Journal of Applied Geophysics 108 (2014) 140–151

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Journal of Applied Geophysics

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Rayleigh wave modeling: A study of dispersion curve sensitivity andmethodology for calculating an initial model to be included in aninversion algorithm

Rodrigo F. de Lucena ⁎, Fabio TaioliInstituto de Geociências, Universidade de São Paulo, Rua do Lago 562, CEP 05508-080 São Paulo, SP, Brasil

⁎ Corresponding author. Tel.: +55 11983145572.E-mail address: rodrigo.lucena@usp.br (R.F. de Lucena

http://dx.doi.org/10.1016/j.jappgeo.2014.07.0070926-9851/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 30 January 2014Accepted 1 July 2014Available online 10 July 2014

Keywords:Rayleigh wave modelingSecular functionDispersion curveMASWSeismic

This paper presents a study on Rayleigh wavemodeling. Aftermodel implementation usingMatlab software, un-published studies were conducted of dispersion curve sensitivity to percentage changes in parameter values, in-cluding S- and P-wave velocities, substrate density, and layer thickness. The study of the sensitivity of dispersioncurves demonstrated that parameters such as S-wave velocity and layer thickness cannot be ignored as inversionparameters, while P-wave velocity and density can be considered as known parameters since their influence isminimal. However, the results showed limitations that should be considered and overcome when choosing theknown and unknownparameters throughdetermining a good initialmodel or/and by gathering a priori informa-tion. A methodology considering the sensitivity study of dispersion curves was developed and evaluated to gen-erate initial values (initialmodel) to be included in the local search inversion algorithm, clearly establishing initialfavorable conditions for data inversion.

).

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The term seismic brings to mind the seismic methods of refractionand reflection, which are the most popular of those employed in geo-physics. However, other seismicmethods have been developed in recentdecades, such as the seismic analysis of surface waves examined in thispaper. In the early 1980s, Nazarian and Stokoe (1984) developed a newseismic method called the spectral analysis of surface waves (SASW),based on the study of Rayleigh waves (surface waves) normally presentas noise in seismic recordingsmade during refraction and reflection sur-veys. A decade and a half later, Park et al. (1999) proposed a new seismicmethod based on the same physical principles as SASW but that soughtto eliminate or reduce the deficiencies of the previous method, namely,to improve the signal-to-noise ratio, increase the efficiency of field sur-veys, and improve data processing. This method was called the multi-channel analysis of surface waves (MASW) and is distinguished by thesimultaneous use of 12 or more receivers in a linear array to ensuregreater efficiency and flexibility of data acquisition, as well as a highersignal-to-noise ratio. MASW acquisition is similar to a refraction seismicsurvey but is distinguished by low-frequency (4.5-Hz) receivers in itslinear array. The data acquired are presented as an offset–time (x–t)seismogram, where the data processing step begins. To separate Ray-leigh waves from other waves, a two-dimensional Fourier transform isperformed to obtain a frequency–wave number (f–k) amplitude

spectrum (Horike, 1985) that can be mathematically transformed intoa graph of the phase velocity as a function of the frequency, called thedispersion curve.

Because Rayleigh waves characteristically have greater energy thanother waves, their greater amplitude enables them to be readily identi-fied on an f–k amplitude spectrum. In vertically heterogeneous media,Rayleigh waves exhibit dispersive behavior, wherein the differentwavelengths of the wave components penetrate to different depths, ac-quiring different phase velocities depending on variations in the me-chanical properties of the medium. In other words, the velocity of asurface wave undergoing dispersion is not unique but, rather, is charac-terized by different values that are dependent upon the frequencies ofthe wave components. In addition, this dispersive characteristic causedby the heterogeneity of the medium is visible in the dispersion curve aswhat we refer to as its modal nature, which is a succession of curveswherein the first one that appears as the frequency increases, calledthe fundamental mode, is the curve normally used for modeling anddata inversion. Because Rayleigh waves have this dispersion character-istic, they are not just a noise feature present in most seismic methodsbut can serve as a source of information in methods such as MASW.

Information on the substrate layers are contained in the dispersioncurve, which, throughmodeling (the direct problem), allows the inver-sion (the inverse problem) of field data possible. Inversion entails solv-ing a nonlinear problem using, for example, an iterative least-squaresprocedure as the Levenberg–Marquardt algorithm (Levenberg, 1944;Marquardt, 1963), which produces a model with layers of differentthicknesses along with their respective shear velocities.

Fig. 1. Dispersion curves (modal nature).

