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A novel methodology to quantify the spatial resolution in 2-D seismic surface wave tomographic problems is proposed in this study. It is based on both the resolving kernels computed via full resolution matrix and the concept of Full Width at Half Maximum (FWHM) of a Gaussian function. This method allows estimating quantitatively the spatial resolution at any cell of a gridded area. It was applied in the northeastern Brazil and the estimated spatial resolution range is in agreement with all previous surface wave investigations in the South America continent.

In the recent years, the general aspects of the lithospheric shear wave velocity structure of the South America continent has been studied via surface waves [

Let’s consider an area, defined by a pair of geographycal coordinates, which is discretized into square cells. Suppose that there are

where

The result of the procedure aforementioned is a dispersion curve (phase and/or group velocity as a function of period) for each cell of the target area. It is also important to remember that only the cells visited by at least one surface wave path are considered in such computation. Equation (1) can be rewritten in a matrix form as

where

where

where

The best and more accurate way of getting the resolution of the estimated phase and/or group velocities in any place of the earth’s planet, i.e., the spatial extent of the smallest phase and/or group velocity anomaly that can be imaged, is through the computation of resolution matrix. However, the computation of this matrix is not a simply task, because it involves operations with very large matrices. For this reason, the major part of these kinds of studies does not compute the resolution matrix. In general, those studies use a different approach, called checkerboard tests, which are based on the inversion of synthetic data constructed as from subjective assumptions about input and output models [

In this study, we calculated the exact resolution matrix (Equation (4)), by using some efficient computational techniques into a Personal Computer [

According to [

In this study, a novel methodology to quantify the spatial resolution in 2-D seismic surface wave tomographic problems is proposed. It is based on both the resolving kernels obtained via computation of the full resolution matrix (Equation (4)) and a basic property of a Gaussian function, because a resolving kernel shape can be approximated to a Gaussian function. In fact, as

According to [

where

where

where

and

Applying the natural logarithm in both sides of Equation (8)

The Equation (11) is a second order polynomial function or a quadratic function, which is represented graphically by a parabola. In this case, a quadratic least-squares fit can be used to get the three coefficients of the quadratic function and, consequently, the parameter

Extracting s from Equation (10), we get

Substituting Equation (12) in Equation (6), one gets an expression for the width of the Gaussian at half maximum

Equation (13) provides a quantitative spatial resolution measurement of the unknown parameter at any cell of the grid under investigation. Here, the unknown parameter is the estimated phase and/or group velocity either at a specific cell or at a particular period (Equation (1)). As the periods of the surface waves dispersion curves are related to depths (i.e. short periods are associated with shallow depths while long periods are related to deep parts of the earth’s interior), then the Equation (13) also allows the evaluation of the spatial resolution in depth.

The Equation (4) shows that the resolution matrix is a square matrix, i.e., if the number of unknown parameters is

The region limited by the geographical coordinates (−125, 45) and (25, −75) is, initially, divided into square cells of 2˚ × 2˚ (

The digital seismograms, related to seismic events with focal depth shallower than 120 km and magnitude higher than 5.0 m_{b}, were requested and received from IRIS. The Seismic Analysis Code (SAC) package [

The surface wave tomography methodology, described in Section 2, was applied in data set aforementioned to get the Rayleigh wave dispersion curves of all cells (visited by the surface waves) in

Seismic stations (red triangles) and epicenters of the earthquakes (blue circles) used in the present study. The area is limited by the geographical coordinates (−125, 45) and (25, −75). The heavy square box represents northeastern Brazilian region

investigate northeastern Brazilian region, thus we selected only the Rayleigh wave dispersion curves associated with each cell of such region (

The width of the Gaussian at half maximum (W—Equation (13)) was used to get the spatial resolution of the estimated Rayleigh wave group-velocities at each cell of the northeastern Brazil. As the region under investigation is formed by 49 cells (

A few cases displayed in

The grid of the Figure 1 was divided into cells of 2˚ × 2˚ and each cell on it was numbered (from top to bottom and from left to right) so that any region inside northeastern Brazil must be associated with the numbers shown on the map

Source-station Rayleigh wave path distributions (for periods of 10.04 and 12.05 s) in the area of the northeastern Brazil. The black lines are the great-circle paths connecting epicenters and stations. The target area is relatively well covered by the Rayleigh wave path distribution. A clear decrease in the quantity of the source-station paths, when the period increases, is observed in these maps

process. The red square is the theoretical Gaussian function calculated with the Gaussian fit coefficients. In the upper right side is shown the width of the Gaussian at half maximum (W—Equation (13)) and the correlation coefficient (CC) between the resolving values (computed via Equation (4)) and the theoretical Gaussian function calculated with the Gaussian fit coefficients. An analysis of the correlation coefficients values displayed in

Source-station Rayleigh wave path distributions (for periods of 16.00 and 20.08 s) in the area of the northeastern Brazil. The black lines are the great-circle paths connecting epicenters and stations. The target area is relatively well covered by the Rayleigh wave path distribution. A clear decrease in the quantity of the source-station paths, when the period increases, is observed in these maps

Source-station Rayleigh wave path distributions (for periods of 24.38 and 32.00 s) in the area of the northeastern Brazil. The black lines are the great-circle paths connecting epicenters and stations. The target area is relatively well covered by the Rayleigh wave path distribution. A clear decrease in the quantity of the source-station paths, when the period increases, is observed in these maps

shows that 70% of them are equal and higher than 0.7. The large quantity of high CC values indicates that Gaussian function seems to be a good representation of the resolving kernels’ shape.

