Q–Str2–Models: A software in PyQGIS to obtain Stress–Strain models from GNSS geodynamic velocities

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Computers & Geosciences 172 (2023) 105308

Available online 30 January 2023

Contents lists available atScienceDirect

Computers and Geosciences

journal homepage:www.elsevier.com/locate/cageo

Research paper

Q–Str2–Models: A software in PyQGIS to obtain Stress–Strain models from GNSS geodynamic velocities

Javier Ramírez-Zelaya

^∗

, Luis Miguel Peci, Alberto Fernández-Ros, Belén Rosado, Alejandro Pérez-Peña, Jorge Gárate, Manuel Berrocoso

Laboratory of Astronomy, Geodesy and Cartography, Department of Mathematics, Faculty of Sciences, Campus of Puerto Real, University of Cadiz, 11510 Puerto Real, Spain

A R T I C L E I N F O

Keywords:

Strain rate Stress–Strain models Earth deformation Interpolation

Geodetic deformation software

A B S T R A C T

Q–Str2–Models is an opensource application developed in Python (PyQGIS) and integrated into QGIS that provides us with several tools for creating Stress–Strain models in geodynamically active areas, from the calculation of the horizontal velocities of many points from a temporal or permanent GNSS network. This software generates the results of the maximum geodetic deformation, shear deformation, rotation and dilatation of the points, which are later translated into graphic products; heatmaps, displacement vectors and Stress–

Strain tensors. In this article we show the results of the application of Q–Str2–Models for the analysis of 65 GNSS stations of SPINA (South of the Iberian Peninsula and North Africa) network to know the geodynamic behavior of the Andalusian region and its surroundings.

1. Introduction

A geodetic network is a set of stations used for continuous monitoring of Global Navigation Satellite System (GNSS) called Continuously Operating Reference Stations (CORS) strategically distributed in specific areas, that provide, to multiple users, real–time and post–process services to solve the problem of absolute geodetic positioning. The temporary continuity of GNSS observations allows the definition of geodetic time series whose treatment and analysis allows calculation of displacement and deformation models in geodynamically active areas.

These models allow studying the tectonic–volcanic behavior through the analysis of geospatial data and/or GNSS time series in complex geodynamic scenarios, where significant seismic or volcanic activity oc- curs; contributing to the creation of forecasts on the geodetic behavior of the area under study for the management and prevention of possible risks.

Currently, there is a wide variety of GIS software dedicated to the treatment and analysis of geospatial data. In this case, we used the free software (GNU,2019) Quantum Geographical Information System (QGIS), QGIS (2020) and QGIS Manual (2020), because it is opensource (Opensource,2019), multi–platform and offers advanced tools for the treatment of these kind of data. In addition, it is compatible with the Python language (Python, 2019) using the Application Program- ming Interface (API) Python and Quantum Geographical Information System (PyQGIS) (PyQGIS, 2021), required for development of new applications capable of executing specific tasks through custom scripts.

∗ Corresponding author.

E-mail address: [email protected](J. Ramírez-Zelaya).

In this manuscript we present the application Quantum Stress–

Strain Models (Q–Str2–Models) that allows us to perform different mathematical calculations to obtain displacement models and geodetic deformation of a specific area from a series of data defined by the user. These calculations are converted to graphical products such as: heatmaps, displacement vectors and stress–strain tensors. They have been developed under the opensource tools: PyQGIS, PyQt and Qt–Designer to be integrated into QGIS like a plugin, (PyQt, 2021;

QtDesigner,2021).

The software proposed has great versatility because it is developed for the two most used platforms (operating systems): Windows and Linux; It integrates into one of the free GIS environments and more used such as QGIS; It offers the generation of vectors shapes and customizable raster images; It has no limitations regarding the number of Continuously Operating Global Navigation Satellite Systems (cGNSS) stations used, to obtain the Stress–Strain models; As a free and open source application it can be modified according to the user’s needs, and supports the mitigation of possible risks in the face of certain geological phenomena, more specifically in the tectonic activity and geodetic deformation in active areas.

Q–Str2–Models provides the calculation and graphical representation of a series of geospatial data translated into geodetic velocity and Stress–Strain models, from georeferenced points, which contain horizontal velocities. These velocities are subjected to different interpola- tion techniques (Delaunay, Inverse Distance Weighting (IDW), Exponential

https://doi.org/10.1016/j.cageo.2023.105308

Received 20 September 2022; Received in revised form 25 January 2023; Accepted 25 January 2023

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and Around) considered within a grid of regular points. To illustrate the operability and versatility of the developed software, South of the Iberian Península and North África (SPINA) region has been considered.

As a result we have obtained the models of displacement and Stress–

Strain from the analysis of 65 GNSS stations (1), (2) located in the said region belonging to different regional, national and international networks.

This manuscript is organized as follows: the first Section2.1presents the mathematical background of the algorithms used (interpolations (2.2), geodetic deformation, rotation, dilatation and Stress–Strain tech- niques). In the next Section3we will find a description of the software developed, the libraries and methods used, as well as a diagram that explains the modules that compose it (Processing Module (3.2) and Graphics Module (3.3)). We will also find a brief description of the functionalities that Q–Str2–Models provides to users and the possible results that we will obtain.

In the final Section 4 we describe the results obtained from the use of this software in the analysis of the geodynamic behavior using 65 GNSS stations of the SPINA network (1), (2). These results show heatmaps that contain maximum geodetic deformation, shear deformation, dilatation, rotation, and Stress–Strain tensors. This article focuses on the development of the software, its operation, and implementation in a specific case (SPINA Region) as well as on the analysis of the results obtained and assessment of the contributions and advantages of this application, in the study of the tectonic and geodynamic activity.

2. Methods to estimate a stress–strain rates fields from GNSS geodetic data

2.1. Mathematical background

Currently there are numerous studies of geodynamic deformation modeling based on displacement models obtained from GNSS time series. The accuracy of these models depends on the rigor of the methods used in processing the GNSS observations and the analysis techniques of the calculated geodetic time series (Pagani et al.,2021;

Ali-Goudarzi et al., 2015; Kreemer et al., 2014; Xiong et al., 2021).

The deformation experienced by a solid, characterized as a continuous medium, is based on the study of the spatial variation of the position of each particle of the solid over time (Oliver and Agelet de Saracíbar, 2000;Segall,2010;Valliappan,1981;Marsden and Hughes,1983;Cai et al.,2005;Ali-Goudarzi et al.,2015;Lleó,2000;Allmendinger et al., 2007).

Taking the previous authors as a reference, the mathematical process underlying the characterization of the Stress–Strain parameters from the variability of the position of the particles of a solid is described below. Let {𝑂𝑅; 𝑋₁, 𝑋₂, 𝑋₃}be a three-dimensional Euclidean system.