141R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

Rayleigh surface wave methods enable the exploration of depths onthe order of 100m and have been successfully used in engineering/con-struction because, in contrast to refraction and reflection seismicmethods, they allow inversion as a function of the S-wave velocity,which, in combination with substrate density, enables calculation ofthe substrate coefficient of rigidity. These methods, furthermore, cancomplement a refraction survey because the seismogram obtained bythe latter method can be used to process and invert surface wave data.

1.1. Rayleigh wave modeling

Modeling Rayleigh waves involves the generation of dispersioncurves (Fig. 1) from amodel of heterogeneous layers with known phys-ical parameters, such as density (ρ), P-wave velocity (α), S-wave veloc-ity (β), and layer thickness (d). The modeling basically involves findingthe roots of a nonlinear secular function

F c;ω;ρ;α;β;dð Þ ¼ 0 ð1Þ

obtained from the equations derived from the theory of Rayleigh wavesin a heterogeneous medium. Implementation of the theory used mostfor Rayleigh wave modeling is based on the method described byDunkin (1965), a method modified from earlier approaches byThomson (1950) and Haskell (1953).

Tofinding the roots of the secular function (Eq. (1)), the values of theparameters ρ, α, β, and d must be known and included in the equationas constants. In this way, for each frequency value (ω) included in theequation, the output will be one or more phase velocity values (c)representing the fundamental mode and, when they exist, the othermodes present in the dispersion curves (Fig. 1).

Based on the theory of waves in a heterogeneous elastic mediumwith the imposition of appropriate boundary conditions, the secularfunction generated by the Dunkin approach is a recursive function.That is, its form depends on the number of layers in the model, sinceeach layer represents a matrix of values and the secular function repre-sents the multiplication of all of these matrices:

F c;ωð Þ ¼

T1212T1213T1214T1223T1224T1234

0BBBBBB@

1CCCCCCA

n

T G1212 G1213 G1214 G1223 G1224 G1234G1312 G1313 G1314 G1323 G1324 G1334G1412 G1413 G1414 G1423 G1424 G1434G2312 G2313 G2314 G2323 G2324 G2334G2412 G2413 G2414 G2423 G2424 G2434G3412 G3413 G3414 G3423 G3424 G3434

0BBBBBB@

1CCCCCCA

n‐1

…

G1212G1312G1412G2312G2412G3412

0BBBBBB@

1CCCCCCA

1

:

Layers

ð2Þ

Each matrix of the equation contains the elements calculated foreach layer thickness, beginning with the half-space and ending withthe top layer. The T elements, for example, are the equations for thehalf-space and the G elements are the equations for the remaininglayers.

The theoretical dispersion curve is obtained by finding the zeros ofthe secular function and is essential for subsequent data inversion. Thedifference between the real and theoretical dispersion curves yieldswhat we call the object function, the minimum of which represents agood fit between theoretical and real data. Therefore, we utilize this ob-ject function to perform data inversion, which, being an iterative proce-dure, requires a set of initial values to start the inversion algorithm.

The data inversion of surface waves can be performed by means ofglobal and local search procedures. Global search procedures, well stud-ied by Yamanaka and Ishida (1996), Wathelet et al. (2004), and Rydenand Park (2006), offer the advantage of low dependence of the initialmodel. On the other hand, higher computational costs are involvedwhen a large number of parameters are unknown. Local search proce-dures, in turn, require a good initial model to obtain appropriate dataconvergence. Therefore, once adopted, a good initial model becomes agreat inversion procedure, accurate and fast.

For local methods, such as the Levenberg–Marquardt (Levenberg,1944;Marquardt, 1963), the better the initial model, the better and fasterit will converge to the finalmodel thatwill be linked to the real geologicalsubstrate. According to Abbiss (1981), the maximum penetration depthof a shear wave is estimated to be between one-third and one-half of itswavelength; that is, the greatest wavelength (v/f) in a dispersion curvegives an estimate of themaximum investigation depth,with this estimaterequiring conversion from phase velocity to S-wave velocity. The layerthickness parameter thereby becomes known and the S-wave velocitieswill be the parameter to be inverted, starting out from a set of initialvalues. Those values can be definedusing the estimate provided byAbbissfor the maximum investigation depth

Z ¼ Pλ ð3Þ

and the values from the dispersion curve, where the value of parameter Plies between approximately 0.33 and 0.5.

2. Methodology

2.1. Rayleigh wave modeling

To be studied, the Rayleigh wavemodels needed to be implementedin Matlab programming language. Dunkin’s (1965) algorithm wasemployed to generate the secular function. The fundamental mode ofthe dispersion curve was found by determining the first zero of the sec-ular function for each frequency chosen. To achieve this, the numericalbisection procedure (Press et al., 1992) was implemented.

