{array}{cc}{c}_a{h}_p& c_a{h}_q\\ c_d{h}_p& c_d{h}_q\end{array}\right]\right)\left[\begin{array}{c}{v}_p\\ v_q\end{array}\right]=\mathbf{A}\left[\begin{array}{c}{\sigma }_a\\ {\sigma }_d\end{array}\right],$$

(10)

by doing the corresponding algebraic steps. Thus, the last system is the same as

$$\mathbf{w}=\left[\begin{array}{c}{v}_p\\ v_q\end{array}\right]={(\mathbf{I}+\mathbf{A}\mathbf{C})}^{-1}\mathbf{A}\left[\begin{array}{c}{\sigma }_a\\ {\sigma }_d\end{array}\right],$$

(11)

by considering

$$\mathbf{I}:=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathbf{C}:=\left[\begin{array}{cc}{c}_a{h}_p& c_a{h}_q\\ c_d{h}_p& c_d{h}_q\end{array}\right],$$

(12)

respectively. On the other hand, if we substitute ${(v_{y_a},v_{y_d})}^{\top }$ given by Equation (6) into Equation (7), we obtain $\mathbf{w}=\mathbf{A}\mathbf{B}{(\mathbf{w}·{\mathbf{y}}_a,\mathbf{w}·{\mathbf{y}}_d)}^{\top }$ or

$$\begin{array}{cc}\hfill \mathbf{w}& {\displaystyle =\mathbf{A}\mathbf{B}\left[\begin{array}{c}{v}_{y_a}+v_{y_d}cos\left(\alpha \right)\\ v_{y_d}+v_{y_a}cos\left(\alpha \right)\end{array}\right]}\hfill \\ \hfill & \\ \hfill & {\displaystyle =\mathbf{A}\mathbf{B}\left[\begin{array}{c}{\sigma }_a-v_z{c}_a+v_{y_d}cos\left(\alpha \right)\\ {\sigma }_d-v_z{c}_d+v_{y_a}cos\left(\alpha \right)\end{array}\right]}\hfill \end{array}$$

(13)

from Equations (3) and (4). The latter is the same as

$$\mathbf{w}=\mathbf{A}\mathbf{B}\left[\begin{array}{c}{\sigma }_a-v_z{c}_a\\ {\sigma }_d-v_z{c}_d\end{array}\right]+cos\left(\alpha \right)\mathbf{A}\mathbf{B}\left[\begin{array}{c}{v}_{y_d}\\ v_{y_a}\end{array}\right],$$

(14)

where the second matrix term with $cos\left(\alpha \right)$ was omitted (ambiguously) in [6], inducing a system given by $\mathbf{w}+v_z\mathbf{A}\mathbf{B}{(c_a,c_d)}^{\top }=\mathbf{A}\mathbf{B}{({\sigma }_a,{\sigma }_d)}^{\top }$ with solution

$$\mathbf{w}=\left[\begin{array}{c}{v}_p\\ v_q\end{array}\right]={(\mathbf{I}+\mathbf{A}\mathbf{B}\mathbf{C})}^{-1}\mathbf{A}\mathbf{B}\left[\begin{array}{c}{\sigma }_a\\ {\sigma }_d\end{array}\right],$$

(15)

derived again from the same SPF assumption (see Equation (21) in [6]). In JKF’s work, matrix $\mathbf{I}+\mathbf{A}\mathbf{B}\mathbf{C}$ in Equation (15) was reported as $\mathbf{I}-\mathbf{A}\mathbf{B}\mathbf{C}$, because Equation (2) was taken with a switched sign, i.e., $\sigma \approx v_y-v_{z}cot\left(\theta \right)$ (see Equation (6) in [6]). Independently of this minor detail, it seems that the use of Equation (6) is unnecessary for the practical purposes of expressing $\mathbf{w}$ in terms of ${({\sigma }_a,{\sigma }_d)}^{\top }$. Thus, our corrected solution corresponds to Equation (11), and to the best of our knowledge, this reformulation has not been considered in any of the current works on the topic. With respect to the detail found, this is not properly an algebraic error, because the authors in [6] were conscious in their Equation (13) that

$$\mathsf{\text{Equation (13):}}\left\{\begin{array}{c} v_a\ne {\mathbf{v}}_h^T\widehat{\mathbf{a}}\leftarrow v_{y_a}\ne \mathbf{w}·{\mathbf{y}}_a=v_{y_a}+v_{y_d}cos\left(\alpha \right)\\ v_d\ne {\mathbf{v}}_h^T\widehat{\mathbf{d}}\leftarrow v_{y_d}\ne \mathbf{w}·{\mathbf{y}}_d=v_{y_a}cos\left(\alpha \right)+v_{y_d},\end{array}\right.$$

(16)

which is correct, according to our Equation (4). Nevertheless, we could have $v_{y_a}=\mathbf{w}·{\mathbf{y}}_a$ and $v_{y_d}=\mathbf{w}·{\mathbf{y}}_d$, only when considering $\alpha =90$ [deg], and consequently (from our Equation (3)), we would be able to write

$$\mathsf{\text{equation (12):}}\left\{\begin{array}{c}{\displaystyle ...\leftarrow \mathbf{w}·{\mathbf{y}}_a=v_{y_a}=\frac{{\varphi }_a}{2k\Delta t_{a}sin\left({\theta }_a\right)}-v_z{c}_a}\\ \\ {\displaystyle ...\leftarrow \mathbf{w}·{\mathbf{y}}_d=v_{y_d}=\frac{{\varphi }_d}{2k\Delta t_{d}sin\left({\theta }_d\right)}-v_z{c}_d}\end{array}\right.$$