The results displayed in

Source-station Rayleigh wave path distributions (for periods of 46.55 and 68.27 s) in the area of the northeastern Brazil. The black lines are the great-circle paths connecting epicenters and stations. The target area is relatively well covered by the Rayleigh wave path distribution. A clear decrease in the quantity of the source-station paths, when the period increases, is observed in these maps

Source-station Rayleigh wave path distributions (for periods of 85.33 and 102.40 s) in the area of the northeastern Brazil. The black lines are the great-circle paths connecting epicenters and stations. The target area is relatively well covered by the Rayleigh wave path distribution. A clear decrease in the quantity of the source-station paths, when the period increases, is observed in these maps

Despite the fast reduction in the Rayleigh wave paths quantity, when the period increases (

Examples of spatial estimates (at cells 1764 and 1842; and periods 10.04 and 18.29 s, respectively) displayed in Table 1. The black circles are the five elements extracted from the resolution matrix (Equation (4)) at a particular cell (central element— position 6) and period. The red squares are the theoretical values computed with the coefficients of the Gaussian fit. In the upper left side, the legends of the symbols are: res-re- solving values from resolution matrix; and gau-theoretical Gaussian values. In the upper right side (and for a specific cell), one can find both the estimated spatial resolution value (W) and the Correlation Coefficient (CC) between the estimated resolving kernels via resolution matrix (Equation (4)) and the theoretical values calculated with the coefficients of the Gaussian fit. In all cases, the theoretical Gaussian curve is very close to the estimated resolution value obtained with the resolution matrix (Equation (4)). This is also confirmed by the large quantity of high CC values in Table 1. It shows that a Gaussian function represents very well the resolving kernel’s shape

Examples of spatial estimates (at cells 1916 and 1993; and periods of 32.00 and 51.20 s, respectively) displayed in Table 1. The black circles are the five elements extracted from the resolution matrix (Equation (4)) at a particular cell (central element— position 6) and period. The red squares are the theoretical values computed with the coefficients of the Gaussian fit. In the upper left side, the legends of the symbols are: res-re- solving values from resolution matrix; and gau-theoretical Gaussian values. In the upper right side (and for a specific cell), one can find both the estimated spatial resolution value (W) and the Correlation Coefficient (CC) between the estimated resolving kernels via resolution matrix (Equation (4)) and the theoretical values calculated with the coefficients of the Gaussian fit. In all cases, the theoretical Gaussian curve is very close to the estimated resolution value obtained with the resolution matrix (Equation (4)). This is also confirmed by the large quantity of high CC values in Table 1. It shows that a Gaussian function represents very well the resolving kernel’s shape

Examples of spatial estimates (at cells 2070 and 2142; and periods of 73.14 and 102.40 s, respectively) displayed in Table 1. The black circles are the five elements extracted from the resolution matrix (Equation (4)) at a particular cell (central element— position 6) and period. The red squares are the theoretical values computed with the coefficients of the Gaussian fit. In the upper left side, the legends of the symbols are: res-re- solving values from resolution matrix; and gau-theoretical Gaussian values. In the upper right side (and for a specific cell), one can find both the estimated spatial resolution value (W) and the Correlation Coefficient (CC) between the estimated resolving kernels via resolution matrix (Equation (4)) and the theoretical values calculated with the coefficients of the Gaussian fit. In all cases, the theoretical Gaussian curve is very close to the estimated resolution value obtained with the resolution matrix (Equation (4)). This is also confirmed by the large quantity of high CC values in Table 1. It shows that a Gaussian function represents very well the resolving kernel’s shape

decrease in a spatial resolution) or an almost constant throughout the entire period range (or depth range) displayed in

Another important point observed in the proposed methodology is related to the improvement of the spatial resolution in surface wave tomography studies. In general, in the major part of the surface wave studies in the seismological literature, the improvement of spatial resolution is associated with the increase of data quantity. In fact, the improvement of the spatial resolution should be related to the cell size. Let’s consider the case of a photo. In order to increase the resolution of a photo we must decrease the size of the discretization element, i.e., in this case, we increase the number of dots per inch (or increase the density of dots and, consequently, decrease the dot size). Thus, to get a better spatial resolution we must either decrease the cell size or increase the number of source-station paths with a good azimuthal coverage. The previous discussions show that the proposed methodology describes the main features of the seismological problem.

A novel methodology to quantify the spatial resolution in 2-D seismic surface wave tomographic problems is proposed. It is based on both the resolving kernels computed via full resolution matrix and the concept of Full Width at Half Maximum (FWHM) of a Gaussian function. The method allows estimating quantitatively the spatial resolution at any cell of a gridded area. The spatial resolution range estimates with the proposed method in northeastern Brazil is in agreement with those obtained with all previous surface wave investigations in the South America continent.

This research was developed by using data from GSN and GTSN seismic networks available at IRIS. The author would like to thank these institutions for providing all seismic data. He also thanks Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) for financial support during the development of this study (Process #: E-26/170.407/2000) and MCTI/Observatório Nacional for all support. Several figures were prepared with the software Generic Mapping Tools [