2.2. Dispersion curve sensitivity study

Dispersion curve sensitivity can be studied by calculating partial de-rivatives (Takeuchi et al., 1964; Xia et al., 1999) to check which param-eter variations influence the dispersion curves more. However, themethodology implemented in this article enables further examinationof the different trends of sensitivity to the variation of a single physicalparameter. In addition, the graphical presentation of the results betterillustrates the variations.

To study the sensitivity of the fundamental mode of dispersioncurves, parameter sets including S-wave velocity (βn), P-wave velocity(αn), density (ρn), and layer thickness (dn) were modified individuallyand jointly. Each parameterwas varied by several percentages, allowingeffects on dispersion curves to be checked.

142 R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

The study was conducted on more than 50 models and the resultswere similar to those presented here, characterized by the studies ofModels 1 to 3 reported in Table 1.

The substratemodels described in Table 1 generate synthetic disper-sion curves without noise. The curves were plotted between 0.5 Hz and50 Hz at logarithmic intervals on the frequency axis (the greatest con-centration of data points was at the lowest frequencies). It is importantto emphasize two points here: The first relates to the absence of noise.We know that it is impossible to completely eliminate all noise fromreal data, such that the region represented by the real dispersioncurve is not as well defined as in the synthetic curve. This influencesthe selection of points that will make up the real dispersion curve tobe used for data inversion. In addition, the heterogeneity of each layermaking up the real layered medium, however subtle it may be, will in-fluence the appearance of the real dispersion curve. The second point isthe discontinuity of the dispersion curve selected to perform the inver-sion. It would be ideal to work with a continuous curve, since no infor-mation would be lost, but this is not possible because, computationally,weworkwith discrete values. Strobbia (2003) discusses the importanceof selecting a higher data point density at lower frequencies, because ifconstant intervals are chosen, then, upon changing the scale from fre-quency to wavelength, the intervals for greater wavelengths will tendto increase. Therefore, opting for a greater point density at low frequen-cies than at high frequencies is recommended to avoid compromisingthe resolution of the final geophysical model.

2.3. Implementation of the initial model for data inversion

The thicknesses of the layers of the initialmodel to be used to initiatedata inversion using a local search algorithm are predefined as a total of10 logarithmically spaced layers (where the thicknesses increase withdepth), where the maximum depth of the scale is defined by Eq. (3),as is the depth of each of the 10 layer thicknesses.

Determination of the S-wave velocities of the initial models wasbased on Eq. (3) and on the scan of the synthetic dispersion curve byapplying

βn ¼ cn λnð ÞQ

ð4Þ

in a method similar to that used by Xia et al. (1999).Where βn is the value for the S-wave velocity in layer n of the initial

model, cn is the phase velocity (Rayleigh wave component) in layer nand Q is a conversion factor for converting between phase velocityand S-wave velocity, with a value between 0.87 and 0.96 (calculatedfrom the equation of Richart et al. (1970), which relates the P-, S-, andRayleigh wave velocities to a Poisson interval between 0 and 0.5 thatis acceptable in a geological context). The parameter λn is the wave-length able to reach layer n and is calculated by Eq. (3) for the 10predefined layer thicknesses. When the dispersion curve frequencyaxis is converted towavelengths, it is possible to search for phase veloc-ities to be inserted into Eq. (4) using wavelengths calculated by Eq. (1).

Table 1Models 1 to 3.

Models N αn (m/s) βn (m/s) ρn (kg/m3) dn (m) αn/βn

Model 1 1 420 200 1350 3 2.12 1100 300 2800 5 3.73 1440 900 2100 ∞ 1.6

Model 2 1 1350 250 1900 4 5.42 2150 580 2200 5 3.73 2500 850 2500 ∞ 2.9

Model 3 1 600 280 1280 3 2.12 860 430 1500 5 2.03 1530 850 1950 ∞ 1.8

More than 50 different models were tested to evaluate this procedureand to define the best values for P and Q.

Since the dispersion curve to be inverted has discrete values, it ispossible for Eq. (4) to generate a βn value between two points on thedispersion curve. In this situation, the value closest to the calculatedpoint can be chosen or interpolated. In this study, the point with thelower velocity value was selected, because as shown in the dispersioncurve sensitivity study, convergence during data inversion turns out tobe better for lower S-wave velocity values.

3. Results and discussion

3.1. Dispersion curve sensitivity study

3.1.1. Alteration of the S-wave velocity parameter (βn)As noted earlier there is a very tight relation between the Raleigh

wave and the S-wave velocities, which allows us to infer that a changein S-wave velocity will appreciably affect the dispersion curve. To con-firm this, S-wave velocities in each layer were altered individually aswell as simultaneously as a group, including all velocities of the n layers.This procedurewas carried out onModel 1 (Table 1),where the individ-ual alterations in each layer can be seen in the graphs of Fig. 2a to c forlayers 1 to 3, respectively, while the alteration of the velocities of thethree layers as a group can be seen in Fig. 2d. The percentage variationsin velocities were up to ±25%, with the black curve representing theunaltered Model 1.