(17)

in a very similar way to what was expressed by Equation (12) in JKF’s work. Nevertheless, the correct expressions for a general angle $\alpha$ must be

$$\begin{array}{c}{\displaystyle \mathbf{w}·{\mathbf{y}}_a=\frac{{\varphi }_a}{2k\Delta t_{a}sin\left({\theta }_a\right)}-v_{z}cot\left({\theta }_a\right)+v_{y_d}cos\left(\alpha \right),}\\ {\displaystyle \mathbf{w}·{\mathbf{y}}_d=\frac{{\varphi }_d}{2k\Delta t_{d}sin\left({\theta }_d\right)}-v_{z}cot\left({\theta }_d\right)+v_{y_a}cos\left(\alpha \right),}\end{array}$$

(18)

from Equations (3) and (4), respectively. Readers can verify in JKF’s paper the detail of considering Equation (12), only valid for $\alpha =90$ [deg], in connection with Equation (14) (equivalent to our Equation (5)) and Equation (16) (equivalent to our Equation (9)), both correct in general. A review of these equations can also be conducted by considering the notation equivalences between some of the variables used in JKF’s work and the variables employed in our proposal, as displayed in

Table 1. Notation equivalences between some of the variables of JKF’s work in [6] and the corresponding ones in our proposal (Prop.).

JKF ← Prop.	JKF ← Prop.	JKF ← Prop.	JKF ← Prop.
$\mathbf{v}\leftarrow \mathbf{v}$	$\mathbf{x}\leftarrow \mathbf{p}$	$\mathbf{y}\leftarrow \mathbf{q}$	$\mathbf{z}\leftarrow \mathbf{z}$
$z_t(x,y)\leftarrow h(p,q)$	${\displaystyle \frac{\partial z_t}{\partial x}}\leftarrow h_p$	${\displaystyle \frac{\partial z_t}{\partial y}}\leftarrow h_q$	${\mathbf{v}}_h\leftarrow \mathbf{w}$
$\widehat{\mathbf{a}}\leftarrow {\mathbf{y}}_a$	$\widehat{\mathbf{d}}\leftarrow {\mathbf{y}}_d$	$v_a\leftarrow v_{y_a}$	$v_d\leftarrow v_{y_d}$
$v_x\leftarrow v_p$	$v_y\leftarrow v_q$	${\varphi }_{d,a}\leftarrow {\varphi }_a$	${\varphi }_{d,d}\leftarrow {\varphi }_d$
${\psi }_a\leftarrow {\theta }_a$	${\psi }_d\leftarrow {\theta }_d$	$\delta T_a\leftarrow \Delta t_a$	$\delta T_d\leftarrow \Delta t_d$
$\alpha ,\beta \leftarrow \alpha ,\beta$	$\mathbf{A}\leftarrow \mathbf{A}$	$\mathbf{B}\leftarrow \mathbf{B}$	$\mathbf{C}\leftarrow \mathbf{C}$

.

Therefore, since JKF’s work remains an important reference point on the subject, even in research papers from the last decade (for example [18]), it is very appropriate to analyze the reformulated solution given by Equation (11).

3. Method Proposed and Real Data Experiment: Velocity Vector Estimation

3.1. Materials and Methods

In our reconstruction, we worked with two deformation interferograms (wrapped phases ${\psi }_d$ and ${\psi }_a$, respectively) obtained from a set of interferometric wide-swath (IW)–single-look complex (SLC) images of Sentinel-1 (SAR data with approximated wavelength $\lambda =0.056$ [m]), with HH-polarization. To have some reference information on the phenomena under study, we downloaded some of the images of the work [18]; these images correspond to a region in the Canadian Arctic (as depicted in Figure 3), all of them acquired in February and March 2016. These interferograms were recovered from descending and ascending data, as described in

Table 2. IW-SLC products used to generate the deformation interferograms ${\psi }_d$ and ${\psi }_a$ to be processed. Both interferograms represent LOS displacements per intervals of revisit $\Delta t_d=\Delta t_a\approx 0.0329$ years (12 days).

Interferogram	Acquisition Date	Baseline ${\mathit{l}}_{\mathit{\perp }}$	Sub-Swath and Bursts
${\psi }_d$	29/Feb/2016 (descending)	30.7 [m]	IW2-Bursts 7-8-9
	12/Mar/2016 (descending)
${\psi }_a$	04/Mar/2016 (ascending)	47.06 [m]	IW1-Bursts 4-5-6
	16/Mar/2016 (ascending)

.

The reason for choosing the arctic zone of Canada is because it is characterized as one of the most dynamic in terms of glacial flow velocity; that is, glaciers move with greater volume and speed per year, which is why most of the icebergs in the North Atlantic are generated by glacial flow in northeastern Canada. Although the work could have been carried out in other areas such as Antarctica, the evidence found in this work would also be noticeable, but not as clear as in this area.

