Abstract
We used 33 stations belonging of the Ecuador Continuous Monitoring GNSS Network (REGME) during the period 2008-2014, with aim to contribute with a methodological approach for the estimation of a new velocity model for Continental Ecuador. We used daily solutions to perform the analysis of GNSS time series, to obtain models of the series that best fit, taking into count the trend, seasonal variations and the type of noise. The sum of all these components represent the real-time series, and thus we can have a better estimation of the velocity parameter and its uncertainty.
The velocities were calculated introducing the trend, seasonality and noise that were presented in each series into the overall model, which improved uncertainty by 12% and changed in magnitude up to 1.7 mm/yr and 2.5 mm/yr in the horizontal and vertical components, respectively, with respect to the initial velocities. The velocity field describes the crustal movement of the REGME stations in mainland Ecuador with uncertainty of 1 mm/yr and 2 mm/yr for the horizontal and vertical components, respectively. Finally, a velocity model has been developed using the kriging technique whose geostatistical approach has been based on the data to identify the spatial characteristics by examining the observations by peers. The mean square error (rms) of prediction obtained in this method is 1.78 mm/yr and 1.95 mm/yr in the east and north components, respectivaly. The vertical component could not be modeled due to its chaotic behavior.
1 Introduction
From a tectonic point of view, Ecuador is located in the zone of convergence between the plates of Nazca and South America. This is characterized by a high rate of deformation, high seismic activity and a complicated regime of stress, mainly caused by the subduction of the Nazca oceanic plate and the presence of a complex system of active faults that generate cortical earthquakes.
The tectonic of the zone is controlled by the subduction of the Nazca oceanic plate under the continental plate of South America (Figure 1). The location and geometry of the subducted plate exert a primary control in the active volcanism of the North Central Andes Mountains, the transpresional deformation in the upper plate and the seismicity at different levels of depth [1, 2].
![Figure 1 Seismotectonic frame of Ecuador. Black vectors indicate the relative rate and azimuth between the Nazca and South America plates [3] and the movement of the North Andean Block relative to the South America plate [5]. The eastern and southern limits of the North Andean Block correspond to the major fault system. The black squares show the position of the REGME stations used in this study. The yellow triangles indicate the position of the volcanoes. Red circles show instrumental seismicity (Mw>5 depth<40 km period 2008-2014) from the USGS (United States Geological Survey) catalog of the National Earthquake Information Center (NEIC). Red stars indicate the epicenters of the major subduction earthquakes. Inset shows tectonic setting of the zone, including the plates, microplates and blocks that interact in this region: Caribbean, South American, Nazca and Cocos plates, Panama Micro-plate and North Andean Block. Figure adapted from [21].](/document/doi/10.1515/geo-2017-0054/asset/graphic/j_geo-2017-0054_fig_001.jpg)
Seismotectonic frame of Ecuador. Black vectors indicate the relative rate and azimuth between the Nazca and South America plates [3] and the movement of the North Andean Block relative to the South America plate [5]. The eastern and southern limits of the North Andean Block correspond to the major fault system. The black squares show the position of the REGME stations used in this study. The yellow triangles indicate the position of the volcanoes. Red circles show instrumental seismicity (Mw>5 depth<40 km period 2008-2014) from the USGS (United States Geological Survey) catalog of the National Earthquake Information Center (NEIC). Red stars indicate the epicenters of the major subduction earthquakes. Inset shows tectonic setting of the zone, including the plates, microplates and blocks that interact in this region: Caribbean, South American, Nazca and Cocos plates, Panama Micro-plate and North Andean Block. Figure adapted from [21].
In order to characterize the relative kinematics of Nazca in relation to the apparently stable South America, several studies have been carried out that proposes velocities and orientations regarding the movement of the oceanic plate. Hereby, the Nazca plate converges along the Ecuadorian margin at a velocity of 55-58 mm/yr in the direction of N83°E [3,4,5] (Figure 1).
The oceanic Nazca plate is subducted with an angle slightly oblique to the southern American continent producing an overall active tectonic regime with transpression due to its convergence [6,7,8,9]. However, the geometry of the subduction of the oceanic plate by depth contours has been consistented with the tectonic and seismic characteristics of the region, proposing a subduction start of a low slope, which would be the cause of accumulation of tension and seismicity generation until a depth of 40 to 50 km [10,11,12], to later increase its slope to beyond the East of the Andes (Figure 2).
![Figure 2 Representation of the subduction geometry using depth contours (units in km). Abbreviations are: IS-RN, Subduction Interphase of Northern Region; IS-RC, Subduction Interphase of the Central Region; IS-RS, Subduction Interphase of the Southern Region; SP-RN, Deep Subduction of the Northern Region; SP-RS, Deep Subduction of the Southern Region. Figure taken from [12].](/document/doi/10.1515/geo-2017-0054/asset/graphic/j_geo-2017-0054_fig_002.jpg)
Representation of the subduction geometry using depth contours (units in km). Abbreviations are: IS-RN, Subduction Interphase of Northern Region; IS-RC, Subduction Interphase of the Central Region; IS-RS, Subduction Interphase of the Southern Region; SP-RN, Deep Subduction of the Northern Region; SP-RS, Deep Subduction of the Southern Region. Figure taken from [12].
This part of the continental South American Plate is characterized by the presence of the North Andean Block (NAB) [3, 5] (Figure 1). This block may have resulted from a combination of factors, including the oblique subduction of the Nazca Plate, the coupling of the subduction interface, and the subduction of the Carnegie Ridge, among others. [5] estimate that this block is moving NE at a velocity of 7.5-9.5 mm/yr and describes its kinematic motion as starting from the center of Ecuador and heading towards Colombia.
Ecuador has a history of major earthquakes associated with the subduction zone. The latest of these occurred on April 16, 2016. It was an important Mw 7.8 magnitude seismic event with the epicenter near the city of Pedernales (Ecuador). Several magnitude 7 or greater earthquakes have occurred within 250 km of this event since 1900 (red stars in Figure 1) (1906, Mw = 8.8; 1942, Mw = 7.8; 1956, Mw = 7.0; 1958, Mw = 7.7; 1979, Mw = 8.2; 1987; Mw = 7.2; 1998, Mw = 7.2 and 2016, Mw = 7.8).
In order to understand the causes of these events and their geodynamic processes, it is fundamental to study the kinematic of this region. Over the past 15 years, continuous and campaign-type GPS measurements have been used to determine movement and deformation of the crust in this area and its surroundings. Previous works in the studied area include several regional studies involving different countries [3, 5,13,14,15].
A velocity model for South America and the Caribbean called VEMOS2009 was obtained using the finite element method treating a geophysical model and with an approach of least squares collocation [16]. More specifically, [17] obtained a velocity model for Ecuador using artificial neural networks with velocities obtained by [18], who presented a velocity field based on measurements from the National GPS Network of Ecuador (RENAGE). RENAGE is a passive network performed by landmarks, of which GPS measurement campaigns were carried out in 1994, 1996 and 1998, being taken as one of the criteria for its selection that have a minimum observation time of 3 hours per campaign. Continuous monitoring stations were not included for that time because they had less than one year of operation. As these are relatively time-spaced campaign data, an exhaustive analysis of the time series was not performed either. This leads us to assume that velocities with their uncertainties were calculated by assuming pure white noise or white noise scaled by some empirical value derived from repeatability in the time series [19]. In addition, some typical problems of the time were identified, among others, the establishment of very long baselines due to a reduced number of International GNSS Service (IGS) stations that decreased the precission of the velocity estimation. Recently, [20] presented the latest updated velocity model with new data for South America and the Caribbean called VEMOS2015, which is the current velocity model for the Geocentric Reference System for the Americas (SIRGAS) with a spatial resolution of 1° × 1°.