The percentage variations for β1 (Fig. 2a) show that the dispersioncurve is affected starting at 9 Hz and this sensitivity cannot be ignored,since the effect is quite pronounced, even in the ±10% curves (particu-larly for negative percentage variations). However, in the second layer(Fig. 2b), the dispersion curve is affected by percentage variationswith-in the interval from 4 Hz to 42 Hz and, just as in the first layer, the sen-sitivity of the curve is quite pronounced for any percentage variation. Inthe third layer (Fig. 2c), which represents the infinite half-space ofModel 1, the percentage variations in velocity affect the dispersioncurve at low frequencies (below 22 Hz). This difference between thedispersion curves for the three layers at different frequencies is expect-ed, since we know that the frequency of a wave component is inverselyproportional to the maximum depth to which the wave can travel. Itmakes sense that layer thickness would have effects within these fre-quency intervals, because if we increase, for example, the thickness ofthe first layer of Model 1, the limiting frequency will no longer be9 Hz but, rather, lower, since we are increasing the maximum depth ofthe model's first layer.

In Fig. 2d, the S-wave velocities of the three layers were altered si-multaneously, most importantly because, when the initial estimates ofthese parameters for performing data inversion are made, any errorcommitted will affect all three parameters simultaneously. Note thatupon the simultaneous introduction of the three variations, the disper-sion curve errors worsened, clearly showing that a small percentagevariation (less than ± 25%) considerably alters the dispersion curve,mainly for the low S wave velocity values. This information is veryimportant, since this high sensitivity of the dispersion curve makesthe S-wave velocity parameter an element that should be inverted in asurface wave data inversion, due to better convergence.

During the dispersion curve sensitivity study, upon altering theS-wave velocity, it was found in some cases that the dispersion curveis insensitive to velocity changes in the model’s intermediate layers.To illustrate this behavior, the S-wave velocity of the second layer ofModel 1 was altered from 300 m/s to 700 m/s with the percentage var-iations in S-wave velocity in the three layers as shown in Fig. 3.

In Fig. 3, it can be seen that the dispersion curve is sensitive to alter-ations in β1 andβ3 (Fig. 3a and b), whereas it is essentially unchanged byvariations inβ2 (Fig. 3c). This behaviorwould seema relevant problem inthe data inversion, since under these circumstances inversion can resultin an unrealistic velocity for the second layer, with the same occurring in

Fig. 2. Effects on the dispersion curve of Model 1 in response to percentage variations in S-wave velocity for a) β1, b) β2, and c) β3 and for d) β1, β2, and β3 simultaneously.

143R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

intermediate layers of models with more than three layers. This insensi-tivity phenomenon is observed as the velocity of the intermediate layerincreasingly differs from that of the upper layer, where the thickness ofthe intermediate layer influences the limiting velocity at which the dis-persion curve becomes insensitive. The limiting velocity increases asthe layer thickness increases until the phenomenonvanishes and the dis-persion curve is affected by variations at any S-wave velocity. On theother hand, at low thicknesses, the limiting velocity decreases, that is,the insensitivity phenomenon appears at a low S-wave velocity and thedispersion curve is essentially unaffected for any value between the ve-locities of the layers above and below the intermediate layer.

In Fig. 4, β2 was varied between 300 m/s and 600 m/s, at 50 m/s in-tervals, for two thicknesses (d2 = 5 m and d2 = 10 m), which allowedthe determination of the extent to which layer thickness can influencethe phenomenon described above.

It is noted in Fig. 4 that when the thickness of the second layer is in-creased, the dispersion curve for the second layer becomes sensitive atdifferent S-wave velocities. At a second layer thickness of 5 m (Fig. 4a),the limiting velocity at which the dispersion curve begins to be

appreciably affected by an increase in β2 lies between 300 m/s and400 m/s, but when thickness is increased to 10 m (Fig. 4b) it can beseen that this limiting velocity increases to somewhere between400 m/s and 500 m/s. It is understandable that the dispersion curvewould be more sensitive to variations at low β2 values since, as wehave seen, the anomalous behavior (dispersion curve insensitivity)will becomemore pronounced as the velocities in the lower, intermedi-ate, and upper layers become increasingly different. In other words, atlow β2 values, curves are more similar to the velocities of the upperlayer and tend to be less influenced by the anomalous effect and thethickness of this intermediate layer delimits the limiting velocitywhere the anomaly begins.