The interferograms were constructed with the free software called Sentinel Application Platform (SNAP 9.0.0), according to the general steps described in [19], by considering the following preprocessing steps: S-1 TOPS Split (with the sub-swaths and bursts given in Table 2), Apply Orbit File, S1 TOPS Coregistration—S-1 Back Geocoding (with a Copernicus 30m resolution DEM [20]), Interferogram Formation, and Goldstein Phase Filtering [21]. After recovering the descending and ascending interferograms individually, we added their elevation bands (DEMs $h(x,r)$) and saved them as non-virtual bands. Then, we added the incident angle bands ${\theta }_d$ and ${\theta }_a$ to each interferogram product, respectively. The latter means a conversion of the incident angle tie-point grids onto non-virtual bands, when considering the Band Maths option in SNAP. After these additions, we created a stack product of both interferograms and a subset of that stack, i.e., we followed a coregistration process (another geocoding process) of the interferograms by using the Stack-tools of SNAP (employing a bicubic interpolation method) and took a subset of the stack product. In this coregistration, the primary or master product was the descending interferogram because it was derived from the oldest image captured at Feb/29/2016. The resultant stack subset covered an area, or a possible reference domain $\Omega$, contained in the intersection region of the two original interferometric patterns (see Figure 3).

The data arrays or bands of the stack subset were rewritten in terms of descending azimuth $x:=x_d$ and descending ground range $y:=y_d$, by using the geometric operation Slant Range to Ground Range, available in the SNAP menu. In this way, all data such as $h(x,y):=h_d(x,y)$, ${\theta }_d(x,y)$, ${\theta }_a(x,y)$, ${\psi }_d(x,y)$ and ${\psi }_a(x,y)$ could be described in terms of common orthogonal coordinates $(x,y)=(x_d,y_d)$. Now, we could avoid the use of additional reference coordinates $(p,q)$ in this experiment, because our reference DEM h was already given in descending coordinates $(x_d,y_d)$. This is similar to having $p:=y_d$ and $q:=x_d$, which is equivalent to assuming $\beta =0$ in Figure 1b. Therefore, points $(p,q)=(y_d,x_d)$ can be defined in a reference domain $\Omega :=\{(p,q)\in {\mathbb{R}}^2:0\le p\le (N-1)\Delta p,0\le q\le (M-1)\Delta p\}$. Here, $\Delta p:=\Delta y$ and $\Delta q:=\Delta x$ are the approximated resolution parameters of the SAR system (obtained after all the preprocessing steps), considering images of $N\times M$-pixels. Nevertheless, we normally expressed all these scalar fields (h, ${\theta }_d$, ${\theta }_a$, ${\psi }_a$, etc.) in terms of pixels $(n,m)$, through an abuse of notation, as $h(n,m)=h(p_n,q_m)$, with $p_n,q_m=n\Delta p,m\Delta q$ and $n,m=0,1,\dots ,N-1,M-1$, correspondingly.

After the aforementioned geometric operation, we created a copy of the stack subset for implementing our own code manipulations in Python 2.7, by using the snappy library. This library can be activated by following the steps described in [22]. From this copy, we imported the interferograms ${\psi }_a$ and ${\psi }_d$ (with their corresponding coherence factors ${\gamma }_a$ and ${\gamma }_d$) and proceeded to unwrap these discontinuous surfaces with an appropriate unwrapping routine. By considering W as the wrapping operator [23], the unwrapping stage can be interpreted as the conversion of a DInSAR interferogram $\psi =W\left[\varphi \right]$ to a phase estimate $\widehat{\varphi }=W_{algorithm}^{-1}\left[\psi \right]$, with certain unwrapping algorithm. In SNAP software, the method used can be the statistical cost network flow algorithm for phase unwrapping (SNAPHU) [24]. Nevertheless, in the presence of unwrapping inconsistencies, we can use alternative algorithms, such as the fast Fourier transform (FFT) approach [23,25] (see the details described in Appendix B of this research).

Once the best unwrapped estimates for ${\varphi }_d$ and ${\varphi }_a$ were obtained (see Figure 4d,j), we also estimated the angle $\alpha$ conformed by axes $y_d$ and $y_a$. To do this, we approximated angles ${\theta }_d(p,q)$ and ${\theta }_a(p,q)$ over our reference domain $\Omega$ (see Figure 4a,g), by considering fits with a global plane model of the form $\widehat{\theta }(p,q)=c_0+c_{p}p+c_{q}q$, for all $(p_n,q_m)\in \Omega$ with $n,m=0,1,\dots ,N-1,M-1$, respectively. In this case, the estimated $\alpha$ corresponds to the angle induced by the gradients $\nabla {\widehat{\theta }}_d$ and $\nabla {\widehat{\theta }}_a$, considering the corresponding fits ${\widehat{\theta }}_d$ and ${\widehat{\theta }}_a$, where $\nabla \widehat{\theta }=(c_p,c_q)$ and $cos\left(\alpha \right)=(\nabla {\widehat{\theta }}_d·\nabla {\widehat{\theta }}_a)/\left(\right||\nabla {\widehat{\theta }}_d\left|\right|\left|\right|\nabla {\widehat{\theta }}_a\left|\right|)$. All these reconstructions were carried out on an OMEN Laptop AMD Ryzen 7 4800H with 16 GB of RAM and Windows 11.