In a solid 𝛺, the position vector X of a particle P at a given initial instant with respect to the considered reference system is given by:

𝐗= 𝑋₁𝐞_𝟏+ 𝑋₂𝐞_𝟐+ 𝑋₃𝐞_𝟑, (1)

While at a later instant it is:

𝐱= 𝑥₁𝐞_𝟏+ 𝑥₂𝐞_𝟐+ 𝑥₃𝐞_𝟑. (2)

In addition, between the considered pointP and another sufficiently close pointQ of the solid, the difference between their position vectors is given by the vector 𝑑𝐗, whose module represents the differential distance between the two points (Fig. 1), (Fig. 2).

To describe the movement of particles in a continuous medium, the time evolution of the position vector of each particle of the solid is analyzed. This movement can be separated into translation, rotation and deformation. The translation and rotation functions affect the position of the solid in space but do not imply a change in the relative movement between particles. The strain function is defined by:

𝜑∶ 𝛺 → 𝛺_𝑑, 𝐱= 𝜑(𝐗). (3)

Thus, 𝐏 → 𝐏^′, 𝐐 → 𝐐^′, and 𝐝𝐗 → 𝐝𝐱. The strain gradient tensor is given by:

𝐹(𝐗, 𝑡) =𝜕𝜑(𝐗)

𝜕𝐗 = 𝜕𝑥_𝑖

𝜕𝑋_𝑗, 𝑖, 𝑗= 1, 2, 3. (4)

The strain tensor as a function of the displacement field is given by:

𝑢(𝐗) = 𝜑(𝐗) − 𝐗 ⇒ 𝐱 = 𝐗 + 𝑢(𝐗). (5)

Considering 𝐝𝐗 and 𝐝𝐱, we have:

𝑑𝐱= 𝑑𝐗 + 𝑢(𝐗 + 𝑑𝐗) − 𝑢(𝐗). (6)

Expanding by Taylor and approximating to order 2, we obtain:

𝑑𝐱= 𝑑𝐗 + 𝜕𝑢

𝜕𝐗𝑑𝐗= (𝐈 + 𝜕𝑢

𝜕𝐗)𝑑𝐗 = (𝐈 + 𝐇)𝑑𝐗, (7)

where 𝐈 is the unit tensor and 𝐇 =[

𝜕𝑢

𝜕𝐗

]

is the gradient displacement tensor. It is related to the strain gradient tensor by:

𝐇= 𝜕𝑢

𝜕𝐗= 𝜕

𝜕𝐗(𝜑(𝐗) − 𝐗) =𝜕𝜑(𝐗)

𝜕𝐗 −𝜕𝐗

𝜕𝐗= 𝐅 − 𝐈. (8)

On the other hand:

(𝑑𝑠)²= 𝑑𝐱⋅ 𝑑𝐱 = (𝑑𝐱)^𝑡(𝑑𝐱) = (𝐅𝑑𝐗)^𝑡(𝐅𝑑𝐗) = 𝑑𝐗^𝑡𝐅^𝑡𝐅𝑑𝐗, (9) (𝑑𝑆)²= 𝑑𝐗⋅ 𝑑𝐗 = (𝑑𝐗)^𝑡(𝑑𝐗) = (𝑑𝐗)^𝑡𝐈(𝑑𝐗), (10) where:

(𝑑𝑠)²− (𝑑𝑆)²= 𝑑𝐗^𝑡𝐅^𝑡𝐅𝑑𝐗− 𝑑𝐗^𝑡𝐈𝑑𝐗= 𝑑𝐗^𝑡(𝐅^𝑡𝐅− 𝐈)𝑑𝐗. (11) From this expression the Green–Lagrange strain tensor is obtained by:

𝐄= 1

2(𝐅^𝑡𝐅− 𝐈), (12)

and as a function of the displacement gradient tensor 𝐇, we obtain:

𝐄= 1

2((𝐇 + 𝐈)^𝑡(𝐇 + 𝐈) − 𝐈) =1

2(𝐇 + 𝐇^𝑡) +1

2(𝐇^𝑡𝐇). (13)

Approximating the linear part, we have:

𝐄= 1

2(𝐇 + 𝐇^𝑡). (14)

This second order tensor decomposes into the sum of a symmetric tensor 𝜉 that characterizes the deformation process and another anti- symmetric 𝜔 that represents the relative displacements due to rotation,

𝐄= 𝜉 + 𝜔. (15)

Matrically, the strain tensor as a function of the components of the displacement vector is given by:

𝜉=

⎛⎜

⎜⎜

⎜⎝

𝜕𝑢₁

𝜕𝑥₁

1 2(𝜕𝑢₁

𝜕𝑥₂ +𝜕𝑢₂

𝜕𝑥₁) 1 2(𝜕𝑢₁

𝜕𝑥₃ +𝜕𝑢₃

𝜕𝑥₁) 1

2(𝜕𝑢₁

𝜕𝑥₂ +𝜕𝑢₂

𝜕𝑥₁) 𝜕𝑢₂

𝜕𝑥₂

1 2(𝜕𝑢₂

𝜕𝑥₃ +𝜕𝑢₃

𝜕𝑥₂) 1

2(𝜕𝑢₁

𝜕𝑥₃ +𝜕𝑢₃

𝜕𝑥₁) 1 2(𝜕𝑢₂

𝜕𝑥₃ +𝜕𝑢₃

𝜕𝑥₂) 𝜕𝑢₃

𝜕𝑥₃

⎞⎟

⎟⎟

⎟⎠

, (16)

Considering only the two-dimensional horizontal strain along the east and north directions:

𝜉=

⎛⎜

⎜⎜

⎜⎝

𝜕𝑉_𝑒

𝜕𝑒

1 2

(𝜕𝑉_𝑒

𝜕𝑛 +𝜕𝑉_𝑛

𝜕𝑒 )

1 2

(𝜕𝑉_𝑛

𝜕𝑒 +𝜕𝑉_𝑒

𝜕𝑛

) 𝜕𝑉_𝑛

𝜕𝑛

⎞⎟

⎟⎟

⎟⎠

=

(𝜖_𝑥𝑥 𝜖_𝑥𝑦 𝜖_𝑥𝑦 𝜖_𝑦𝑦 )

. (17)

This strain tensor can be transformed to its canonical form considering an orthonormal basis. Their eigenvalues, obtained as a solution of the associated characteristic equation, represent the modules of the main deformations and the eigenvectors represent the main directions of the deformation. In these directions the maximum and minimum

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Computers and Geosciences 172 (2023) 105308 J. Ramírez-Zelaya et al.

Fig. 1. Kinematics of a deformable solid subjected to a given force. From the initial state 𝛺, we pass to the final state 𝛺_𝑑; that with respect to the considered reference system presents a displacement subjected to processes of translation, rotation and deformation. Only deformation involves a change in the shape and dimensions of the solid.

Source:Figure modified fromOliver and Agelet de Saracíbar(2000).

Fig. 2. Definition of the main directions, semi–axes and orientation of the deformation ellipse as a function of the modules of the eigenvalues of the deformation tensor in its canonical form. Positive values of the eigenvalues represent an expansion process and negative values a compression process, both according to the main directions of the deformation ellipse.