This anomalous effect can be decreased by increasing the number oflayers during data inversion and bymaking a good initial estimate of theS-wave velocity parameter used to initiate the inversion algorithm. In-creasing the number of layers will cause the increase in S-wave velocityas a function of depth to be gradual, with few abrupt velocity changes,and a good initial estimate will facilitate successful convergence duringdata inversion.

Fig. 3. Effects on the dispersion curve of Model 1 with β2 altered (700 m/s) in response to percentage variations in the S-wave velocity for a) β1, b) β3, and c) β2.

Fig. 4. Effects on the dispersion curve of Model 1 in response to varying β2 between 300 m/s and 600 m/s for a) d2 = 5 m and b) d2 = 10 m.

144 R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

Fig. 5. Effects on the dispersion curve of Model 2 in response to percentage variations in the P-wave velocity for a) α1, b) α2, and c) α3 and for d) α1, α2, and α3.

145R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

3.1.2. Alteration of the P-wave velocity parameter (αn)To determine the effects of P-wave velocity changes on a dispersion

curve, initially three graphs, one for each of the three layers (Fig. 5a to c)of Model 2 (Table 1), were plotted, along with one graph for the simul-taneous alteration of the three layers (Fig. 5d). The graphs show the per-centage variations of the altered P-wave velocity relative to the velocityof the predefined model. The maximum percentage variation conformsto theminimumallowable Poisson ratio (γ), that is,αn/βn cannot be lessthan √2.

Fig. 5a shows the percentage variations ofαn in layer 1. All curves ex-cept for the −50% curve are practically superimposed, that is, the dis-persion curve sensitivity of Model 2 is practically insignificant over anα1 alteration range of ±25%, as well as forα1 +50%. At low frequencies(below 7 Hz), all curves are practically superimposed, which is under-standable, since thedispersion curve is affected themost at low frequen-cies by the physical parameters of the deepest layers. In the percentage

variations of the second layer (Fig. 5b) of Model 2 (Table 1), as well as inthe first layer, we observe the curves, for the most part, overlapping,with the exception of the−50% curve. This curve enables the identifica-tion of the frequency rangewithinwhich the dispersion curve is affectedby altering α2, that is, within the interval between 3 Hz and 22 Hz, pre-cisely because it is the intermediate layer. The percentage variations inthe half-space of Model 2 are shown in Fig. 5c. Again, it is noted thatthe curves are very similar, particularly for variations within ±25%,with an exception again for the −50% curve, which stands out fromthe others within the frequency interval below 20 Hz, which is charac-terized by carrying information from the deepest layers.

A simultaneous analysis of the variations of all the layers can beperformed using the dispersion curves of Fig. 5d. It is noted that errorsin the curves worsen when errors from the percentage variations inP-wave velocities within the three layers are considered simultaneous-ly. However, with the exception of the−50% curve, the curves are quite

146 R.F. de Lucena, F. Taioli / Journal of Applied Geophysics 108 (2014) 140–151

close to each other, considering that, in a real situation, it would be verydifficult to distinguish these curves due to the presence of noise in field-acquired data. According toXia et al. (1999), thedispersion curve is littleaffected by changes in P-wave velocity, so long as the percentage varia-tion is less than ±25%. In fact, we observe this behavior among the dis-persion curve P-wave velocities of Model 2, primarily for velocitiesexceeding the model', whose behavior is well characterized within alllayers, where the maximum velocity curves (+50%) are practicallysuperimposed on the model curve, which does not occur with the−50% curve.

Given the low sensitivity of the P-wave velocity dispersion curve(with percentage variations less than 25%), any inversion performedas a function of this velocity would likely be unsuccessful because con-vergence will be compromised, since the dispersion curve is affectedvery little by a change in P-wave velocity, particularly for unrealisticallyhigh velocities. On the other hand, we can consider the P-wave velocity

Fig. 6. Effects on the dispersion curve of Model 3 in response to the percentage variations in

to be known, since it has little effect on the dispersion curve; since theerror will not be greater than ±25%, we need not concern ourselveswith its inversion, thereby decreasing the number of unknown param-eters during data inversion.

Nonetheless, it is clear that the dispersion curve sensitivity at−50% ismuch greater than at+50%, suggesting that the dispersion curve ismoresensitive to αn values below those of the model. By testing variousmodels, it was found that dispersion curve sensitivity as a function ofvariations inαn is directly related to theαn/βn ratio, that is, to the Poissonratio. To confirm this, Model 3 (Table 1) was generated for low αn/βn

ratios and P-wave velocity alterations were again made by a proceduresimilar to that for Model 2, with percentage variations conformingto theminimum allowable Poisson ratio (γ=0orαn/βn=√2). Accord-ingly, we plotted three graphs, one for each of the three layers (Fig. 6ato c) and one graph for the simultaneous alteration of all three layers(Fig. 6d). The maximum percentage variations are different for each

P-wave velocity for a) α1, b) α2, and c) α3, and for d) α1, α2, and α3, simultaneously.