3.2. Estimated Angle $\alpha$ and General Processing Flow Description

After taking the subset described by region $\Omega$, the resultant descending and ascending interferometric images had an approximated resolution parameters in descending azimuth and descending ground range coordinates, $\Delta q=13.76$ [m] and $\Delta p=3.59$ [m], with $N\times M=3984\times 2415$ pixels, respectively. The region $\Omega$ described by the discrete points $(p_n,q_m)$ corresponds to a rectangle with $0\le p\le 14.30256$ [km] and $0\le q\le 33.21664$ [km], approximately. The coregistered interferograms ${\psi }_d$ and ${\psi }_a$ in terms of pixels $(n,m)$ or positions $(p,q)$ in region $\Omega$ are displayed in Figure 4b,h, respectively. The resolution parameters $\Delta q$ and $\Delta p$ are derived from the slant-range-to-ground-range conversion, which, according to the references in the SNAP software, consists of the following major steps: (1) create a warp polynomial of given order that maps ground range pixels to slant range pixels, (2) for each ground range pixel, compute its corresponding pixel position in the slant range image using a warp polynomial, (3) compute pixel value using the user-selected interpolation method (in our case, it was a linear interpolation).

With respect to the coregistration of the descending and ascending DInSAR interferograms (the stacking process in Figure 5), we can follow a similar description, as in the case of a single pair of SAR images; where the values of the secondary (slave) image are resampled on the common georeferenced pixels of the primary (master) image. This coregistration process can be carried out from the geocodes of both interferograms, with their corresponding restored or updated orbit files, independently of their ascending or descending geometries. The description in Section 1 and Section 2 assumes that the ascending interferogram is the primary product, while the descending one is the secondary product, i.e., an ascending–descending configuration to declare the primary–secondary interferometric products, respectively. However, Equations (3)–(12) remain valid, even if we switch the meaning of subscripts a and d in Figure 1a, i.e., subscript a for descending and d for ascending information, respectively. This means that our description in Section 1 and Section 2 could assume a descending–ascending configuration (as in our experimental case) to declare the primary–secondary interferometric products, as suggested in Figure 1b.

Now, in terms of the data for this experiment, an angle of 40 degrees between descending and ascending tracks, with equivalent supplement $\alpha =140$ [deg], was reported in the work [18]. However, the estimated $\alpha$ with our method was $135.01703$ [deg], and according to our convention, $\beta =0$ [deg] in our particular descending–ascending configuration.

From the unwrapped estimates ${\varphi }_d$, ${\varphi }_a$, the partial derivatives of our reference DEM h, the angular data ${\theta }_d$, ${\theta }_a$, and the approximated $\alpha$ value, we are ready to estimate de velocity components $v_p$, $v_q$, and $v_z$, under the SPF hypothesis. A simplified description of our processing flow, from the reading of the image pairs to the calculation of velocity components is provided in Figure 5.

3.3. Velocity Components’ Estimation

By considering the estimated data ${\varphi }_d$, ${\varphi }_a$, $h_p$, $h_q$, $\alpha$, etc., we can recover the velocity components $v_p$, $v_q$, and $v_z$ in two ways: through Equations (11) and (15), in order to compare or view the differences achieved from our proposed solution and the classical one, respectively (see Figure 6). It is necessary to point out that systems in Equations (11) and (15) can be quite sensitive to unwrapped phase errors because matrices $(\mathbf{I}+\mathbf{A}\mathbf{C})$ and $(\mathbf{I}+\mathbf{A}\mathbf{B}\mathbf{C})$, both dependent on DEM h-slopes, angles $\{\alpha ,\beta \}$, and angular distributions $\{{\theta }_a,{\theta }_d\}$, are generally ill-conditioned in some pixels $(n,m)$ (as we see later). As a result, the maps for velocity components, at first glance, are badly scaled in $\Omega$. Thus, appropriate threshold values must be considered before plotting their magnitudes.

In our case, if $v_p^0$ (in [m/yr]) is an initial guess derived directly from Equation (15), or from Equation (11), we defined $v_p(n,m)=v_p^0(n,m)$ for pixels such that $|v_p^0(n,m)|<T$, and $v_p(n,m)=\pm T$ (depending on the sign of $v_p^0(n,m)$) for those pixels that exceed the magnitude of a threshold value $T>0$. Actually, different threshold values (including negative ones) as upper bounds and lower bounds can be estimated from the original data. For example, by considering the mean ${\overline{v}}_p^0$ and standard deviation ${\sigma }_p^0$ of values $v_p^0(n,m)$, the upper and lower bounds can be $T={\overline{v}}_p^0\pm {\sigma }_p^0$, respectively. Otherwise, we can also define the thresholds from true physical reference information when available. But in this case, we chose arbitrary upper–lower bounds $T=\pm 40/\sqrt{3}$ and applied a thresholding step with these bounds to all initial guesses $v_p^0$, $v_q^0$, $v_z^0$, in order to have velocity estimates $v_p$, $v_q$, $v_z$, such that $\left|\right|\mathbf{v}\left|\right|=\sqrt{v_p^2+v_q^2+v_z^2}\le 40$ [m/yr]. This means that pixel values with $\left|\right|\mathbf{v}\left|\right|=40$ have a magnitude of 40 or greater, in a similar way to how information about velocity maps in [18] is displayed.

Thus, the results obtained from Equations (11) and (15), after thresholding, are plotted in Figure 6, respectively. As we can see, these results are very different, and we could ask what improvement is achieved by the correction. Since we did not have true physical reference information (control points) of the phenomena under study, we alternatively implemented a computer simulation of ascending and descending wrapped phases in order to compare the magnitude or the percentage of error achieved from these two main equations. To see other reconstructions in a different zone of the region of analysis (see Figure 7), in a different year like 2017, we processed another set of 4 Sentinel-1 HH-polarized images, described in

Table 3. IW-SLC products used to generate deformation interferograms ${\psi }_a$ and ${\psi }_d$ from images of the Canadian Arctic in 2017.