Source:Figure taken fromPérez-Peña(2007).

deformation occur and define the semi–major and minor axes of the deformation ellipse. These eigenvalues are given by:

𝜆₁= ¹

2

(

𝜖_𝑥𝑥+ 𝜖_𝑦𝑦+√

(𝜖_𝑥𝑥− 𝜖_𝑦𝑦)²+ 4𝜖_𝑥𝑦²)

∈ R, 𝜆₂= ¹₂(

𝜖_𝑥𝑥+ 𝜖_𝑦𝑦−√

(𝜖_𝑥𝑥− 𝜖_𝑦𝑦)²+ 4𝜖_𝑥𝑦²)

∈ R.

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The orientation of the maximum strain axis with respect to the abscissa axis is given by the angle:

𝜃=1 2arctan

( 2𝜖_𝑥𝑦 𝜖_𝑥𝑥− 𝜖_𝑦𝑦

)

. (19)

The eigenvectors of the strain tensor provide the directions of the axes of the strain ellipses, and the eigenvalues provide the semi- major and minor axes of the ellipse, corresponding to the maximum and minimum strain. These eigenvalues are evaluated as an extension or compression depending on whether they are positive or negative, respectively.

The dilatation parameter, 𝛥, is defined by the trace of the strain tensor matrix, and represents the isotropic part of the strain indicating the relative area variation for infinitesimal strains. Positive values indicate extension, and negative values indicate compression (Grafarend and Voosoghi,2003). The dilatation is given by:

𝛥= 𝜖_𝑥𝑥+ 𝜖_𝑦𝑦. (20)

The maximum shear strain represents the anisotropic part of the strain. It is in the direction of the maximum strain value and indicates the geometric strain of the region. It is always a positive value that is given by:

𝛤=√

(𝜖_𝑥𝑥− 𝜖_𝑦𝑦)²+ 4𝜖_𝑥𝑦². (21)

The maximum geodetic deformation represents the deformation when horizontal movements of the crust are considered, always taking

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positive values, and is given by:

max_DEFGEO= {|𝜆1|, |𝜆2|}. (22)

2.2. Interpolation methods for geodynamic modeling

To determine the parameters of the geodynamic deformation from the displacements of the GNSS stations, it is necessary to model the region covered by the stations; i.e. to find a relationship between the deformation parameters and the position and velocities of the stations. Since the GNSS observations and the displacements obtained are discrete functions in time and space, in order to obtain continuous deformation models it is necessary to consider interpolation techniques that estimate this continuous information. To do this, the area is discretized into smaller regions defining a mesh of points. Thus, the deformation is constrained to the interior of the region included by the mesh. For each element of the mesh, a linear interpolation function of the velocities is found, and the deformation parameters are obtained from itPérez-Peña et al.(2007). Although in this work several interpolation strategies are considered: IDW, Delaunay triangulation and exponential method, there is no single solution to choose one method or another.

The IDW interpolation method is based on establishing the value corresponding to the interpolation point by assigning weights to the surrounding data as an inverse function of the distance that separates them. In this method all points with experimental data are chosen. The general expression is given by:

̂ 𝑧_𝑗=

∑𝑛 𝑖=1

𝑘_𝑖𝑗𝑧_𝑖, (23)

where ̂𝑧_𝑗 is the estimated value for point 𝑗; 𝑛 is the number of points used in the interpolation; 𝑧_𝑖 the value at the ‘‘i-th’’ point and 𝑘_𝑖𝑗 the weight associated with data ‘‘i’’ in the calculation of node ‘‘j’’. The weights ‘‘k’’ vary between ‘‘0’’ and ‘‘1’’ for each datum and the total sum of them is the unit. To establish a proportionality function between weight and distance, the general expression of the method is:

̂ 𝑧_𝑗=

∑

𝑖 𝑧𝑖 𝑑^𝑏_𝑖𝑗

∑

𝑖 1 𝑑^𝑏_𝑖𝑗

, (24)

where 𝑑_𝑖𝑗 is the relative distance between points 𝑖 and 𝑗, and 𝑏 is a weighting exponent that controls how the weight decreases with distance.

The exponential interpolation method is analogous to the interpolation method but weights are assigned to the experimental data based on the following exponential function:

𝑊 = 𝑒

−𝑑2 2𝛼2,

where 𝑑 is the distance between the nodes and 𝛼 is a correction factor (Allmendinger et al.,2007).

Another method derived from the IDW Interpolation is the around method. In this method, for each grid points, a restricted set of points with experimental values will be considered.

The Delaunay triangulation consists of dividing the region into a connected and convex network of triangles. These triangles are defined with the condition that there is no vertex of another triangle in the cir- cle circumscribed by the vertices of the said triangle (Nico et al.,2005).

This condition involves to regularizing the angles of the triangle to the maximum, being the closest to equilateral triangles. The interpolation point will be the centroid of each of the triangles of the defined network and its value will be established from the experimental values of the vertices of the considered triangle.

3. Q–Str2–Models software description

3.1. Libraries used for the software develop

Q–Str2–Models was developed as a complement software for QGIS using the mathematical model described above (2.1) under the Python language and PyQGIS API. It is released under the General Public License (GPL) (GPL,2020) and is cross-platform and can be installed on Windows and Linux operating systems. For the design of the Graphical User Interface (GUI) we used the Qt–Designer application, an auxiliary tool included in the official QGIS package.

PyQGIS has the advantage of interpreting and executing Python code, providing us with different libraries for the development of applications through custom scripts that serve as an auxiliary tool to solve specific tasks, e. g: Process automation and implementation of GIS services, metadata editing, image geoprocessing, layers manipulations, reprojections, objects resampling, etc. In addition, it is capable of handling large volumes of data using custom scripts programmed with simple syntax.

Auxiliary libraries were also used in developing Q–Str2–Models, such as PyQt; A cross-platform object-oriented framework that together with Qt–Designer application provides advanced tools for creating graphical environments easy to use. Qt–Designer has the Qt–Widgets function necessary for developing applications with User Interface (UI).

This provides a comfortable and intuitive interoperability between the users and the application (QtWidgets,2021).

QGIS software allows us to manipulate, edit, analyze and represent geospatial data using advanced applications. It is free, versatile, scalable and cross-platform software, compatible with many external libraries such as: Geospatial Data Abstraction Libray (GDAL) (GDAL, 2021), System for Automated Geoscientific Analyses (SAGA) (SAGA, 2021) and Geographic Resources Analysis Support System (GRASS) (GRASS,2021). This software is dedicated to the treatment and processing of geospatial data, representation and analysis of images in multiple formats. These tools were specifically used in data processing and graphic representation of the results. All the applications and libraries described above are opensource; this allows us free manipulation, publication and distribution, ensuring that other developers can modify the code for their own purposes.

Q–Str2–Models was developed under the concept of modular soft- ware and is divided into two specific modules: The first module (Pro- cessing Module)(3.2) is oriented to the insertion and processing of a series of initial or input data (a data file with coordinates (Latitude, Lon- gitude) and velocities (East, North) of GNSS stations of the study zone, see Tables1and2), as well, the generation of results in ‘‘Comma–Separated Values (CSV)’’ format. The second module (Graphics Module) (3.3) is dedicated to transforming the results obtained into graphic products such as; heatmaps with maximum geodetic deformation, dilatation, rotation, shear deformation and Stress–Strain tensors.