Table 2Maximum percentage variations permitted in each layer to constitute the combined per-centage variation of the three layers of Model 3.

Curves Percentage variation of each layer

Layer 1 Layer 2 Layer 3

Blue (+44%) +44% +29% +21%Cyan (+25%) +25% +25% +21%Green (+10%) +10% +10% +10%Yellow (−10%) −10% −10% −10%Magenta (−25%) −25% −25% −21%Red (−44%) −44% −29% −21%

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layer, since theαn/βn ratio decreases with depth and themaximumneg-ative percentage variations (−44%,−29%, and−21%) approximate theratio αn/βn = √2, while the model does not permit negative percentagevariations that are greater (in absolute value) than these for each layer.Due to this differential limitation in each layer, the simultaneous per-centage variations (Fig. 6d) of all layers are compromised; to resolvethis issue, it was established that the percentage variations should beconstrained to conform to the variations summarized in Table 2.

In Model 2, the lowest αn/βn ratio is equal to 2.9, dropping to 2.1 inthe first layer of Model 3, limiting the maximum percentage variationto ±44% (Fig. 6a). However, even with percentage variations below±50%, the dispersion curve is appreciably affected (above 7 Hz),mostlyat low velocities, as was found for Model 2. This change in the shape ofthe dispersion curve was obvious for the −25% variation, supportingthe hypothesis that at low Poisson ratios the dispersion curve is moresensitive to P-wave velocity alterations, even for an error of ±25%. Inlayer 2, the αn/βn ratio dropped to 2.0 and the effects on the dispersioncurve became more pronounced, even for positive variations, and con-siderable variation was observed even below +25%. The −25% curveis completely altered in a uncharacteristic fashion of the referencecurve of Model 3. In the half-space, the αn/βn ratio dropped evenmore (1.8) and the maximum percentage variation permitted a de-crease to ±21% (Fig. 6c). Even so, we see a considerable difference inthe dispersion curve for unrealistically low velocities, particularly at−21% (below 18 Hz).

The results from the simultaneous introduction of the percentagevariations in the three layers ofModel 3 (Fig. 6d) show that the problemhasworsened and all curves can be clearly distinguished, even those for±10%. Again, the curves with the lowest velocities are those affectedthe most by altering αn. However, the curves for positive percentagevariations also altered their shapes, which could be deleterious to datainversion when αn is considered a known parameter.

Considering what was observed in the alterations of Models 2 and 3and in other various models generated, it is obvious that for high αn/βn

values (high Poisson ratios), αn has almost no effect on the dispersioncurve, especially for unrealistically high P-wave velocity values; thatis, in these cases, consideringαn a known parameterwill have very littleeffect on dispersion curve data inversion. However, at the lowest αn/βn

ratios, great care should be taken, since a small variation inαn, especial-ly at unrealistically low values, can compromise data inversion becausethe inversion may converge to a model in which the S-wave velocitiesare unacceptable when compared to the real substrate. Starting from apriori information, obtained in a refraction study, for example, it is rec-ommended that, whenever possible, any error in designating αn as aknown value should be made on the high side, since, as we have seenin Models 2 and 3, errors relating to high velocities in models affect dis-persion curves much less. Thus, based on all the tests performed, theseprecautions should be considered, especially when the αn/βn ratio issuspected to be below about 2.5.

3.1.3. Alteration of the density parameter (ρn)Alterations in ρnwere studied using Model 1. Figs. 7a to 7c show the

individual alterations in layers 1 to 3, respectively, and Fig. 7d shows the

simultaneous alterations in the three layers, the black curverepresenting Model 1 itself. In all cases, the effects of the variations inρn on the dispersion curve were studied with a maximum percentagevariation of ±50%.

The effects on the dispersion curve of Model 1 caused by variationsin ρ1 can be seen in the curves in Fig. 7a. We note that the dispersioncurve is insensitive to density variations, especially within ±25%, and,as with P-wave velocity variations, the dispersion curve is less sensitiveto percentage variations in density above the initial model density.However, in the first layer it is apparent that curve sensitivity occursover a wide frequency interval (between 2 Hz and 37Hz). In the secondlayer (Fig. 7b), a relatively low dispersion curve sensitivity to changes inthe value of ρ2 is also observed. For variations between−25 and+50%,the curve retains the shape of the model curve, with practically negligi-ble change. Given a real dispersion curve with intrinsic noise, for exam-ple, it would be very difficult to distinguish curves with these densityvariations, which would make this parameter a candidate to be aknown value in data inversion. It was also noted in the second layerthat the dispersion curve ismuchmore sensitive in the interval between1 Hz and 37 Hz. It is interesting to note that, within the intermediatelayer, the curves for percentage variations intersect the model curve atthe same point. It is as though the curves were rotating on a fixedpoint as the percentage variation increased, which did not occur forany other parameters under any circumstances tested.