Interferogram	Acquisition Date	Baseline ${\mathit{l}}_{\mathit{\perp }}$	Sub-Swath and Bursts
${\psi }_a$	22/Jan/2017 (ascending)	16.84 [m]	IW1-Bursts 4-5-6
	03/Feb/2017 (ascending)
${\psi }_d$	30/Jan/2017 (descending)	92.56 [m]	IW2-Bursts 7-8-9
	11/Feb/2017 (descending)

.

In this case, the ascending–descending configuration was considered ($p=y_a$, $q=x_a$, $\beta =0$ in Figure 1a), because the oldest image captured on 22 January 2017 was ascending. The region $\Omega$ in this new example is depicted in the ascending domain, where angle $\alpha$ was 135.01266 [deg]. The preprocessing steps were similar as in our previous reconstruction (by using a Copernicus 30 m DEM and FFT unwrapping); however, due to the new values for the orthogonal baselines for this new image set, the resulting resolution parameters were $\Delta q=\Delta x_a=$13.82 m and $\Delta p=\Delta y_a$ = 4.22 m for a delineated subset with $N\times M=3011\times 2040$ pixels. The ascending and the descending deformation interferograms, as well as the estimated velocity components in this new region $\Omega$ are plotted in Figure 8.

3.4. SPF Limitations and General SAR Data Details

As described in detail in [6], ice does not flow parallel to the surface, and there are many limitations and error sources to deal with, not only those properly due to the actual conditions of the region of analysis but especially those of the InSAR data. Consequently, it is very difficult to recover unambiguous measurements. According to the description in JKF’s paper, ice flow is inclined slightly upward from the surface in the ablation zone and is tipped slightly downward in the accumulation one. The authors in [6] explain that in fast-moving areas with bumpy terrain, the vertical component of motion due to SPF is large with respect to the submergence/emergence velocity. The authors also mention that in areas where there is heavy ablation or accumulation, the SPF assumption may yield significant errors in estimates of vertical motion. Nevertheless, estimates of the horizontal components of motion should be relatively unaffected by deviations from the surface-parallel flow [6]. Of course, from our new proposed equation, a possible future analysis of the capabilities and limitations of the SPF hypothesis under real situations, conducted with ground-truth data, would be very valuable. At this stage, we just could say that the limitations of this technique correspond to the same ones as for DInSAR or InSAR applications but focused on the monitoring of this specific target (snow or ice) and including precise angle estimations for $\alpha$ and $\beta$, between ascending and descending tracks, and the reference domain in general $(p,q)$ coordinates.

For future experiments, it would be interesting to analyze other SAR data from different satellite platforms, when considering different aspects, as resolution. In our case (Sentinel-1 images), we worked with the C band, where the wavelength range was the interval [3.75 cm, 7.5 cm]. A better wavelength range for a SAR system, in terms of resolution, could be the X band, whose wavelength range is [2.5 cm, 3.75 cm] or the Ku band in the range [1.67 cm, 2.5 cm]. As is well known in the literature [26], short wavelengths permit high-resolution SAR systems and consequently, more detailed interferograms in SAR applications like this one. Normally, for InSAR or DInSAR applications, co-polarized images such as HH or VV are used, instead of cross-polarized ones such as HV or VH. This is due to the evident effects in the coherence parameter, where lower coherence values take place in DInSAR signals recovered from cross-polarized images. Low coherence values imply poor quality fringes and deficient or impossible unwrapped phase estimates. With respect to general features on the DInSAR interferogram constructions, they entirely depend on the image information, the reference DEM used in the coregistration process, and the possible additional preprocessing steps like the wrapped phase filtration. For example, the fringes recovered for the descending interferogram may look slightly different, if we add a multi-look process after the Goldstein phase filtering. The fringes would be more visible and less noisy but with lower resolution due to the sub-sampling induced by this process. In the case of the image coregistration, a low-resolution DEM (with respect to the native image resolution) makes more sense than trying to use a higher-resolution one. Of course, a higher-resolution DEM (not superior to the native image resolution) may offer a finest interferogram resolution. However, for the data in our experiment with IW-SLC images, a 30 m resolution DEM was enough. For instance, the native finest image resolution for IW-SLC products is in the range direction, which is properly a range spacing given by $\Delta r=$2.33 m. After a coregistration process with the Copernicus 30 m DEM in our experiment, the interferogram range resolution was $\Delta r=$3.57 m, and after a ground range conversion, the resulting ground range resolution was $\Delta y=$3.59 m, something quite acceptable.

4. Simulated Experiment: Accuracy Analysis

4.1. Synthetic Data Generation

Assuming this time the ascending–descending configuration in Figure 1a, with $\beta =0$ [deg], our simulated interferograms ${\psi }_a=W\left[{\varphi }_a\right]$ [rad] and ${\psi }_d=W\left[{\varphi }_d\right]$ [rad], as functions of points $(p,q)$, could be generated by considering a rectangular domain $\Omega =[0,p_{\max }]\times [0,q_{\max }]=[0,1495]\times [0,2990]$ induced by a mesh of discrete points $p_n$ and $q_m$ with sampling factors $\Delta p=5$ [m], $\Delta q=10$ [m], and $N=M=300$, respectively. The angular functions ${\theta }_a(p,q)=0.00006p+29.9541$ [deg] and ${\theta }_d(p,q)=(0.0918/d_0)[cos\left(\alpha \right)p+sin\left(\alpha \right)q]+29.9541$ [deg], where $d_0=\sqrt{p_{\max }^2+q_{\max }^2}$, were introduced to simulate the ascending and descending information of a hypothetical system with satellite eight $H=700000$ [m] and $\alpha =\mathrm{angle}\{{\mathbf{y}}_a,{\mathbf{y}}_d\}=96$ [deg] (see Figure 9a,b).