TheFig. 3show the workflow diagram with the external libraries used for software development (GRASS, SAGA, GDAL, PyQt, PyQGIS and Qt–Designer), the modules that compose it (Processing and Graphics Modules), the entities (Layers, Input and Output Data, Displays, etc.), the task flow (Interpolation Methods, grid and Coordinate Reference System (CRS) selection)and the internal processes (Create Grid, Create Raster, Create Vectors and Create Tensors)which are executed to create the final results (CSV, Tag Image File Format (TIFF) and QGIS Layers)available to users.

3.2. Processing module description

The processing module allows the following options: Input data file insert option, selection of destination path of results, selection of active layer to be analyzed, and selection of the different interpolation methods to use. Initially, the active layer & the entry data file should contains geodetic coordinates and horizontal velocities (East, North)

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Fig. 3. ‘‘Q–Str2–Models’’ Workflow Diagram, show: The libraries and APIs required (externals and internals) for the operation of the application, main modules (processing and graphics), required input files (velocities data file and active layer with points), calculation methods and programmed tasks (creation of interpolations, regular grid, tensors, and raster images), output paths and storage of results.

of GNSS stations (Points). The available interpolation techniques are Delaunay, Exponential, IDW, and Around (2.2). Later, we will create a temporal grid in an interest area. This grid is customizable and the user can define the number of rows and columns and the output CRS of the final layer. This grid constitutes the geographical framework for the subsequent graphical representation of the results (maximum geodetic deformation, shear deformation, rotation, dilatation, displacement models and Stress–Strain tensors)(3.3).

To establish a study area we have the option Select Frame that allows us to select, within the active layer, the frame or interest zone required for the generation of the regular grid, using the option Create Grid, this button also runs the user-defined interpolation process and outputs the results. On the right-hand section of the application, we have the display of maximum and minimum values of the selected geographical framework, as well as the velocities values (E, N, U) of the entry data file, (Fig. 4).

Once defined the values of the temporal grid (e.g. 15 × 15) and the interpolation type on the interest zone, we will obtain a CSV file containing the final results depending on the selected interpolation technique, this file will be imported into QGIS automatically. The re- sults will be: Cartesian coordinates (X, Y); Horizontal velocities (E, N);

Maximum and minimum elevations values (EMAX, EMIN, EMAXABS, EMINABS); Maximum geodetic deformation values (MAXGEO); Shear deformation values (SHEAR); Rotation values (ROTATION); Dilatation values (DILATATION); Values of angles of the axes used to generate the Stress–Strain tensors (THETA, THETA90, THETA180, THETA270).

These results will depend on the number of rows and columns that the user defines in the regular grid option, (Fig. 5).

3.3. Graphics module description

The graphics representation module allows the conversion of the results obtained in the processing module (3.2) to heatmaps, from the values of dilatation, rotation, shear deformation and maximum geodetic deformation. These graphic products are created in the area of interest delimited by the points defined in the regular grid, (Fig. 6).

This module must be used after the creation of the regular grid, to generate the graphs. We simply select one of the default items in the multiple selection tab (Geodetic Deformation, Shear Deformation, Dilatation & Rotation)(Fig. 4), we select the local destination folder to save the raster image (Fig. 4) and then we press the Create Raster button (Fig. 4) to create the raster image in TIFF format; then, this image will be automatically imported into QGIS (Fig. 6). In the same module using the corresponding options (Create Vectors & Create Tensors) we will be able to graph the displacement vectors (E, N, U) and the Stress–Strain tensors on the results generated previously. Both options will apply a default style to the properties (symbology) of the active layer.

It is important to mention that Q–Str2–Models applies a basic default style to generated plots, however these can be customized using multiple QGIS options. The figures presented in this article, except for the Stress–Strain tensors and displacement models, were customized using QGIS (color editing, rendering & transparencies) in order to achieve a better illustration of the results, (Fig. 9).

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Fig. 4. Options description available in Q–Str2–Models software: (A) Option to search the entry data file (.csv); (B) Selection box of the destination path to save the results; (C) Selection box of the interpolation methods; (D) Selection box of the active layer to be processed; (E) Selection boxes of rows and columns that will have the temporal grid and CRS output; (F) Display box with the values (E, N, U) of active layer; (G) Select Frame button to select a interest area; (H) Create Grid button to generate the grid and execute the interpolation; (I) Display box with the maximum and minimum coordinate values of the interest area; (J) Create Vectors & Create Tensors buttons to define Stress–Strain tensors and displacement models on the active layer; (K) Create Raster module to define and save the raster image; (L) Create Raster button to execute the option selected (Maximum geodetic deformation ‘‘MAXGEO’’, Dilatation, Shear deformation ‘‘SHEAR’’ & Rotation)by the user on the QGIS active layer; (M) Close button to program exit.

Fig. 5. Figure showing the final results, the regular grid generated, and the import of the layers into QGIS: (A) Shortcut to Q–Str2-Models; (B) QGIS layer box containing the active layer to processed; (C) QGIS main viewer containing the regular grid (15 × 15) in the interest area; (D) QGIS table viewer with the final results obtained: Cartesian coordinates (X, Y); Horizontal velocities (East, North); maximum and minimum elevations values (EMAX, EMIN, EMAXABS, EMINABS); Rotation values of the study area (ROTATION); Dilatation values of the study area (DILATATION); Shear Deformation values on the study area (SHEAR); Maximum Geodetic Deformation on the study area (MAXGEO); and values of angles of the axes used to generate the Stress–Strain tensors (THETA, THETA90, THETA180, THETA270).

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Fig. 6. Raster image generation on interest area defined by the user: (A) Raster image of the Maximum Geodetic Deformation generated on the grid using Q–Str2–Models; (B) Raster image (temporal layer) automatically imported into QGIS layer box.

4. Application example: Q–Str2–Models applied to the SPINA re- gion

4.1. Geodynamics and seismotectonic settings of SPINA region

The region under study is conditioned by its situation with respect to the great Eurasian and African plates, being of great interest for the study of geodynamic deformation. This region corresponds to the transition between the oceanic edge (between the Azores Islands and the Gorringe bank region) and the continental edge where the Iberian Peninsula and Africa meet in the direction to Tunisia. This region is known as Ibero-Magrebi and is delimited by meridians 6^◦W; 1^◦E and parallels 37^◦N; 34^◦N.

This region includes the Betic mountain ranges, the Gulf of Cadiz, the Alboran Sea and the northern part of Morocco, characterized by a large complex of faults, giving rise to a complex tectonic evolution and moderate seismic activity as a consequence of the convergence process between the Eurasian and African plates (Martín-Dávila et al., 2000;

Jiménez-Munt and Negredo,2003;Dewey et al.,1989;Nocquet,2012).