In this paper, we present a methodological approach to obtain the velocity model for Ecuador, by means of the study of time series taking into account the tendency, the seasonal variations and the type of noise, in order to obtain more reliable results [21]. The estimated velocity field allows modeling with a high spatial resolution (0.25° × 0.25°). Therefore, we propose the Kriging geostatistical method, whose main principle is the correlation of the study variable (station velocities) at adjacent points. The Kriging technique minimizes the variance of the prediction error by introducing weights whose sum must be equal to 1, so that the predictor expectancy is equal to the expectation of the variable [22, 23].
2 Methodology
The current methodology has been subdivided into five parts, namely data processing, analysis of data, additive decomposition of time series (tendency, seasonality, noise-spectral analysis), velocity field and velocity model. The first four parts are developed in more detail in [21].
2.1 Data processing
We used the data from 33 GNSS stations of the REGME network for the period 2008-2014, as well as 17 IGS stations in order to link to the global reference frame IGb08 (ITRF2008). These data have been processed by Bernese 5.0 software, developed by the Astronomical Institute of the University of Bern (AIUB), for the processing of multi-GNSS data [24]. The methodological approach used for the processing of GNSS observations with the Bernese software has been: preparation of files (observations/input data), data pre-processing and finally data processing and estimation of daily coordinates. As a result of this processing, we obtained daily SINEX files [25] with the geocentric cartesian coordinates of the stations and the covariance matrix of variances indicating the precision (more details of the data processing in [21]).
2.2 Analysis of data
The final solution of the processing is in cartesian coordinates (X, Y, Z) relative to IGb08 reference frame, located in the SINEX files, where the variance-covariance matrix of the final solution appears. In order to analyze and interpret the time series, we convert them from global geocentric cartesian coordinates (X, Y, Z) to a local cartesian topocentric coordinates (e, n, u). This allows to obtain a clearer idea of the behavior of each series in their respective components and the advantage of working with small values, since these can be centered in zero, which facilitates their study. Prior to the analysis of the time series it is necessary to work with clean series, meaning that they are free of atypical values and offsets (more details in [21]).
2.3 Additive decomposition of time series
One of the purposes of this study has been to obtain models of series that best fits them, taking into account the trend, seasonal variations and the type of noise, so that the sum of all these components represent the real time-series. The theoretical model may be expressed as:
where yt are the observed values, Tt the trend, St the seasonality and Nt the noise or irregular component. Below we detail each of them.
Trend
The velocity is described by a linear trend, so the functional model will be expressed as:
where y0 represents the ordinate in the origin, r the slope, t the time instant and ε the adjustment error. Each coordinate has an uncertainty associated with the variance-covariance matrix of the data processing. We have used this uncertainty to obtain by a weighted linear regression the trend of the series, of which the weights are given by:
The solution is determined by the least squares method using the following matrix system:
Where X is the matrix of the parameters to be determined (y0, r), A is the matrix of the coefficients, W is the diagonal matrix of weights and Y is the matrix of the observed values.
Seasonality
We eliminate the trend of the series for the determination of the periodogram and calculation of the harmonics, what means to operate with the residuals. From the previous equation, we were able to calculate r and yo, and therefore the value of the residuals would be expressed as:
Two techniques may be used, such as the Fourier Transform and the least squares, depending on whether or not the data are equally spaced [25]. The power spectrum by means of a periodogram for equally spaced discrete series is defined by the Fourier Transform as [27]:
Where fn = n/T, T is the fundamental period, vi is the residue and n = 1, 2,… N/2
All our series have data that are not equally spaced, for this case [26] recommend following the procedure proposed by [28] where:
The power spectrum peaks P(f) represent the predominant frequencies and periods. The seasonal variations are given by:
Each component has a different fundamental period. For the analysis of our series we have taken only one harmonic of the seasonal period (p = 1), in order to not overestimate the magnitude and precision of the velocities.
Noise - Spectral Analysis
The study of the power spectrum has been used to determine the type of noise in each series. In recent years, the temporal correlation of the coordinates has been detected and studied, since its determination is affected by numerous geophysical and or meteorological effects that are time-dependent. Several studies determined the presence of noise correlated with the time and its effect in the estimation of the uncertainty in the series, indicating that these are underestimated if only white noise is considered in the series [26, 29,30,31,32]. The power spectrum P of many geophysical phenomena is well approximated by a form-dependent power law:
where f is the time frequency, PO and f0 are normalization constants and α is the spectral index [33]. Generally, the spectral index is not an integer and its values vary between –0 < α < –3, which is the one that appears in most geophysical phenomena. If –1 < α < 0 than it is called the fractional Gaussian noise and it is usually considered to be stationary, meaning that its statistical properties are invariant over time.
There are three types of specific noise: white noise, flicker and random walk when the spectral index reaches values of 0, –1 and –2 respectively. The knowledge of the spectral index is fundamental because it allows to identify the type of noise present in the series and thus to be able to model it and to consider it for the estimation of the velocities and improve its precision. The method for obtaining the index is by fitting a line to log (P (f)) – log (f), where the slope is the value corresponding to the spectral index.
2.4 Velocity field
Any periodic signal consists of the main frequency as well as its highest harmonics. Due to this fact, we assume that the time series contains both the deterministic part through the functional model, which includes trend and seasonality, as well as background noise [34]. Once we obtain the trend, seasonality and noise type, they are introduced in the general model of the series to estimate together the parameters of the periodic variations models, the trend of the series and noise, so that a better estimate is realized in its magnitude and uncertainty. Then if we replace the equations (2) and (9) in (1) we have:
where yi corresponds to the observed values, ti is the time and f is the frequency of the fundamental period of the series that are obtained in the spectral analysis and ∊i corresponds to the noise model.
By means of a least square fit, the parameters ordered on the origin yo, slope r, sinusoidal terms A, B and discontinuities are calculated.
2.5 Velocity model
The solution of the kriging spatial prediction problem requires knowledge of the self-correlation structure for any possible distance between sites within the study area. This technique uses as fundamental element the analysis of the spatial distribution of the information. That means that the estimation of data in points not sampled is not only done in function of the distance, but also its self-correlation structure intervenes, which is analyzed by the calculation of semivariograms:
Where Z(x) is the variable (velocity) in the site x, Z (x + h) is the value of the variable separated from the previous with the distance h, and n is the number of pairs that are separated by a distance h. The semivariance function is calculated for several distances, resulting in an experimental semivariogram that must be fitted to a theoretical semivariogram model. The parameters to be determined are the theoretical model of semivariogram, nugget (C0), sill (C1) and range (a), which are the common parameters of any function of semivariation (Figure 3).