Now, the artificial DEM can be

$$h(p,q)=500exp(-\{4\times {10}^{-6}[{(p-p_c)}^2+{(q-q_c)}^2]\})\left[\mathrm{m}\right]$$

(19)

with velocity components $v_p(p,q)=7.5sin\left[0.005(p-p_c)\right]$ [m/yr] and $v_q(p,q)=0.005p+0.001q$ [m/yr], where $p_c,q_c=p_{\max }/2,q_{\max }/2$ [m], respectively. From the SPF hypothesis, the velocity component $v_z(p,q)$ [m/yr] is given by Equation (1), and, from the explicit dot products on the left in Equation (4), it can be deduced that

$$\begin{array}{c}{v}_{p}cos\left(\beta \right)+v_{q}sin\left(\beta \right)=v_{y_a}+v_{y_d}cos\left(\alpha \right),\\ v_{p}cos(\alpha +\beta )+v_{q}sin(\alpha +\beta )=v_{y_a}cos\left(\alpha \right)+v_{y_d},\end{array}$$

(20)

or equivalently that

$$\begin{array}{c}{v}_{y_a}=v_p[cos\left(\beta \right)+sin\left(\beta \right)cot\left(\alpha \right)]+v_q[sin\left(\beta \right)-cos\left(\beta \right)cot\left(\alpha \right)],\\ v_{y_d}=-v_p[sin\left(\beta \right)/sin\left(\alpha \right)]+v_q[cos\left(\beta \right)/sin\left(\alpha \right)],\end{array}$$

(21)

for general angles $\alpha$, $\beta$. The system in Equation (21) induces a way to simulate artificial components $v_{y_a}(p,q)$, $v_{y_d}(p,q)$ in terms of $v_p(p,q)$, $v_q(p,q)$, which in our case ($\beta =0$) would be $v_{y_a}=v_p-v_{q}cot\left(\alpha \right)$ and $v_{y_d}=v_q/sin\left(\alpha \right)$. These last components can be used to declare from Equation (3) that

$$\begin{array}{c}{\varphi }_a=[(4\pi /\lambda )\Delta t_{a}sin\left({\theta }_a\right)][v_{y_a}+v_{z}cot\left({\theta }_a\right)],\\ {\varphi }_d=[(4\pi /\lambda )\Delta t_{d}sin\left({\theta }_d\right)][v_{y_d}+v_{z}cot\left({\theta }_d\right)],\end{array}$$

(22)

for all $(p,q)=(p_n,q_m)$, in terms of our synthetic data ${\theta }_a$, ${\theta }_d$, $v_p$, $v_q$, $v_z$, with $\Delta t_a=\Delta t_d=0.0329$ [yr] and $\lambda =0.056$ [m].

From functions ${\varphi }_a$ and ${\varphi }_d$, artificial SAR interferograms can be simulated by considering the addition of noise in the signal $\mu =cos\left(\varphi \right)+isin\left(\varphi \right)=:c+is$, where $\varphi ={\varphi }_a,{\varphi }_d$. For example, the cosine term can be rewritten as $c_{\eta }(n,m)=c(n,m)+(\eta /100)[-1+2\mathrm{rand}(n,m)]$, where $0\le \mathrm{rand}(n,m)\le 1$ is a random noise distribution, and $\eta$ is the percentage of noise added. In our case, we worked with signals ${\mu }_{\eta }=c_{\eta }+i{s}_{\eta }$ with $\eta =15$. Thus, any noisy version of ${\psi }_a$ or ${\psi }_d$ was derived from an extended inverse tangent $\psi =\mathrm{atan}2[s_{\eta },c_{\eta }]$. The noisy interferograms ${\psi }_a$ and ${\psi }_d$ are shown in Figure 9c,d, and the simulated velocity components $v_p$, $v_q$, and $v_z$ are displayed in Figure 10a–c, respectively.

4.2. Errors Achieved by Different Ascending–Descending Across-Track Angles

Now, considering scalar fields $v=v(n,m)$ as ${\mathbb{R}}^{NM}$ vectors with Euclidean norm $\left|\right|v\left|\right|$, if we denote the estimations of velocity components as ${\widehat{v}}_{•}$, where $•=p,q,z$, we can use the normalized error $0\le E\left[{\widehat{v}}_{•}\right]=\left|\right|v_{•}-{\widehat{v}}_{•}\left|\right|/\left(\right||v_{•}\left|\right|+\left|\right|{\widehat{v}}_{•}\left|\right|)\le 1$ (dimensionless) to analyze the accuracy of our estimations with Equations (11) and (15), after applying a thresholding step with the corresponding bounds for each synthetic, true data point $v_{•}$. The same normalized error formula can be used to analyze the accuracy of our unwrapped estimates, which, in this case, combine a preprocessing step consisting of applying an average filter to signal ${\mu }_{\eta }$, the FFT unwrapping method, and a post-processing step of congruence [23]. The resultant FFT estimates $\widehat{\varphi }={\widehat{\varphi }}_a,{\widehat{\varphi }}_d$ are plotted in Figure 9e,f, where their re-wrapped versions $W\left[\widehat{\varphi }\right]=\mathrm{atan}2[sin\left(\widehat{\varphi }\right),cos\left(\widehat{\varphi }\right)]$ correspond to items (g) and (h), respectively. The normalized errors in unwrapped phases and velocity components through Equations (11) and (15) are displayed in