In the third layer (Fig. 7c), it is noted that the dispersion curve is sen-sitive to changes in ρ3 between 1 Hz and 23 Hz, but, as in the otherlayers, this sensitivity is low, particularly for variations within ±25%.

Model 1 (Table 1) is characterized by layers with low and highαn/βn

ratios. Looking at the dispersion curves in Fig. 7a to c, we find norelationship between this ratio and curve sensitivity to percentagevariations in density. We note only that the dispersion curve is charac-terized by greater sensitivity at lower densities, independent of thePoisson ratio.

In the study examining the simultaneous introduction of threepercentage variations in the densities of the three layers, an interestingdetail was found: When the densities of all layers were increasedor decreased proportionately at the same time, we obtain exactly thesame dispersion curve. In Fig. 7d we observe that all curves where thepercentage variations in all the layers are identical are superimposed,which shows that, in contrast to the other parameters, in response tothe simultaneous introduction of three percentage variations, dispersioncurve errors are not augmented but, rather, diminished. In other words,as the percentages of variation for the layers approach each other, thedispersion curve loses sensitivity, changing progressively less, which isyet another reason to consider density as a known parameter.

3.1.4. Alteration of the layer thickness parameter (dn)The layer thickness parameter was studied by considering Model 1

(Table 1). To do so, a percentage variation of up to±40%was applied in-dividually to the parameters of the first and second layers (Fig. 8a and b)and simultaneously to all layers (Fig. 8c) of themodel. In the study of si-multaneous layer variations, the graph in Fig. 8d was plotted for highpercentage variations (between +300% and +450%), showing the be-havior of the dispersion curve for large layer thicknesses (Fig. 8d).

Alteration of d1 in Model 1 shows that the dispersion curve, startingat 3 Hz, is quite sensitive when layer thicknesses are varied (Fig. 8a). Upto a variation of ±10%, the effect on the curve is small, but at a variationof ±25% the effect becomes sufficient to influence data inversion. To alesser extent, due to limited curve sensitivity between 1 Hz and 32 Hz(Fig. 8b), similar behavior is observed in the second layer (d2) in re-sponse to percentage variations in thickness. Between 18 Hz and32 Hz, the curves separate in a manner uncharacteristic of the disper-sion curve, starting at ±25%.

With the simultaneous introduction of the percentage variations inthe two layers (Fig. 8c), the effects on the dispersion curve are augment-ed and it becomes obvious that a small variation in layer thickness can

Fig. 7. Effects on the dispersion curve of Model 1 in response to the percentage variations in P-wave velocity for a) ρ1, b) ρ2, and c) ρ3 and for d) ρ1, ρ2, and ρ3, simultaneously.

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greatly modify the dispersion curve. Note that with increased layerthickness, the curve tends to have a more abrupt fall in phase veloc-ity as a function of frequency; we can infer from an abrupt drop inthe dispersion curve as frequency increases that one or more layersare very thick. It was also observed that, as layer thickness increased,the dispersion curve became less sensitive to the variation. This canbe seen in Fig. 8d, where the thicknesses of the two layers were var-ied simultaneously from+300% to+450%, and the dispersion curveshows only a very slight change in response to the large percentagevariation.

Analysis of dispersion curve response to layer thickness variationshows that this parameter cannot be ignored in data inversion; we con-sequently cannot consider it as a known parameter. An alternative is toconsider the problem as a model with several thin layers. Doing so

would enable the examination of S-wave velocity variations as depth in-creases, with depth discretized into small layer thicknesses.

3.2. Implementation of the initial model for data conversion

As previously noted, defining the initial model close to geologicalreality is essential for successful data inversion. Methodology forimplementing an initial model for values of S-wave velocity as a func-tion of depth was applied in a large number of models to evaluatetheir efficacy, aswell as to optimize the values of the P andQ factors. Re-garding the latter, it was found that the optimum P and Q values for themethodology employed were 0.5 and 0.9, respectively. Fig. 9 shows thegraphs comparing the initial model to be utilized to start an inversionprocedure and the knownmodel,which is to be approximated as closely

Fig. 8. Effects on the dispersion curve of Model 1 in response to percentage variations in layer thickness for a) d1, b) d2, c and for d) d1 and d2, simultaneously.