Parameters of a theoretical semivariogram model. The red solid line represents the theoretical semivariogram, while the blue dot represents the experimental semivariogram.
Due to the different behaviors of the components east and north we perform independent models for each one of them. Both components present a tendency with respect to the latitude and longitude coordinates, since we use the Universal Kriging method. This method proposes that the value of the variable can be predicted as a linear combination of the n random variables as:
Being λi are the weights and Zi the measured data of the variable of interest in the sampled points. When the data is characterized by a trend, it is common to decompose the variable Z (x) as the sum of the trend, treated as a deterministic function, plus a random zero-mean stationary component. That is:
With E (∊ (x)) = 0, V (∊ (x)) = σ2 and therefore E (Z (x)) = m(x).
The trend can be expressed by:
The deterministic functions fi (x) are known and p is the number of terms used to adjust m (x).
Obtaining the weights λi results from the matrix solution:
where γi0 corresponds to the values of the semivariograms that are calculated as a function of the distance between the sampled points and the point where the corresponding prediction needs to be made, γij is the semivariogram value for two separated points in a distance h and μ is the Lagrange parameter, which is used to minimize the prediction variance. The functions fin and fi0 are the deterministic functions evaluated at the sampled points and the points to predict respectively. The prediction variance is given by:
3 Results and Discussion
3.1 Seasonality
The CUEC station has an annual period (1.03 years) for both horizontal components, as the majority of the stations this annual seasonal period (~ 365 days) is present, while there are stations that contain semi-annual (~ 180 days) and quarterly (~ 90 days) seasonal periods (Figure 4). This is more evidently in the spectral analysis, in order to determine the noise and in which we observed the highest values of the spectral power density (PSD), which are in the annual period (Figure 5). Table 1 shows the values of the fundamental periods in the three components, together with the observation time of each series.
Decomposition of time series. The values on the abscissa correspond to the date of observation in years. The black dots represent the observed values and the red line seasonality. The upper graph corresponds to the East component and the lower graph to the North component, both have the same seasonal value.
Spectral index for each component of the CUEC stations. The blue, green and yellow lines represent the annual, semiannual and quarterly periods, respectively. The spectral indices have been obtained by an estimation of the power law noise.
Fundamental periods and observation times of the REGME stations.
| East | North | Up | ||
|---|---|---|---|---|
| Observation period | ||||
| STATION | 1stPeriod Years | 1st Period Years | 1st Period Years | |
| Years | ||||
| ALEC | 0,52 | 1,03 | 1,03 | 2,06 |
| AUCA | 0,77 | 0,51 | 1,54 | 1,54 |
| BAHI | 0,03 | 1,13 | 1,13 | 1,13 |
| CHEC | 1,06 | 0,02 | 1,06 | 1,06 |
| CLEC | 1,45 | 0,73 | 0,73 | 1,45 |
| COEC | 1,05 | 0,35 | 1,05 | 2,10 |
| CUEC | 1,03 | 1,03 | 1,03 | 6,18 |
| CXEC | 0,54 | 1,07 | 1,07 | 1,07 |
| ECEC | 0,53 | 1,05 | 1,05 | 2,10 |
| EPEC | 0,56 | 1,11 | 1,11 | 1,11 |
| EREC | 0,78 | 0,52 | 1,55 | 1,55 |
| ESMR | 1,89 | 5,67 | 1,13 | 5,65 |
| GYEC | 5,75 | 5,75 | 5,75 | 5,75 |
| GZEC | 0,52 | 2,07 | 1,03 | 2,06 |
| IBEC | 2,61 | 2,61 | 0,87 | 2,60 |
| LJEC | 0,99 | 1,99 | 0,99 | 5,95 |
| LREC | 0,65 | 0,65 | 0,65 | 0,65 |
| MAEC | 0,49 | 2,22 | 1,11 | 4,45 |
| MHEC | 0,49 | 0,49 | 1,97 | 1,97 |
| MTEC | 0,50 | 0,50 | 1,00 | 1,20 |
| NJEC | 2,12 | 2,12 | 0,21 | 2,12 |
| PDEC | 1,03 | 0,34 | 0,69 | 2,06 |
| PJEC | 0,53 | 2,10 | 1,05 | 2,10 |
| PREC | 1,56 | 0,52 | 1,56 | 1,56 |
| PTEC | 0,91 | 2,27 | 2,27 | 4,53 |
| QUEM | 2,38 | 0,79 | 0,59 | 2,37 |
| QUI1 | 0,41 | 0,17 | 0,41 | 0,82 |
| QVEC | 0,99 | 3,95 | 0,99 | 3,95 |
| RIOP | 5,46 | 2,73 | 0,91 | 6,20 |
| SEEC | 0,52 | 0,16 | 1,03 | 2,06 |
| SNLR | 0,54 | 0,54 | 0,06 | 1,07 |
| STEC | 1,64 | 1,64 | 0,82 | 1,84 |
| TNEC | 1,01 | 3,02 | 1,01 | 3,22 |
According to those results, we have classified the sites into three categories according to their periodicity, namely the unmodelled periodic ground motion, the periodic variation of the estimated time series and the periodic variations that are correlated with the years of observation.
The unmodelled periodic ground motion is the category with the series of annual and semi-annual seasonality are found and refers to that the site is moving periodically [34]. In our study, it becomes more evident in the vertical component. The stations with annual seasonality are: ALEC (1.03 years), BAHI (1.13), CHEC (1.06), COEC (1.05), CUEC (1.03), CXEC (1.07), ECEC (1.05), EPEC (1.11), ESMR (1.13), GZEC (1.03), IBEC (0.87), LJEC (0.99), MAEC (1.11), PDEC (1.03), PJEC (1.05), RIOP (1.03), SEEC (1.03), STEC (0.92) and TNEC (1.08). The stations that have semi-annual seasonality are CLEC (0.73), LREC (0.65), QUEM (0.59) and QUI1 (0.41).
The periodic variation of the estimated time series corresponds to stations that “apparently” move periodically, but they are the product of systematic errors such as the tidal effect produced by time series with systematic spurious effects [34,35,36]. Within this category, there are the series AUCA (1.54 years), EREC (1.55), MHEC (1.97), MTEC (1.20), NJEC (0.21), PREC (1.56 years) and SNLR (0.06).
The periodic variations that are correlated with the years of observation, is the category where the stations have the first fundamental period at half the time of the observation or the total time of the observation. Within this category are the stations GYEC (period and years of observation = 5.75 years), PTEC (period = 2.27 years, observation years = 4.53 years) and QVEC (period and years of observation = 3.95 years). The second fundamental period for these stations corresponds to shorter times. Thus, we obtained for GYEC a period of 1.92 years, PTEC of about 0.91 years and for QVEC a period of 1.32 years. Other reasons for different periodicity values are due to the presence of power law noise and multipath effect [33].
3.2 Noise - Spectral Analysis
The values of spectral index of the station CUEC have a rank of –1 <α < 0 that corresponds to the fractional gaussian noise, considered stationary, meaning being an uncorrelated noise with time (Figure 5). Several authors suggest that white noise and flicker noise dominate the GPS coordinate noise spectrum for the time series and in a smaller extend the random walk [26,30, 31,34].