Table 4. Normalized errors in velocity estimates $E\left[{\widehat{v}}_{•}\right]$ ($•=p,q,z$) and in unwrapped phases $E\left[\widehat{\varphi }\right]$ ($\widehat{\varphi }={\widehat{\varphi }}_a,{\widehat{\varphi }}_d$), for different angles $\alpha$ [deg], Equations (11) and (15), and a noise factor $\eta =15$.

$\mathit{\alpha }$	Equation	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{p}}\right]$	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{q}}\right]$	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{z}}\right]$	$\mathit{E}\left[{\widehat{\mathit{\varphi }}}_{\mathit{a}}\right]$	$\mathit{E}\left[{\widehat{\mathit{\varphi }}}_{\mathit{d}}\right]$
96	Equation (15)	0.1518	0.1161	0.2250	0.0047	0.0075
	Equation (11)	0.0424	0.0323	0.0646
100	Equation (15)	0.2027	0.1557	0.2980	0.0073	0.0045
	Equation (11)	0.0356	0.0274	0.0562
135	Equation (15)	0.4741	0.3098	0.5981	0.0076	0.0039
	Equation (11)	0.0259	0.0252	0.0597

for $\alpha =96$ [deg] and $\eta =15$. As we can see, if $\alpha$ was close to 90 [deg], matrices $(\mathbf{I}+\mathbf{A}\mathbf{B}\mathbf{C})$ and $\mathbf{B}$ would be the same as $(\mathbf{I}+\mathbf{A}\mathbf{C})$ and $\mathbf{I}$, respectively, and good results from Equation (15) would be plausible. However, when considering $\alpha$ angles different from 90, such as 96, 100, or 135 (see Figure 9i–p), the accuracy of velocity estimates with Equation (15) decreased, in comparison to our correction (see Figure 10 and Figure 11, and the corresponding errors in Table 4).

As previously mentioned, velocity estimations may be quite sensitive to unwrapped phase errors and the ill-conditioning of matrices $(\mathbf{I}+\mathbf{A}\mathbf{B}\mathbf{C})$ and $(\mathbf{I}+\mathbf{A}\mathbf{C})$. That is why we plotted in Figure 10 and Figure 11 the condition number of these matrices, in natural logarithmic scale, i.e., $\ln \left(\right|{\lambda }_{+}|/|{\lambda }_{-}\left|\right)$, where $|{\lambda }_{+}|$, $|{\lambda }_{-}|$ are the module’s maximum and minimum matrix eigenvalues, respectively. To have an idea of the sensitivity of the velocity estimates with respect to the unwrapping errors, we also repeated these reconstructions with different $\alpha$ angles, by keeping the same velocity components but working with an increased noise factor to $\eta =20$, as reported in

Table 5. Normalized errors in velocity estimates $E\left[{\widehat{v}}_{•}\right]$ ($•=p,q,z$) and in unwrapped phases $E\left[\widehat{\varphi }\right]$ ($\widehat{\varphi }={\widehat{\varphi }}_a,{\widehat{\varphi }}_d$), for different angles $\alpha$ [deg], Equations (11) and (15), and a noise factor $\eta =20$.

$\mathit{\alpha }$	Equation	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{p}}\right]$	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{q}}\right]$	$\mathit{E}\left[{\widehat{\mathit{v}}}_{\mathit{z}}\right]$	$\mathit{E}\left[{\widehat{\mathit{\varphi }}}_{\mathit{a}}\right]$	$\mathit{E}\left[{\widehat{\mathit{\varphi }}}_{\mathit{d}}\right]$
96	Equation (15)	0.1311	0.1123	0.2125	0.0425	0.0116
	Equation (11)	0.0913	0.0664	0.1296
100	Equation (15)	0.3079	0.1918	0.4168	0.1321	0.0252
	Equation (11)	0.2097	0.1289	0.2956
135	Equation (15)	0.5228	0.3199	0.6366	0.0982	0.0067
	Equation (11)	0.1725	0.1129	0.2835

.

Clearly visible errors for the velocity estimations in the pixels of high condition number can be observed as “rings” in these particular examples (see items (h) and (l) in Figure 10 and Figure 11). Therefore, the need to consider some thresholds (information known in these simulations) is another important step for general applications with real data.

The results reported in Table 4 and Table 5 can be interpreted as follows: For $\alpha =96$ [deg], the worst normalized errors observed in Table 4 correspond to the estimates of $v_z$ ($E=0.225,0.0646$), while the worst phase estimate due to unwrapping is given by ${\widehat{\varphi }}_d$ ($E=0.0075$). This means that for the worst error in the unwrapping stage, at about 0.75%, the worst error induced by Equation (15) is 22.5%, while with our correction, it is 6.46%.