At a detailed level opposing movements are produced, fundamentally due to the difference in oceanic opening velocities in the Atlantic and the structural complexity of the Alboran domain. The representation of these tectonic units can be seen inFig. 7. The plate border is very well delimited in the oceanic part, from the Azores Islands to 12^◦W, along the Azores–Gibraltar fault in an approximate West–East direction, from 11^◦W to 4^◦E. In this region, the border is diffuse (Buforn et al.,2004), (Fig. 7).

The tectonic evolution of the Ibero–Maghreb region is determined by the oblique horizontal compression process, in a NW–SE direction, as a consequence of the convergence of the Eurasian and African plates and the consequent subduction of the oceanic lithosphere (McClusky et al.,2003;Serpelloni et al.,2007). A perpendicular extension ENE–

WSW is associated with the aforementioned convergence, so that currently there are zones in which shortening is fundamentally observed, while there are others in which the extension predominates.

In the area of the Gulf of Cadiz, the compression follows an ap- proximate direction between NW–SE and N–S (approximately 5 mm/yr), while the extension oscillates between SW–NE and W–E. In the Alboran

Sea area, it is observed that the compression follows a N–S direction and an extension in a W–E direction. As we move eastward, compression turns NW–SE and extension SW–NE with displacement values ranging from 1.3 to 1.5 mm/yr. The eastern zone of the peninsula is characterized by a compressive stress regime in a NNW–SSE direction, with a rate of 2.6 mm/yr and a W–E extension turning to NW–SE (Pérez-Peña et al.,2007;Rosado et al.,2017). The seismic activity in the region is characterized as moderate, although very diffuse, and extends along the plate boundary, from south of Saint Vincent Cape to north of Algeria (Buforn et al.,1988). As shown inFig. 7, we can dis- tinguish several areas of deformation with very different characteristics throughout this region, highlighting its complexity (Jiménez-Munt and Negredo,2003).

In the Gulf of Cadiz, the epicenters are distributed in an E–W direction along a 100 km wide band located to the north of the gulf. In the Alboran Sea region, north of Morocco and southeastern peninsular, seismicity is diffuse, which makes it difficult to identify the plate boundary. Most of the seismic activity is concentrated in the shear zones of the Eastern Betic ‘‘EBSZ’’ and its prolongation in the Alboran Sea along the ‘‘TASZ’’. Therefore, the plate boundary is not narrow and clean, but is distributed in a wide deformation zone that includes North Africa, the Alboran Sea and the Betic mountain range.

Numerous geophysical studies have focused on explaining the geodynamic model of this region, especially the evolution of the current Alboran Sea, introducing numerous hypotheses, many of them antag- onistic (Alonso-Chaves et al.,2004). Some of these models are based on the displacement of the Alboran domain as a deformable tectonic terrain and others on the gravitational collapse of an area located in the current Alboran Sea. These studies have proposed different tectonic models based on different mechanisms that would explain the appearance of an extension within the compressive framework between the Eurasian and African plates: On the one hand, those that imply the detachment or delamination of the subcontinental lithosphere under the Alborán domain and, on the other, those that propose the subduction of the oceanic lithosphere associated with a roll-back process, and/or detachment of the subducted plate (Pedrera et al.,2011).

In this study, have been used data from GNSS stations located in the southern region of the Iberian Peninsula and North Africa (SPINA

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Fig. 7. Map of the geodynamic context, seismic activity and main faults of the southern region of the Iberian Peninsula and North Africa, the GNSS networks that were considered in this study are:REGAM,ROA,REP,RENEP,RAP,MERISTEMUM,IGS,IGNandERVA. The most important faults and fractures in the SPINA region are: Gorringe Bank Region, Gulf of Cadiz, Azores–Gibraltar Fault, Saint Vincent Cape, Alboran Sea, Betic Mountain Ranges, Eastern Betic Shear ZoneEBSZ, and Trans–Alboran Shear ZoneTASZ.

Fig. 8. Map of the horizontal geodynamic model of the Southern Region of the Iberian Peninsula and North Africa, based on the results of 65 GNSS stations of the SPINA network, stations were processed using ‘‘IGS2014’’, seeTables 1and2.

Network)(Fig. 7). This network is made up of 65 stations belonging to the regional networks: RAP, RENEP, ERVA, ROA, MERISTEMUM, REP, REGAM, National networks: IGN, and the International networks IGS;

Tables 1and2.

The data from the SPINA network have been processed with the sci- entific software GIPSY–OASIS using the ‘‘PPP’’ (Precise Point Positioning) processing technique (Zumberge et al.,1997), obtaining time series of

geocentric coordinates (X, Y, Z) for the period between 2005 and 2022.

These solutions are transformed to topocentric coordinates (E, N, U) using the Ferrari method (Berrocoso et al.,2004). The analysis of these time series (topocentric coordinates) allows to eliminating the anomalous values that each series presents (Barba et al.,2021), obtaining the hor- izontal displacement vectors (East & North). The horizontal velocities (Fig. 8) constitute the input values of Q–Str2–Models software, (3.2).

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Table 1

Table with coordinates (latitude and longitude), and horizontal velocities (mm/year) of GNSS stations of the SPINA network, considered for the study described above.

Stations Longitude Latitude Ve (mm/yr) Vn (mm/yr)

ALGC −5.44418 36.11104 16.950 18.580

ALMR −2.44088 36.86267 19.087 17.207

ANDU −4.03031 38.04038 18.918 17.517

ARAC −6.56541 37.89388 17.936 18.008

CAAL −2.54761 37.22109 19.226 17.226

CABR −4.42424 37.46795 17.739 17.161

CAZA −5.75978 37.93788 18.705 18.773

CRDB −4.78783 37.87740 19.034 17.444

GRA1 −3.59641 37.18990 17.865 18.531

HULV −6.91354 37.28028 18.582 17.969

HUOV −1.94212 37.40156 19.688 17.985

LEBR −6.08193 36.92242 17.302 19.229

MLGA −4.43541 36.71560 17.506 16.981

MOTR −3.52052 36.75476 17.223 16.302

OSUN −5.09517 37.23231 18.650 17.415

PALC −2.93240 37.70234 17.347 17.960

POZO −4.84933 38.38347 18.801 17.166

ROND −5.14348 36.75404 17.296 17.987

SEVI −5.97156 37.34571 18.822 17.351

UCAD −6.21050 36.53167 17.454 18.482

UJAE −3.78173 37.78776 18.857 17.182

VIAR −3.01248 38.16764 18.800 16.979

ALCO −0.47355 38.69798 19.595 17.992

TORR −0.68089 37.97531 19.858 18.195

ALAC −0.48123 38.33892 20.485 17.433

CEU1 −5.30639 35.89197 15.874 18.477

MELI −2.95166 35.28119 13.381 23.338

TARI −5.60262 36.00851 13.919 20.500

ZFRA −6.41003 38.42601 18.316 16.608

ALBO −3.03581 35.93820 18.955 18.680

SFER −6.12202 36.27515 16.216 17.685

RABT −6.51154 33.59532 16.715 18.587

CASC −9.41853 38.69342 18.613 17.527

LAGO −8.66838 37.09894 17.714 18.037

CABO −0.69839 37.63086 17.552 19.658

CRVC −1.86864 38.11458 16.325 19.554

JUMR −1.32716 38.47122 19.231 18.012

LORR −1.68677 37.65389 28.953 13.296

MURR −1.12469 37.99216 19.379 18.268

MAZA −1.31049 37.59344 18.954 19.553

MORA −1.99883 38.24753 18.929 17.800

MULA −1.44885 38.04111 18.701 17.979

AMAR −7.22788 38.20870 19.243 18.303

Table 2

Table with the rest of GNSS stations of the SPINA network, considered for the study described above.