Similarly, it has been demonstrated that for shorter observation times white noise is the dominant noise contribution, whereas for longer observations it is the random walk [32]. In all stations, spectral index values have been determined between 0 and –1 corresponding to the fractional white noise (Table 2). Several authors suggest that white noise and pink noise dominate the GPS coordinate noise spectrum for the time series and in a smaller extent Brownian noise [26,30,31,34].
Spectral index values of the REGME stations.
| STATION | East, α | North, α | Up, α | ||||||
|---|---|---|---|---|---|---|---|---|---|
| ALEC | –0,51 | ± | 0,14 | –0,54 | ± | 0,13 | –0,48 | ± | 0,14 |
| AUCA | –0,31 | ± | 0,18 | –0,52 | ± | 0,19 | –0,29 | ± | 0,17 |
| BAHI | –0,60 | ± | 0,42 | –0,14 | ± | 0,42 | –0,21 | ± | 0,42 |
| CHEC | –0,31 | ± | 0,27 | –0,50 | ± | 0,22 | –0,47 | ± | 0,23 |
| CLEC | –0,50 | ± | 0,18 | –0,41 | ± | 0,15 | –0,55 | ± | 0,17 |
| COEC | –0,59 | ± | 0,14 | –0,52 | ± | 0,13 | –0,48 | ± | 0,14 |
| CUEC | –0,46 | ± | 0,09 | –0,55 | ± | 0,09 | –0,45 | ± | 0,09 |
| CXEC | –0,33 | ± | 0,22 | –0,64 | ± | 0,19 | –0,47 | ± | 0,20 |
| ECEC | –0,54 | ± | 0,13 | –0,68 | ± | 0,13 | –0,42 | ± | 0,13 |
| EPEC | –0,42 | ± | 0,23 | –0,61 | ± | 0,18 | –0,39 | ± | 0,19 |
| EREC | –0,61 | ± | 0,15 | –0,35 | ± | 0,18 | –0,52 | ± | 0,16 |
| ESMR | –0,48 | ± | 0,09 | –0,55 | ± | 0,08 | –0,41 | ± | 0,09 |
| GYEC | –0,64 | ± | 0,08 | –0,74 | ± | 0,08 | –0,47 | ± | 0,08 |
| GZEC | –0,41 | ± | 0,13 | –0,44 | ± | 0,15 | –0,42 | ± | 0,14 |
| IBEC | –0,64 | ± | 0,13 | –0,48 | ± | 0,13 | –0,64 | ± | 0,12 |
| LJEC | –0,60 | ± | 0,08 | –0,47 | ± | 0,08 | –0,67 | ± | 0,08 |
| LREC | –0,41 | ± | 0,29 | –0,38 | ± | 0,24 | –0,03 | ± | 0,27 |
| MAEC | –0,55 | ± | 0,11 | –0,49 | ± | 0,11 | –0,40 | ± | 0,12 |
| MHEC | –0,38 | ± | 0,15 | –0,41 | ± | 0,15 | –0,58 | ± | 0,15 |
| MTEC | –0,38 | ± | 0,23 | –0,23 | ± | 0,23 | –0,35 | ± | 0,22 |
| NJEC | –0,33 | ± | 0,16 | –0,31 | ± | 0,16 | –0,40 | ± | 0,16 |
| PDEC | –0,42 | ± | 0,14 | –0,39 | ± | 0,14 | –0,56 | ± | 0,15 |
| PJEC | –0,36 | ± | 0,14 | –0,67 | ± | 0,15 | –0,53 | ± | 0,15 |
| PREC | –0,57 | ± | 0,17 | –0,47 | ± | 0,17 | –0,71 | ± | 0,15 |
| PTEC | –0,63 | ± | 0,12 | –0,58 | ± | 0,12 | –0,51 | ± | 0,13 |
| QUEM | –0,29 | ± | 0,20 | –0,30 | ± | 0,17 | –0,25 | ± | 0,19 |
| QUI1 | –0,49 | ± | 0,21 | –0,43 | ± | 0,21 | –0,65 | ± | 0,22 |
| QVEC | –0,58 | ± | 0,14 | –0,62 | ± | 0,13 | –0,47 | ± | 0,12 |
| RIOP | –0,52 | ± | 0,08 | –0,46 | ± | 0,08 | –0,56 | ± | 0,09 |
| SEEC | –0,82 | ± | 0,15 | –0,57 | ± | 0,15 | –0,47 | ± | 0,13 |
| SNLR | –0,42 | ± | 0,12 | –0,25 | ± | 0,12 | –0,02 | ± | 0,12 |
| STEC | –0,44 | ± | 0,18 | –0,58 | ± | 0,20 | –0,45 | ± | 0,16 |
| TNEC | –0,38 | ± | 0,12 | –0,44 | ± | 0,12 | –0,40 | ± | 0,12 |
In our study, we expected to encounter at least pink noise, since the series have periods of observation ranging from two to seven years. In addition, 26 stations have been on the roofs of public buildings and seven on the ground, where also no correlation of time has been found in terms of observation, noise and location of the stations. The time series with the obtained GPS positions have a complex nonlinear behavior and, therefore, the statistical model of the time series is more complex than the simple white noise. Therefore, we assume that the obtained noise value does not reflect the true type of noise, mainly due to the gaps observed in each series. For example, the CUEC, ESMR, GYEC, LJEC, PTEC and RIOP stations start operations in 2008 and 2009 but they have a data loss between 22% and 37%. The LREC station has an 8% data loss but it is a relatively short series, with less than two years of observation. It has been demonstrated that for short time observations the white noise is the dominant noise contribution, whereas for the longer observations it is Brownian noise [32]. We may compare our data with the spectral indices determined by [30] who analyzed in ten stations with daily measurements in a period of 1.6 years of observation a data variation between –0.05 and –0.52 with an average of –0.40.
For stations with longer time observations it is verified that the northern component is noisier than the other components. In general, the horizontal component has a higher magnitude of noise than the vertical component. For zones tectonically less complex it is observed that the horizontal components are less noisy than the vertical component [37]. [26] used time series of daily positions of 23 globally distributed stations with three years of data. There, the determined spectral indices ranged between –0.51 and –2.17 and concluded that there is no significant difference in the spectral nature of the noise in the north, east and up component. Table 3 presents the analysis of the variance by comparing the spectral index values of all the stations of the different components. The results obtained indicate that statistically there are no significant differences between the different components.
Analysis of variance of spectral index values (e, n, u).
| Sum of | Degrees of | Average of | Critical value for | |||
|---|---|---|---|---|---|---|
| Origin ofvariations | F | Probability | ||||
| squares | freedom | squares | F | |||
| Between groups | 0,027 | 2 | 0,013 | 0,698 | 0,4999 | 3,091 |
| Within groups | 1,846 | 96 | 0,019 | |||
| Total | 1,873 | 98 |
3.3 Velocity field
The station velocities have been determined by linear regression where trend, periodic variations and noise were included in the model. Figure 6 presents the series of the CUEC station with the trend, seasonality and adjusted values in the components, where the adjustment is more visible for those data that have greater dispersion. The final velocities with their respective precisions of the REGME and IGS network stations were transformed into a module and orientation to represent the velocity field of Ecuador (Figure 7).