5. Discussion and Conclusions

A mathematical review of the method proposed by Joughin, Kwok, and Fahnestock (JKF) in [6], for 3D ice-flow estimation from DInSAR data, was described. Part of our contribution in this work was the way to implement a simplified version of JKF’s method in one of the most popular free software tools: the SNAP(9.0.0) or Sentinel-1 toolbox [19] and the snappy library [22]. However, when investigating this method, a subtle detail in one of the main equations was found; specifically, Equation (15). Thus, a reformulation of this equation (given by Equation (11)), apparently unreported yet, was provided. Following this theory, a simplification of 3D ice-flow estimation in a satellite’s ascending or descending orthogonal coordinates (across-track, azimuth, and vertical directions), was also proposed, without the need to consider additional reference of a third coordinate system. This means that only information of the Sentinel-1 image captures is required.

We want to emphasize that JKF’s work has been one of the most important references on the subject since 1998; it is a very interesting approach, and a very valuable tool for the recovery of ice-flow velocity estimates from SAR data, when properly applied. Moreover, this technique could be generalized to other possible InSAR or DInSAR applications, by considering a similar hypothesis to the SPF Equation (1), which basically relates the velocity components through a linear model. Nevertheless, some references such as [27] have pointed out the accuracy differences between InSAR measurements based on the SPF hypothesis and alternative ones based on a GPS method, suggesting that the InSAR velocity estimates could be improved. Therefore, independently of the sensitivity of this technique, with respect to unwrapped phase errors, or the ill-conditioning state of the matrix $(\mathbf{I}+\mathbf{A}\mathbf{C})$ for some pixels, we believe that possible improved velocity estimates could be achieved with the reformulation proposed in this research. Thus, the current reformulated equation based on the original work of [6] could be tested again, by considering reference or in situ data in some future investigations, including possible comparisons with entirely independent techniques, such as the offset tracking [28].

The errors in our computer simulations may give us an idea of the precision achieved with our correction proposed. For example, from Table 5 in the case where $\alpha =135$ [deg], we could say that for the worst error in the unwrapping stage, at about 9.82%, the worst error induced by Equation (15) was 63.66%, while with our correction it was 28.36%. However, these results are only valid when assuming the SPF hypothesis as true, i.e., this is just a preliminary work where future investigations with ground-truth data will be required. Currently, our proposal purports to improve JKF’s work on its own and not to test it or compare it with other methods. For that reason, we believe that computer simulations under the SPF hypothesis are the best way to see the improvement achieved at this stage. Our work can be considered as initial research, that eventually may be supported with more experiments applied to other regions of analysis by considering ground-truth data from open databases, such as EPOS [29].

It is important to remark that our study is a simplified version of JKF’s method, because our velocity vector estimation took place in the ascending or the descending image domain (azimuth and ground range), i.e., the region $\Omega$. Since our current proposal is a 3D velocity estimation on the SAR image domain $\Omega$, where the horizontal components of the flow are in azimuth and ground range directions, it would be very difficult to find an in situ reference for a validation in the same directions as those of the SAR image’s independent variables. Thus, part of our future research work is to analyze the complete and corrected JKF method, by using a third reference coordinate system (when $\beta \ne 0$), where the possible in situ or reference data of the phenomena under study are available. Nevertheless, as a practical experiment in SNAP software, like those given in many of the InSAR or DInSAR application guide manuals [19], and with no more reference data, the velocity vector estimation in the ascending or descending image domain $\Omega$ may be quite useful, as preliminary monitoring.

For testing the corrected JKF approach with other methods, as well as its capabilities and limitations in complex scenarios, we may apply our new formula, by working with DInSAR signals from a fixed region of analysis, in different seasons and with many image pairs as possible, captured at closed periods. However, the simplest technique consists in using just two pairs of images (for recovering the ascending and the descending DInSAR signals) for estimating an instantaneous velocity vector. As described in reference [30], the use of at least two pairs is enough for a 3D velocity recovery. The use of more pairs in closed periods may be useful in order to expand the estimation, in a similar way to a time series analysis [31], whose main purpose would be the estimation of an average velocity vector. However, the latter is a little bit different from an instantaneous measure. The problem of estimating an instantaneous velocity vector from many pairs in closed periods is that we cannot have DInSAR signals at periods close enough. A period of 6 days is the limit for Sentinel-1, but unfortunately, the intersection of the ascending and descending tracks from that period is sometimes too small or impossible to analyze for certain regions. Subsequent estimations at closer periods may be practical to analyze the evolution or the change in the velocity vector along time and conducting at most a semi-local time analysis. But normally, when fixing a region of analysis, the availability of more than two useful Sentinel-1 image pairs for recovering a local or an instantaneous estimation is not always possible.

The implementation of the corrected JKF method would be very useful, not only to analyze its capabilities, but for future more balanced or equitable comparisons with other methods. Thus, independently of the complex factors or limitations of the SPF assumption in connection with the phenomena observed through ascending and descending DInSAR signals, the performance of this technique, at first glance, depends on the ideal conditions required for general InSAR or DInSAR applications that are not always satisfied in some cases. The absence of these conditions is translated in the common drawbacks, which we will deal with in our future analysis: low coherence in some regions of noisy or high-frequency fringes, not enough short baselines for the DInSAR data, and not enough long ones for the InSAR case, small intersections between ascending and descending tracks (where the monitoring of specific regions is not possible), imprecise angle $\alpha$ estimation (including $\beta$, if an additional reference coordinate system is required), bad scaling due to matrix ill-conditioning, unwrapping inconsistencies, etc.