Stations Longitude Latitude Ve (mm/yr) Vn (mm/yr)

BEJA −7.87285 38.01288 19.029 18.722

BENA −8.12670 37.23237 18.777 17.744

MERT −7.66005 37.64654 19.043 18.979

MESS −8.24469 37.83463 19.271 18.514

ODEM −8.63125 37.59866 18.744 20.971

PALM −8.90345 38.57146 19.855 17.678

SCAC −8.69260 38.01878 19.142 18.259

SMAN −7.75199 38.45922 19.854 18.347

TAVI −7.64091 37.13230 18.794 19.353

JERE −6.77943 38.32005 18.042 17.027

LLER −6.01071 38.23675 18.234 16.761

CHAF −2.43109 35.18348 16.954 18.308

VELE −4.30048 35.17299 15.828 16.585

TETN −5.36301 35.56164 14.558 16.983

BENI −6.31859 32.37679 16.502 18.200

IFRN −5.12552 33.51393 17.156 17.183

TAZA −3.99644 34.22952 15.846 18.699

ALJI −5.64938 36.52988 16.739 17.083

CAST −6.53214 38.12270 17.411 16.750

ALBA −1.85639 38.97791 18.801 16.469

CACE −6.34178 39.47886 18.222 16.585

VALE −0.33764 39.48082 19.375 16.593

4.2. GNSS stations used in the study of the SPINA region

In this section we show the Tables 1 and 2 with the description (station names, coordinates and horizontal velocities) of the 65 GNSS stations of SPINA network used for the study of the geodynamic behavior of Andalusian and surroundings.

5. Conclusions

Q–Str2–Models is an opensource software used like a plugin into QGIS software, that allows us to obtain the calculations resulting from the maximum geodetic deformation, shear deformation, rotation, dilatation, displacement and Stress–Strain tensors of an area, from the horizontal velocities (East, North) of several points (GNSS stations).

These results come from the execution of the interpolation methods:

IDW, Delaunay, Exponential and Around in the study points using a regular grid over an area of interest, which will later allow us to create surface deformation and displacement models to know the geodynamic behavior of the area.

Q–Str2–Models offers the following advantages: it is a multiplatform application; it is free to use and edit; it is developed with the Python programming language (one of the most used in the scientific field) and the PyQGIS API; it has a friendly and easy-to-use graphical interface; it is simply executed by entering a layer in QGIS containing horizontal velocities; with its graphic module it is capable of graphically rep- resenting the results obtained (Maximum geodetic deformation, Shear Deformation, Dilatation, Rotation, Displacement vectors & Stress–Strain Tensors); it calculates the maximum and minimum elevation parameters of the points along with their displacement angles to subsequently cre- ate Stress–Strain tensors; it allows us to export the results in (CSV, TIFF) formats compatible with other current GIS software; being free software, it will receive constant updates that will improve both modules (Graphic Representation & Processing Modules)and the computational performance in the execution of tasks. With the results obtained, it will be possible to create geodynamic models of deformation and displacement, which in turn will help to forecast and monitor dangerous geological phenomena (tectonic–volcanic deformation, landslides, lahars, etc.). Below we show some results obtained from the study carried out on the SPINA region, generated with Q–Str2–Models and QGIS.

TheFig. 9(C) shows the distribution of dilatation rates, negative values of dilatation indicate areas under compression and positive values areas where there are extensions. The Gulf of Cadiz and Alboran Sea zones are characterized by compression, being more pronounced in the North of Morocco (in the area of Al Hoceima). This compression is supposed to be produced by the convergence between the Eurasian and African plates. The Internal area of the Béticas is characterized by positive values, very similar to each other, and therefore it is an area subject to an extension regime. In the eastern part of the Peninsula, values corresponding to a compression regime appear again. TheFig. 9 (A, B) shows the maximum geodetic deformation and maximum shear deformation, both represent strain increase towards the South, being more pronounced in the Gulf of Cadiz zone, Gibraltar Strait and in the North of Morocco (in the zone of Al Hoceima). The intensity of this deformation progressively diminishes towards the center of the Iberian Peninsula. Maximum Shear strain and maximum geodetic deformation are greater in the zone between the tectonic plates. Finally, Fig. 9 (D) shows the magnitude and orientation of principal strains axes in ‘‘𝜇𝑠𝑡𝑟𝑎𝑖𝑛∕𝑦𝑒𝑎𝑟’’, the extensional axes are shown as red color and compressional axes as dark blue color (inside the circles).

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Fig. 9. Representation of the Maximum Geodetic Deformation (A); Shear Deformation (B); Dilatation (C); and Stress–Strain Tensors (D); of the SPINA Region, using ‘‘IDW’’

interpolation method.

Fig. 10. Representation of the Maximum Geodetic Deformation (A); Shear Deformation (B); Dilatation (C); and Stress–Strain Tensors (D); of the SPINA Region, using ‘‘Around’’

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Fig. 11. Representation of the Maximum Geodetic Deformation (A); Shear Deformation (B); Dilatation (C); and Stress–Strain Tensors (D); of the SPINA Region, using ‘‘Delaunay’’

Fig. 12. Representation of the Maximum Geodetic Deformation (A); Shear Deformation (B); Dilatation (C); and Stress–Strain Tensors (D); of the SPINA Region, using ‘‘Exponential’’

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The Gulf of Cadiz area the compression follows an approximated direction between NW–SE and N–S, whereas the extension oscillates between SW–NE and W–E. In the Alboran Sea zone, it is observed that the compression follows a direction N–S and an extension in direction W–E. As we moved towards the east, the compression turns towards direction NW–SE and the extension towards SW–NE. The internal Betic zone is characterized by a field of extension in directions NW–SE turning towards W–E, being the compression minimum. Finally in the Eastern part of the Iberian Peninsula it is observed a compression in direction N–S and an extension W–E having turned NW–SE. In the southeastern zone, an important geodetic deformation is observed due to the consequences produced by the magnitude 5.1 (Mw) earthquake that occurred on May 11, 2011 in the municipality of Lorca (Rosado et al.,2017).

In this manuscript we also show the Figs. 10, 11, and 12 with the graphic results of the geodynamic behavior ((A) Maximum geodetic deformation, (B) Shear deformation, (C) Dilatation, (D) Stress–Strain Ten- sors), of the same region, using the remaining interpolation methods (Around, Delaunay and Exponential).Figs. 10and11show homologous results, except theFig. 12(A, B) in which it is observed that the model does not fit correctly.