Decomposition of time series for the determination of the velocity. The values in the abscissa correspond to the date of observation in years and the values in the ordinates are in millimeters. The black dots correspond to the observed values, the blue line is the trend of the series, the green line the seasonality that presents and the red line is the series adjusted by means of least squares introducing these parameters. The velocities are presented at the top with their respective uncertainties.
![Figure 7 GNSS horizontal velocity field (black vectors) of REGME stations in ITRF2008 with 95% confidence error ellipses. ITRF2008 velocities were derived from position time series produced by Bernese 5.0 software after time series analysis considering trend, seasonality and noise. Figure adapted from [21].](/document/doi/10.1515/geo-2017-0054/asset/graphic/j_geo-2017-0054_fig_007.jpg)
GNSS horizontal velocity field (black vectors) of REGME stations in ITRF2008 with 95% confidence error ellipses. ITRF2008 velocities were derived from position time series produced by Bernese 5.0 software after time series analysis considering trend, seasonality and noise. Figure adapted from [21].
Our velocity field is consistent with others obtained in previous studies [3, 15, 20]. We compared our velocities with those obtained by [15] in the seven common stations of both studies, obtaining differences inferior to 1.4 mm/yr of average for the horizontal component. The maximum difference has been obtained in the GYEC station with –2.9 mm/yr and 3.9 mm/yr in the east and north component, respectively. (More details of the velocity field in [21]).
3.4 Velocity model
For the velocity model, we have used only the horizontal components, due to the chaotic dispersion of the vertical component. We have tested several theoretical models of semivariograms such as spherical, pentaspherical, circular, stable, exponential, gaussian and cubic model to obtain the model that best fits the experimental semivariogram determined in equation (13) [22]. By means of iteration of the parameters, we have chosen those that had the lowest average square error value. The result obtained for both the east component and the north component corresponds to a spherical semivariogram model whose mathematical expression is:
The parameter values for the east component are:
The function that determines the trend is:
The parameter values for the north component are:
The function that determines the trend in this component is:
Once the theoretical models of semivariograms and the functions that determine the trend for the different components are obtained, they were replaced in the matrix equation (17) to obtain the respective weights and finally to replace in equation (14) to obtain the values of the velocities in the not sampled points. The mean accuracy of the prediction errors has been 1.37 mm for the east component and 0.88 mm for the north component. Figure 8 shows the velocities obtained in a grid of 0.25° × 0.25°.
Velocity model for Ecuador. The red arrows are the REGME horizontal velocities estimated in this study. Black arrows are the velocities obtained by the kriging model.
Comparing our model using the KRIGING geostatistical technique and the VEMOS2015 model, obtained from the SIRGAS multi-year solution SIR15P01 [20], we found diferencies than not exceed 2 mm/yr on the horizontal component (Table 4). The greatest differences are found in the stations located in the central zone of Ecuador (Andes), which is the deformation zone between North Andean Block and Southamerica plate.
Comparison of velocities between different solutions.
| SIR15P 01 | VEMOS 2015 | KRIGING MODEL | DIFFERENCE | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| STATION | ||||||||||
| Ve | Vn | Ve | Vn | Ve | Vn | VEMOS 2015 | KRIGING MODEL | |||
| (mm/y) | (mm/y) | (mm/y) | (mm/y) | (mm/y) | (mm/y) | Ve | Vn | Ve | Vn | |
| ALEC | 7,60 | 9,30 | 10,06 | 2,56 | 5,27 | 6,38 | -2,46 | 6,74 | 2,33 | 2,92 |
| COEC | 2,60 | 13,60 | 10,25 | 3,25 | 5,67 | 11,15 | -7,65 | 10,35 | -3,07 | 2,45 |
| CUEC | 0,60 | 8,30 | 5,45 | -1,68 | -0,26 | 7,96 | -4,85 | 9,98 | 0,86 | 0,34 |
| ECEC | 12,90 | 18,20 | 10,86 | 4,38 | 12,32 | 14,50 | 2,04 | 13,82 | 0,58 | 3,70 |
| ESMR | 16,30 | 16,90 | 13,24 | 5,42 | 17,21 | 15,92 | 3,06 | 11,48 | -0,91 | 0,98 |
| GYEC | 5,90 | 7,50 | 11,48 | 3,13 | 5,90 | 9,45 | -5,58 | 4,37 | 0,00 | -1,95 |
| GZEC | –0,30 | 7,40 | 5,55 | –1,86 | –0,24 | 6,33 | -5,85 | 9,26 | -0,06 | 1,07 |
| IBEC | 1,80 | 10,20 | 6,06 | 1,37 | 4,94 | 13,08 | -4,26 | 8,83 | -3,14 | –2,88 |
| LJEC | –1,30 | 8,30 | 6,06 | –1,71 | –1,02 | 7,82 | -7,36 | 10,01 | -0,28 | 0,48 |
| MAEC | –0,30 | 8,70 | 4,59 | –0,58 | 0,68 | 8,47 | -4,89 | 9,28 | -0,98 | 0,23 |
| NJEC | 6,10 | 5,70 | 7,43 | –0,55 | 5,57 | 6,80 | -1,33 | 6,25 | 0,53 | –1,10 |
| PJEC | 11,10 | 9,70 | 11,94 | 3,63 | 16,05 | 9,75 | -0,84 | 6,07 | -4,95 | –0,05 |
| PTEC | 6,90 | 13,30 | 11,76 | 3,70 | 8,22 | 13,79 | -4,86 | 9,60 | -1,32 | –0,49 |
| QUEM | 6,20 | 10,70 | 6,68 | 1,88 | 6,99 | 10,39 | -0,48 | 8,82 | -0,79 | 0,31 |
| QVEC | 8,60 | 12,40 | 11,17 | 3,54 | 8,74 | 8,13 | -2,57 | 8,86 | -0,14 | 4,27 |
| RIOP | 1,30 | 6,80 | 8,57 | 2,18 | 1,63 | 6,92 | -7,27 | 4,62 | -0,33 | –0,12 |
| STEC | –1,00 | 7,00 | 5,10 | –2,79 | 0,50 | 8,27 | -6,10 | 9,79 | -1,50 | –1,27 |
| TNEC | –1,00 | 9,90 | 2,97 | –0,30 | –1,12 | 12,00 | -3,97 | 10,20 | 0,12 | -2,10 |
| rms | 4,73 | 9,10 | 1,78 | 1,95 | ||||||
The mean squared errors of our model for the east and north component are 1.78 mm/yr and 1.95 mm/yr, respectively (Table 4). Compared to the mean squared errors of the VEMOS2015 model of 4.72 mm/yr for the east component and 9.10 mm/yr for the northern component (Table 4). However, it should be taken into consideration that, although the VEMOS2015 model is based on the SIR15P01 solution, which considers a large number of REGME stations, the times of observation of the time series used to obtain such solution are lower than those used in the time series of the present study. In addition, in this work we have estimated a local model for Ecuador, while VEMOS2015 is a regional model that includes South America and Caribbean.