Fig. 9. Initialmodels calculated (lines drawn in red) in comparison to knownmodels (continuous blue lines) a)Model 1, b)Model 2, and c)Model 3. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

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Fig. 10. a) The altered dispersion curve of Model 3 (β1 N β2 b β3) and b) the initial model calculated (dotted red line), with a comparison to the knownmodel (continuous blue line). (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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as possible in data inversion. The methodology employed was appliedto Models 1 to 3 (Table 1) to exemplify their efficacy and the resultsare shown in Fig. 9.

Fig. 9 shows the initial models calculated (lines drawn in red) usingEqs. (3) and (4) applied toModels 1 to 3 (Table 1). For purposes of com-parison, the known models are plotted with continuous blue lines(Fig. 9a to c, respectively). These initial models were calculated andplotted for 10 layers, enabling a gradual increase in S-wave velocityand greater flexibility in accommodating small thicknesses in knownmodels with few layers. Note that this methodology works quite well,since in all cases conversion in subsequent data inversion is well guidedat the outset, avoiding a large number of iterations of the inversion algo-rithm. In addition, convergence tends to be ensured in the correct direc-tion, since the minimum of the object function is relatively close to theinitial model.

In a model with layers whose S-wave velocities are inverted, for ex-ample, where β1 N β2 b β3, the initial model suffers from the perturba-tions of dispersion curve inflections. To observe this phenomenon, theparameters β2, ρ2, and α2 of Model 3 were decreased by 50%, with β2

(215m/s) remainingbelowβ1 (280m/s) and then thedispersion curves(Fig. 10a) for this new model and for the initial model (Fig. 10a) werecalculated using the same methodology as in the previous models andplotted.

The inflection of the dispersion curve (Fig. 10a), around 17 Hz and218 m/s is caused by the inversion of the S-wave velocity in the inter-mediate layer (Fig. 10b). The velocities of lower and upper layers aregreater than that of the intermediate layer and “force” the dispersioncurve to resume increasing velocity around the inflection point beforethe curve reaches the phase velocity corresponding to the S-wave veloc-ity of the second layer (the S-wave velocity calculated at the inflectionpoint is between 227 m/s and 250 m/s, greater than the 215 m/s ofthe known model). Therefore, in the initial model, as noted in Fig. 8b,the S-wave velocity will not drop as sharply as in the known model,but even so the initialmodel is satisfactory for initiating the inversion al-gorithm. Amore detailed study on the inversion of S-wave velocitywithdepth can be found in Pan et al. (2013).

By gathering a priori information through knowledge of local/globalgeology and seismic refraction applications and by taking advantage ofsuitable surface wave surveys and other sources of information, suchas those studied here, on the sensitivities of dispersion curves, known

parameters can be includedwherever appropriate in the inversion algo-rithm, along with the unknown parameter to be inverted.

The initialmodel for the inversion procedure can be calculated usingthe methodology given in this paper. In this way, conditions leading toerroneous convergences can be avoided, increasing the probability ofsuccessful inversion.

4. Conclusions

The study of the sensitivity of dispersion curves demonstrated thatparameters such as the S-wave velocity and layer thickness cannot beignored as inversion parameters, due to the high sensitivity of disper-sion curves to variance in the values of these parameters, particularlythe S-wave velocity. However, the P-wave velocity and density can beconsidered as known parameters, so long as they are not problematicwith regard to errors resulting from percentage variations in initial pa-rameters and certain precautions are taken regarding the substratestudied. In this respect, the a priori information discussed in this papermay help reduce undesirable effects in the absence of information onthe method itself. The dispersion curve was also observed to be moresensitive to the variation of all parameters below the known dispersioncurve. This fact is particularly positivewhen the S-wave velocity param-eter is considered, since inversion initiated under these conditions willresult in greater convergence efficiency. Despite that strategy beingnegative when P-wave velocity and density are considered, their influ-ence is small and could be considered as known parameters during in-version. Implementation of an initial model for the S-wave velocityand for a depth scale based on a model with several layers reducessome of the problems identified in the study of dispersion curve sensi-tivity and provides an initial model that is relatively close to the realmodel, making convergence of the inversion algorithm faster andmore reliable.

Thus it is clear that the surfacewave data pre-inversion step is criticalto the successful convergence of the local search inversion algorithm to afinal model. The studies conducted demonstrate appropriate character-istics for the parameters considered to be known and unknown, which,when combinedwithmethodology to implement an initialmodel for in-version and with a priori information, will provide favorable initial con-ditions for subsequent data inversion.

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Acknowledgments

The authors thank the Geosciences Institute of the University of SãoPaulo for supporting the study of surfacewaves, theNational Council forScience and Technology (CNPq), the CAPES–PROAP - RMH-2013 by thefinancial support and Dr. Choon Park for his comments and suggestionson the manuscript.

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