All figures shown in the conclusions of this manuscript were cre- ated with Q–Str2–Models (mathematical calculations, interpolations, grids, raster generation, displacement vectors, and Stress–Strain tensors)and they were customized (styles, colors and transparencies) with ‘‘QGIS 3.22 LTR’’.

List of acronyms

API Application Programming Interface

cGNSS Continuously Operating Global Navigation Satellite Systems CORS Continuously Operating Reference Stations

CRS Coordinate Reference System CSV Comma–Separated Values EBSZ Eastern Betic Shear Zone

ERVA Red de Estaciones de Referencia GNSS de Valencia GDAL Geospatial Data Abstraction Libray

GNSS Global Navigation Satellite System GPL General Public License

GRASS Geographic Resources Analysis Support System GUI Graphical User Interface

IDW Inverse Distance Weighting IGN Instituto Geográfico Nacional IGS International GNSS Service

MERISTEMUM Red GNSS MERISTEMUM de Murcia

PyQGIS Python and Quantum Geographical Information System QGIS Quantum Geographical Information System

Q–Str2–Models Quantum Stress–Strain Models RAP Red Andaluza de Posicionamiento

REGAM Red de estaciones GNSS activas de la región de Murcia ROA Real Instituto y Observatorio de la Armada Española REP Red Extremeña de Posicionamiento

RENEP Rede Nacional de Estações Permanentes GNSS SAGA System for Automated Geoscientific Analyses SPINA South of the Iberian Península and North África TASZ Trans–Alboran Shear Zone

TIFF Tag Image File Format UI User Interface

CRediT authorship contribution statement

Javier Ramírez-Zelaya: Formal analysis, Conceptualization, Methodology, Investigation, Writing – original draft, Software developer, Writing – review & editing, Data curation. Luis Miguel Peci: Software developer. Alberto Fernández-Ros: Methodology, Software developer. Belén Rosado: Formal analysis, Investigation, Writing – original draft. Alejandro Pérez-Peña: Formal analysis, Supervision, Validation. Jorge Gárate: Data curation, Supervision, Validation. Manuel Berrocoso: Conceptualization, Formal analysis, Investigation, Supervision, Methodology, Project administration, Resources, Funding acquisition, Writing – review & editing.

Declaration of competing interest

The authors declare the following financial interests/personal rela- tionships which may be considered as potential competing interests:

Manuel Berrocoso reports financial support was provided by Ministry of Education and Science of Spain.

Data availability

Data will be made available on request.

Acknowledgments

This research has been carried out with the support of the Ministry of Education and Science of Spain through the following research project; ‘‘Monitoring and surveillance of active geodynamic processes by geodetic GNSS deformation in different places (Antarctica, Gulf of Cadiz and Latin America) (CTM2017-84210-R)’’.

We thank the anonymous reviewers and editors for their correc- tions and suggestions that have helped us significantly improve this document.

We also thank each of the institutions listed below for making the data used in this study publicly available:

Red Geodésica de Estaciones de Referencia de Valencia (ERVA) https://icvficherosweb.icv.gva.es/ICV/geova/erva/Datos/

International GNSS Service (IGS) https://igs.org/

Instituto Geográfico Nacional (IGN)

http://www.ign.es/web/ign/portal/gds-area-geodesia

Red GNSS MERISTEMUM de Murcia (MERISTEMUM) http://gps.medioambiente.carm.es/

Red Andaluza de Posicionamiento (RAP)

https://www.juntadeandalucia.es/institutodeestadisticaycarto grafia/rap/

Red Extremeña de Posicionamiento (REP) http://www.rep-gnss.es/

Red de estaciones GNSS activas de la región de Murcia (REGAM) https://portaleslr.carm.es/web/sitmurcia/regam-presentacion

Real Instituto y Observatorio de la Armada Española (ROA) https://armada.defensa.gob.es/ArmadaPortal/page/Portal Rede Nacional de Estações Permanentes GNSS (RENEP)

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https://www.dgterritorio.gov.pt/cartografia_e_geodesia/geodesia/r edes_geodesicas/renep/

Code availability section

The developers are: Javier Ramírez-Zelaya, Luis Miguel Peci and Alberto Fernández–Ros; contact address: Laboratory of Astronomy, Geodesy and Cartography, Department of Mathematics, Faculty of Sci- ences, Campus of Puerto Real, University of Cadiz, 11510 Puerto Real, Spain; telephone number: (+34) 956012830; e–mail (corresponding author):[email protected].

The minimum hardware required is a computer with dual core processor, 2 GB RAM, 3 GB of free hard disk space. Software required is ‘‘QGIS 3 LTR’’ (3.22 or later), the main programming languages are

‘‘Python 3’’ and ‘‘PyQGIS API’’, the program size is 350 kB approximately. The source code, user guide and plugin test data, are hosted at:

Official Website of the LAGC–UCA:

https://lagc.uca.es/servicios-homologados/software/

Official GitHub Repository of the LAGC–UCA:

https://github.com/lagc-uca/Q-Str2-Models.git

The output data files with the results (maximum geodetic deformation, shear deformation, rotation, dilatation and stress–strain tensors values)of each interpolation performed (IDW, exponential, Delaunay, and Around) at the 65 GNSS stations of the SPINA region; they are available in the folder ‘‘SPINA Results’’ located in the aforementioned web repositories.

References

Ali-Goudarzi, M., Cocard, M., Santerre, R., 2015. GeoStrain: An opensource software for calculating crustal strain rates. Comput. Geosci. 82, 1–12.

Allmendinger, R., Reilinger, R., Loveless, J., 2007. Strain and rotation rate from GPS in Tibet, Anatolia, and the Altiplano. Tectonics 26 (3), 1–18.

Alonso-Chaves, F., Soto, J., Orozco, M., Kilias, A., Tranos, M., 2004. Tectonic evolution of the betic cordillera: an overview. In: Proceedings of the 10th International Congress, Bulletin of the Geological Society of Greece, Vol. XXXVI. Thessaloniki, Greece.

Barba, P., Rosado, B., Ramírez-Zelaya, J., Berrocoso, M., 2021. Comparative analysis of statistical and analytical techniques for the study of GNSS geodetic time series. Eng.

Proc. 5 (1), http://dx.doi.org/10.3390/engproc2021005021, URL: https://www.

mdpi.com/2673-4591/5/1/21.

Berrocoso, M., Ramírez, M., Pérez-Peña, A., Enríquez-Salamanca, J., Fernández, A., Tor- recillas, C., 2004. El Sistema de Posicionamiento Global. Servicio de Publicaciones de la Universidad de Cádiz.

Buforn, E., Bezzeghoud, M., Udías, A., Pro, C., 2004. Seismic sources on the Iberia- African plate boundary and their tectonic implications. Pure Appl. Geophys. 161, 623–646.

Buforn, E., Udías, A., Colombás, M., 1988. Seismicity, source mechanisms and tectonics of the Azores-Gibraltar Plate Boundary. Tectonophysics 152, 89–118.

Cai, J., Grafarend, E., Schaffrin, B., 2005. Statistical inference of the eigenspace components of a two-dimensional, symmetric rank-two andow tensor. J. Geod. 78, 425–436.