4 Conclusions
A new GNSS velocity field has been used for Ecuador, based on the analysis of the time series of 33 REGME stations in the period 2008-2014. The analysis of these data confirms and quantifies the current tectonic activity of Ecuador.
There are a considerable percentage of gaps in the time series and data losses being on average of 32%, which does not allow determining the actual type of noise in the series. From the spectral analysis, it is determined that the preponderant noise in the time series is the fractional white noise. The introduction of cyclical variations, trend and noise in series with relatively short periods of time (approximately two years) improves uncertainty and velocity values approaching series with longer periods. By introducing trend, seasonal and noise parameters, uncertainty has been improved by 12% and changed in magnitude up to 1.7 mm/yr in the horizontal component and 2.5 mm/yr in the vertical component, with respect to the velocities estimated initially.
The prediction model obtained in this study using the kriging technique estimates the velocities with an accuracy of ±1.78 mm/yr and ±1.95 mm/yr in the in the east and north component, respectivally. The predictions are valid only for continental Ecuador and present a much higher precision than the VEMOS2015 model. This is due to our model is local respect to the regional model of VEMOS2015, as well as that we have a greater amount of information for the period of 2008-2014 in most of the stations of the REGME network.
Supplementary data
Horizontal velocities obtained from the Kriging velocity model (in mm/yr) for Ecuador, with a grid of 0,25° × 0,25°.
| λ° | ϕ° | Ve | Vn | λ° | ϕ° | Ve | Vn | λ° | ϕ° | Ve | Vn |
|---|---|---|---|---|---|---|---|---|---|---|---|
| -80,75 | -2,25 | 5,90 | 11,77 | -80,00 | -0,75 | 7,41 | 10,95 | -79,50 | -2,50 | 0,85 | 9,94 |
| -80,75 | -1,75 | 11,95 | 11,04 | -80,00 | -0,50 | 10,15 | 11,26 | -79,50 | -2,25 | -0,34 | 11,28 |
| -80,75 | -1,50 | 13,17 | 11,00 | -80,00 | -0,25 | 12,64 | 12,08 | -79,50 | -2,00 | 2,36 | 10,64 |
| -80,75 | -1,25 | 11,59 | 11,98 | -80,00 | 0,00 | 13,86 | 13,09 | -79,50 | -1,75 | 4,53 | 10,47 |
| -80,75 | -1,00 | 10,88 | 13,63 | -80,00 | 0,25 | 13,93 | 12,66 | -79,50 | -1,50 | 5,99 | 10,40 |
| -80,50 | -2,50 | 4,29 | 11,03 | -80,00 | 0,50 | 13,81 | 12,73 | -79,50 | -1,25 | 5,40 | 11,23 |
| -80,50 | -2,25 | 5,56 | 11,30 | -80,00 | 0,75 | 17,58 | 13,67 | -79,50 | -1,00 | 7,31 | 9,23 |
| -80,50 | -2,00 | 6,98 | 11,13 | -79,75 | -4,25 | -1,55 | 7,16 | -79,50 | -0,75 | 2,94 | 11,22 |
| -80,50 | -1,75 | 11,14 | 10,75 | -79,75 | -4,00 | -2,14 | 6,86 | -79,50 | -0,50 | 8,43 | 11,62 |
| -80,50 | -1,50 | 14,48 | 10,57 | -79,75 | -3,75 | -1,71 | 7,43 | -79,50 | -0,25 | 10,73 | 13,26 |
| -80,50 | -1,25 | 8,08 | 11,69 | -79,75 | -3,50 | -0,12 | 8,59 | -79,50 | 0,00 | 10,86 | 11,49 |
| -80,50 | -1,00 | 6,94 | 12,21 | -79,75 | -3,25 | 0,72 | 9,05 | -79,50 | 0,25 | 10,71 | 11,24 |
| -80,50 | -0,75 | 9,12 | 11,61 | -79,75 | -3,00 | 1,46 | 9,26 | -79,50 | 0,50 | 12,39 | 11,71 |
| -80,25 | -4,25 | -0,96 | 9,09 | -79,75 | -2,75 | 3,79 | 8,87 | -79,50 | 0,75 | 14,50 | 12,23 |
| -80,25 | -4,00 | -0,91 | 8,93 | -79,75 | -2,50 | 3,05 | 10,09 | -79,50 | 1,00 | 16,66 | 13,16 |
| -80,25 | -2,50 | 3,85 | 10,73 | -79,75 | -2,25 | 2,67 | 11,13 | -79,25 | -4,75 | 2,46 | 8,15 |
| -80,25 | -2,25 | 4,62 | 11,17 | -79,75 | -2,00 | 3,43 | 10,79 | -79,25 | -4,50 | 1,71 | 7,76 |
| -80,25 | -2,00 | 3,53 | 11,06 | -79,75 | -1,75 | 4,21 | 10,86 | -79,25 | -4,25 | 0,63 | 8,17 |
| -80,25 | -1,75 | 5,42 | 10,84 | -79,75 | -1,50 | 5,54 | 10,76 | -79,25 | -4,00 | -0,35 | 8,24 |
| -80,25 | -1,50 | 9,52 | 10,73 | -79,75 | -1,25 | 5,80 | 11,05 | -79,25 | -3,75 | -0,49 | 8,26 |
| -80,25 | -1,25 | 5,68 | 11,28 | -79,75 | -1,00 | 5,59 | 11,13 | -79,25 | -3,50 | 0,40 | 7,78 |
| -80,25 | -1,00 | 4,85 | 11,98 | -79,75 | -0,75 | 6,05 | 11,03 | -79,25 | -3,25 | 1,13 | 7,15 |
| -80,25 | -0,75 | 7,75 | 11,38 | -79,75 | -0,50 | 9,41 | 11,18 | -79,25 | -3,00 | -1,26 | 9,12 |
| -80,25 | -0,50 | 10,94 | 11,63 | -79,75 | -0,25 | 11,59 | 11,62 | -79,25 | -2,75 | -0,28 | 8,17 |
| -80,25 | -0,25 | 13,63 | 13,02 | -79,75 | 0,00 | 12,44 | 11,94 | -79,25 | -2,50 | 0,10 | 9,34 |
| -80,00 | -4,25 | -1,62 | 7,27 | -79,75 | 0,25 | 12,83 | 12,03 | -79,25 | -2,25 | 0,80 | 10,04 |
| -80,00 | -4,00 | -1,79 | 6,66 | -79,75 | 0,50 | 13,87 | 12,27 | -79,25 | -2,00 | 2,41 | 10,31 |
| -80,00 | -3,75 | -0,95 | 8,35 | -79,75 | 0,75 | 16,14 | 13,16 | -79,25 | -1,75 | 4,01 | 10,27 |
| -80,00 | -3,50 | 0,85 | 8,69 | -79,50 | -4,50 | 0,48 | 7,76 | -79,25 | -1,50 | 4,76 | 10,35 |
| -80,00 | -3,25 | 3,38 | 9,30 | -79,50 | -4,25 | -0,40 | 7,40 | -79,25 | -1,25 | 4,93 | 10,50 |
| -80,00 | -2,25 | 4,32 | 11,40 | -79,50 | -4,00 | -1,60 | 7,12 | -79,25 | -1,00 | 4,97 | 10,37 |
| -80,00 | -2,00 | 3,26 | 11,32 | -79,50 | -3,75 | -1,73 | 6,69 | -79,25 | -0,75 | 4,49 | 10,83 |
| -80,00 | -1,75 | 3,44 | 11,45 | -79,50 | -3,50 | -0,35 | 8,11 | -79,25 | -0,50 | 6,90 | 11,31 |
| -80,00 | -1,50 | 5,79 | 10,91 | -79,50 | -3,25 | -0,80 | 8,08 | -79,25 | -0,25 | 8,09 | 11,79 |
| -80,00 | -1,25 | 5,86 | 11,12 | -79,50 | -3,00 | -0,48 | 8,68 | -79,25 | 0,00 | 9,34 | 10,68 |
| -80,00 | -1,00 | 6,13 | 10,70 | -79,50 | -2,75 | 2,66 | 8,16 | -79,25 | 0,25 | 9,85 | 10,38 |
| -79,25 | 0,50 | 10,65 | 10,92 | -78,75 | -2,50 | 0,64 | 8,26 | -78,25 | -3,25 | 1,27 | 9,36 |
| -79,25 | 0,75 | 11,62 | 11,33 | -78,75 | -2,25 | 2,68 | 7,20 | -78,25 | -3,00 | 1,26 | 9,16 |
| -79,25 | 1,00 | 12,74 | 11,97 | -78,75 | -2,00 | 3,39 | 8,62 | -78,25 | -2,75 | 1,24 | 8,39 |
| -79,00 | -4,75 | 3,04 | 8,66 | -78,75 | -1,75 | 2,56 | 8,69 | -78,25 | -2,50 | 0,95 | 8,83 |
| -79,00 | -4,50 | 2,38 | 8,53 | -78,75 | -1,50 | 2,76 | 9,20 | -78,25 | -2,25 | 0,94 | 8,89 |
| -79,00 | -4,25 | 1,30 | 8,66 | -78,75 | -1,25 | 3,43 | 9,95 | -78,25 | -2,00 | 1,38 | 9,35 |
| -79,00 | -4,00 | 0,46 | 8,17 | -78,75 | -1,00 | 4,59 | 10,09 | -78,25 | -1,75 | 1,95 | 9,28 |
| -79,00 | -3,75 | 0,17 | 8,80 | -78,75 | -0,75 | 7,00 | 10,61 | -78,25 | -1,50 | 2,50 | 9,09 |
| -79,00 | -3,50 | 0,04 | 8,60 | -78,75 | -0,50 | 9,75 | 11,25 | -78,25 | -1,25 | 2,36 | 9,75 |
| -79,00 | -3,25 | -1,07 | 9,19 | -78,75 | -0,25 | 7,97 | 10,83 | -78,25 | -1,00 | 2,33 | 10,19 |
| -79,00 | -3,00 | -3,42 | 10,98 | -78,75 | 0,00 | 7,90 | 10,70 | -78,25 | -0,75 | 3,38 | 10,48 |
| -79,00 | -2,75 | -1,48 | 9,23 | -78,75 | 0,25 | 9,36 | 10,42 | -78,25 | -0,50 | 6,11 | 11,65 |
| -79,00 | -2,50 | 0,29 | 8,04 | -78,75 | 0,50 | 9,80 | 10,37 | -78,25 | -0,25 | 5,04 | 11,24 |
| -79,00 | -2,25 | 3,31 | 8,52 | -78,75 | 0,75 | 7,85 | 10,80 | -78,25 | 0,00 | 2,70 | 11,50 |
| -79,00 | -2,00 | 3,71 | 9,43 | -78,75 | 1,00 | 5,39 | 11,58 | -78,25 | 0,25 | 4,94 | 11,40 |
| -79,00 | -1,75 | 3,60 | 9,83 | -78,75 | 1,25 | 8,07 | 11,31 | -78,25 | 0,50 | 6,80 | 11,22 |
| -79,00 | -1,50 | 3,64 | 10,00 | -78,50 | -3,75 | 1,10 | 9,47 | -78,25 | 0,75 | 6,21 | 11,70 |
| -79,00 | -1,25 | 3,64 | 10,33 | -78,50 | -3,50 | 0,73 | 8,33 | -78,25 | 1,00 | 4,31 | 11,89 |
| -79,00 | -1,00 | 4,44 | 10,48 | -78,50 | -3,25 | 0,30 | 8,59 | -78,00 | -3,00 | 1,93 | 9,04 |
| -79,00 | -0,75 | 6,09 | 10,67 | -78,50 | -3,00 | 0,12 | 9,15 | -78,00 | -2,75 | 1,50 | 9,44 |
| -79,00 | -0,50 | 7,44 | 10,96 | -78,50 | -2,75 | 0,39 | 8,66 | -78,00 | -2,50 | 1,16 | 9,37 |
| -79,00 | -0,25 | 7,52 | 10,95 | -78,50 | -2,50 | 0,49 | 8,62 | -78,00 | -2,25 | 1,05 | 9,21 |
| -79,00 | 0,00 | 8,51 | 10,62 | -78,50 | -2,25 | 0,65 | 8,53 | -78,00 | -2,00 | 1,26 | 9,62 |
| -79,00 | 0,25 | 9,45 | 10,45 | -78,50 | -2,00 | 1,69 | 8,97 | -78,00 | -1,75 | 1,68 | 9,42 |
| -79,00 | 0,50 | 9,66 | 10,60 | -78,50 | -1,75 | 1,99 | 8,82 | -78,00 | -1,50 | 1,97 | 8,41 |
| -79,00 | 0,75 | 8,57 | 10,77 | -78,50 | -1,50 | 2,79 | 9,18 | -78,00 | -1,25 | 1,22 | 9,83 |
| -79,00 | 1,00 | 6,45 | 10,40 | -78,50 | -1,25 | 3,17 | 9,81 | -78,00 | -1,00 | 0,56 | 10,51 |
| -79,00 | 1,25 | 9,34 | 10,58 | -78,50 | -1,00 | 3,62 | 9,93 | -78,00 | -0,75 | 2,12 | 10,12 |
| -78,75 | -4,50 | 2,49 | 9,57 | -78,50 | -0,75 | 4,50 | 10,45 | -78,00 | -0,50 | 5,00 | 10,55 |
| -78,75 | -4,25 | 1,73 | 9,35 | -78,50 | -0,50 | 6,69 | 11,12 | -78,00 | -0,25 | 4,34 | 10,86 |
| -78,75 | -4,00 | 1,05 | 9,08 | -78,50 | -0,25 | 5,77 | 10,51 | -78,00 | 0,00 | 2,23 | 10,88 |
| -78,75 | -3,75 | 0,60 | 9,14 | -78,50 | 0,00 | 5,34 | 11,31 | -78,00 | 0,25 | 3,00 | 11,26 |