Dewey, J.F., Helman, M.L., Turco, E., Hutton, D.H., Knott, S.D., 1989. Kinematics of the Western Mediterranean. Eds. Alpine tectonics. The Geological Society, London, Special Publication 45.

GDAL, 2021. GDAL official website, GDAL documentation. (accessed on 11-03-2021).

URL:https://gdal.org/.

GNU, 2019. šQué es el Software Libre?, GNU official website. (accessed on 04-01-2019).

URL:https://www.gnu.org/philosophy/free-sw.html.

GPL, 2020. A quick guide to GPL, GNU official website. (accessed on 10-05-2020).

URL:https://www.gnu.org/licenses/quick-guide-gplv3.html.

Grafarend, E., Voosoghi, B., 2003. Intrinsic deformation analysis of the Earth’s surface based on displacement fields derived from space geodetic measurements. Case studies: present-day deformation patterns of Europe and of the Mediterranean area (ITRF dada sets). J. Geod. 77, 303–326.

GRASS, 2021. GRASS GIS official website. (accessed on 10-05-2021). URL: https:

//grass.osgeo.org/learn/.

Jiménez-Munt, I., Negredo, A., 2003. Neotectonic modelling of the western part of the Africa-Eurasia plate boundary: from the Mid-Atlantic ridge to Algeria. Earth Planet.

Sci. Lett. 205, 257–271.

Kreemer, C., Blewitt, G., Klein, E., 2014. A geodetic plate motion and Global Strain Rate Model. Geochem. Geophys. Geosyst. 15, 3849–3889.

Lleó, A., 2000. Tensores Cartesianos. Teoría y Problemas. Universidad Nacional de Educación a Distancia, Madrid.

Marsden, J., Hughes, T., 1983. Mathematical Foundations of Elacticity. Prentice-Hall Englewood Cliffs.

Martín-Dávila, J., Gárate, J., Pazos, A., 2000. Geophysical/Geodetical activities of the ROA to study the Eurasia-Africa plate boundary zone in the Ibero-Maghrebian region. In: The Tenth General Assembly of the Wegener Project. Extended Abstracts Book. Ministerio de Defensa.

McClusky, S., Reilinger, R., Mahmoud, S., Ben Sari, D., Tealeb, A., 2003. GPS constraints on Africa (Nubia) and Arabia plate motions. Geophys. J. Int. 155, 126–138.

Nico, G., Rutigliano, P., Benedetto, C., Vespe, F., 2005. Terrain modelling by kinematical GPS survey. Nat. Hazards Earth Syst. Sci. 5, 293–299.

Nocquet, J., 2012. Present-day kinematics of the mediterranean: a comprehensive overview of GPS results. Tectonophysics 579, 220–242.

Oliver, X., Agelet de Saracíbar, C., 2000. Mecánica de Medios Continuos Para Ingenieros. Ediciones UPC, Barcelona, España.

Opensource, 2019. Open Source Software official website, What is opensource software? (accessed on 11-05-2019). URL: https://opensource.com/resources/what- open-source.

Pagani, C., Bodin, T., Métois, M., Lasserre, C., 2021. Bayesian estimation of surface strain rates from global navigation satellite system measurements: Application to the Southwestern United States. J. Geophys. Res.: Solid Earth 126, 1–25.

Pedrera, A., Ruiz-Constán, A., Galindo-Zaldivar, J., Chalouan, A., Sanz de Galdeano, C., Marín-Lechado, C., Ruano, P., Benmakhlouf, M., Akil, M., López-Garrido, A., Chabli, A., Ahmamou, M., Gonzalez Castillo, L., 2011. Is there an active subduction beneath the Gibraltar orogenic arc?. Constraints from Pliocene to present-day stress field. J. Geodyn. 52, 83–96.

Pérez-Peña, A., 2007. Modelización de las Deformaciones Corticales en el Sur de España y Norte de África a Partir de Observaciones de Satélites (GPS) (Ph.D. thesis).

Facultad de Ciencias, Universidad de Cádiz, Cádiz, Spain.

Pérez-Peña, A., Martín-Davila, J., Gárate, J., Berrocoso, M., Buforn, E., 2007. Velocity field and tectonic strain in Southern Spain and surrounding areas derived from GPS episodic measurements. J. Geodyn. 48, 232–240.

PyQGIS, 2021. Pyqgis official website, PyQGIS Developer Cookbook. (accessed on 10-03-2021). URL:https://docs.qgis.org/3.16/en/docs/pyqgis_developer_cookbook/

index.html.

PyQt, 2021. PyQt official website, Qt 5 Documentation. (accessed on 11-03-2021). URL:

https://doc.qt.io/qt-5.15/.

Python, 2019. Python official website, Python3 Documentation. (accessed on 15-05-2019). URL:https://docs.python.org/3/.

QGIS, 2020. QGIS official website. (accessed on 02-01-2020). URL:https://docs.qgis.

org/3.16/es/docs/user_manual/.

QGIS Manual, 2020. QGIS Training Manual, official website. (accessed on 02-01-2020).

URL:https://docs.qgis.org/3.16/en/docs/training_manual/index.html.

QtDesigner, 2021. Qtdesigner official website, Qt5 Designer Manual. (accessed on 11-03-2021). URL:https://doc.qt.io/qt-5/qtdesigner-manual.html.

QtWidgets, 2021. Qtwidgets official website, QtWidgets Documentation. (accessed on 11-03-2021). URL:https://doc.qt.io/qt-5/qtwidgets-index.html.

Rosado, B., Fernández-Ros, A., Jiménez, A., Berrocoso, M., 2017. GPS horizontal deformation model in the southern region of the Iberian Peninsula and Northern Africa (SPINA). Bol. Geol. Min. 128, 1, 141–156.

SAGA, 2021. SAGA GIS official website, Tutorial for the SAGA GIS API for Python.

(accessed on 12-03-2021). URL: https://saga-gis.sourceforge.io/saga_api_python/

index.html.

Segall, P., 2010. Earthquake and Volcano Deformation. Princeton University Press, New Jersey, USA.

Serpelloni, E., Vannucci, G., Pondrelli, S., Argnani, A., Casula, G., Anzidei, M., Baldi, P., Gasperini, P., 2007. Kinematics of the western Africa–Eurasia plate boundary from focal mechanisms and GPS data. Geophys. J. Int. 169, 1180–1200.

Valliappan, S., 1981. Continuum Mechanics. Fundamentals. Ed. A.A. Balkema, Rotterdam.

Xiong, Z., Zhuang, J., Zhou, S., Matsuura, M., Hao, M., Wang, Q., 2021. Crustal strain- rate fields estimated from GNSS data with a Bayesian approach and its correlation to seismic activity in Mainland China. Tectonophysics 815, 1–17.

Zumberge, J., Heflin, M., Jefferson, D., Watkins, M., Webb, F., 1997. Precise point positioning for the efficient and robust analysis of GPS data from large networks.

J. Geophys. Res. 102, 83, 5005–5017.