| -78,75 | -3,50 | 0,11 | 8,58 | -78,50 | 0,25 | 8,11 | 10,64 | -78,00 | 0,50 | 4,74 | 11,35 |
| -78,75 | -3,25 | -0,79 | 8,91 | -78,50 | 0,50 | 9,80 | 10,17 | -78,00 | 0,75 | 5,05 | 11,43 |
| -78,75 | -3,00 | -1,45 | 9,48 | -78,50 | 0,75 | 8,23 | 11,11 | -77,75 | -2,75 | 1,34 | 9,94 |
| -78,75 | -2,75 | -0,55 | 8,51 | -78,50 | 1,00 | 6,20 | 11,74 | -77,75 | -2,50 | 1,00 | 9,93 |
| -77,75 | -2,25 | 0,84 | 9,91 | -77,25 | -0,50 | -1,01 | 9,96 | -76,50 | -0,25 | 0,09 | 9,99 |
| -77,75 | -2,00 | 0,91 | 9,95 | -77,25 | -0,25 | -1,26 | 9,84 | -76,50 | 0,00 | 1,49 | 9,68 |
| -77,75 | -1,75 | 1,13 | 9,87 | -77,25 | 0,00 | -0,91 | 9,80 | -76,25 | -2,25 | -3,93 | 10,42 |
| -77,75 | -1,50 | 1,21 | 9,78 | -77,25 | 0,25 | -0,21 | 10,24 | -76,25 | -2,00 | -4,85 | 10,48 |
| -77,75 | -1,25 | 0,77 | 10,03 | -77,00 | -2,50 | -0,87 | 10,22 | -76,25 | -1,75 | -5,54 | 10,60 |
| -77,75 | -1,00 | -0,47 | 10,85 | -77,00 | -2,25 | -1,30 | 9,81 | -76,25 | -1,50 | -5,43 | 10,62 |
| -77,75 | -0,75 | 0,31 | 9,19 | -77,00 | -2,00 | -1,56 | 9,33 | -76,25 | -1,25 | -4,46 | 10,54 |
| -77,75 | -0,50 | 1,44 | 9,96 | -77,00 | -1,75 | -1,54 | 10,08 | -76,25 | -1,00 | -3,24 | 10,55 |
| -77,75 | -0,25 | -1,72 | 10,10 | -77,00 | -1,50 | -1,32 | 10,23 | -76,25 | -0,75 | -2,01 | 10,80 |
| -77,75 | 0,00 | -0,49 | 9,96 | -77,00 | -1,25 | -1,01 | 10,21 | -76,25 | -0,50 | -0,92 | 11,98 |
| -77,75 | 0,25 | 0,91 | 10,56 | -77,00 | -1,00 | -0,82 | 10,04 | -76,25 | -0,25 | -1,81 | 10,51 |
| -77,75 | 0,50 | 2,64 | 11,10 | -77,00 | -0,75 | -0,77 | 9,65 | -76,25 | 0,00 | -2,29 | 9,37 |
| -77,75 | 0,75 | 4,07 | 11,56 | -77,00 | -0,50 | -0,47 | 9,91 | -76,25 | 0,25 | 0,69 | 10,13 |
| -77,50 | -2,75 | 0,88 | 10,20 | -77,00 | -0,25 | 0,18 | 9,97 | -76,00 | -2,00 | -5,47 | 10,55 |
| -77,50 | -2,50 | 0,55 | 10,16 | -77,00 | 0,00 | 1,02 | 9,97 | -76,00 | -1,75 | -6,59 | 10,87 |
| -77,50 | -2,25 | 0,34 | 10,12 | -77,00 | 0,25 | 1,55 | 10,10 | -76,00 | -1,50 | -6,57 | 10,95 |
| -77,50 | -2,00 | 0,31 | 10,10 | -76,75 | -2,50 | -1,66 | 10,28 | -76,00 | -1,25 | -5,20 | 10,61 |
| -77,50 | -1,75 | 0,45 | 10,10 | -76,75 | -2,25 | -2,24 | 10,09 | -76,00 | -1,00 | -3,87 | 10,58 |
| -77,50 | -1,50 | 0,69 | 10,06 | -76,75 | -2,00 | -2,66 | 10,00 | -76,00 | -0,75 | -2,70 | 10,73 |
| -77,50 | -1,25 | 1,06 | 9,83 | -76,75 | -1,75 | -2,79 | 10,21 | -76,00 | -0,50 | -1,99 | 10,98 |
| -77,50 | -1,00 | 1,13 | 9,38 | -76,75 | -1,50 | -2,62 | 10,32 | -76,00 | -0,25 | -2,19 | 10,61 |
| -77,50 | -0,75 | -0,12 | 9,83 | -76,75 | -1,25 | -2,19 | 10,31 | -76,00 | 0,00 | -1,99 | 10,24 |
| -77,50 | -0,50 | -1,14 | 9,90 | -76,75 | -1,00 | -1,66 | 10,17 | -76,00 | 0,25 | -0,58 | 10,29 |
| -77,50 | -0,25 | -2,34 | 9,50 | -76,75 | -0,75 | -1,07 | 9,85 | -75,75 | -1,75 | -6,18 | 10,66 |
| -77,50 | 0,00 | -1,98 | 8,45 | -76,75 | -0,50 | -0,20 | 10,02 | -75,75 | -1,50 | -6,14 | 10,68 |
| -77,50 | 0,25 | -0,63 | 10,21 | -76,75 | -0,25 | 1,16 | 9,78 | -75,75 | -1,25 | -5,21 | 10,57 |
| -77,50 | 0,50 | -0,22 | 11,20 | -76,75 | 0,00 | 2,65 | 9,74 | -75,75 | -1,00 | -4,11 | 10,54 |
| -77,25 | -2,75 | 0,29 | 10,30 | -76,75 | 0,25 | 3,41 | 9,96 | -75,75 | -0,75 | -3,15 | 10,57 |
| -77,25 | -2,50 | -0,11 | 10,21 | -76,50 | -2,25 | -3,16 | 10,32 | -75,75 | -0,50 | -2,51 | 10,59 |
| -77,25 | -2,25 | -0,41 | 10,05 | -76,50 | -2,00 | -3,80 | 10,33 | -75,75 | -0,25 | -2,19 | 10,51 |
| -77,25 | -2,00 | -0,53 | 9,97 | -76,50 | -1,75 | -4,15 | 10,40 | -75,75 | 0,00 | -1,71 | 10,41 |
| -77,25 | -1,75 | -0,44 | 10,11 | -76,50 | -1,50 | -3,99 | 10,44 | -75,50 | -1,25 | -4,71 | 10,50 |
| -77,25 | -1,50 | -0,17 | 10,16 | -76,50 | -1,25 | -3,35 | 10,42 | -75,50 | -1,00 | -3,94 | 10,49 |
| -77,25 | -1,25 | 0,18 | 10,07 | -76,50 | -1,00 | -2,45 | 10,40 | -75,50 | -0,75 | -3,18 | 10,49 |
| -77,25 | -1,00 | 0,16 | 9,94 | -76,50 | -0,75 | -1,45 | 10,45 | -75,50 | -0,50 | -2,56 | 10,49 |
| -77,25 | -0,75 | -0,43 | 9,94 | -76,50 | -0,50 | -0,53 | 10,54 | -75,50 | -0,25 | -2,05 | 10,47 |
Acknowledgement
We thank the Military Geographic Institute of Ecuador IGM, for the data obtained from the REGME. We also thank the Universidad de las Fuerzas Armadas ESPE for logistic and financial support.
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