IMAGING OF NEAR SURFACE VELOCITY HETEROGENEITIES BY MEANS OF REFRACTION TOMOGRAPHY

Transcription

1 ROZPRAWA DOKTORSKA IMAGING OF NEAR SURFACE VELOCITY HETEROGENEITIES BY MEANS OF REFRACTION TOMOGRAPHY mgr inŝ. Nasar El Zawam Promotor: prof. dr hab. inŝ. Zbigniew Kasina KRAKÓW 2007

2 - 2 - Each type of seismic data Required it s own Specialized processing strategy.

3 - 3 - Author s Acknowledgments I would like to thank Prof. Zbigniew Kasina for his insight and direction throughout this scientific process. He has aided me in gaining a tremendous amount of knowledge, but more importantly, he has shown me what it takes to be a true scientist. Thanks goes to the many workers in the Department of Geophysics who have helped me get things done in all my four years at AGH. Special thanks goes to workers of Zakład Surowców Energetycznych, who have provided me opportunity with countless hours of computer support to realize most of the calculations using their facility ProMAX system. Very special and heart felt Thank You goes to Mohamed and Aisha, my parents, for instilling the values of courage and trust, and for making me believe that anything is possible if you just put your mind to it. Thank You goes to my Brothers and Sisters, for without them nothing is possible. Last, but definitely not least, I would like to thank my wife Madiha for Her kind words of encouragement and her smiling face gave me a reason to fight on.

4 - 4 - CONTENTS INTRODUCTION... 5 I. DIFFERENT APPROACHES TO IMAGING THE NEAR SURFACE VELOCITY HETEROGENEITIES IN SEISMIC EXPLORATION... 8 II. THE DESCRIPTION OF THE APPLIED PROGRAMS II.1. Promax s programs for velocity model editing, seismic modeling, turning ray tracing, turning ray tomography, interactive first break picking and database operations II.2. Fortran programs for direct and inverse problem of the head wave tomography II.2.1. The description of the program #RAYEDH for direct problem of the head wave tomography II.2.2. The description of the program # INVERSDH for inverse problem of the head wave tomography II.3. Auxiliary Fortran programs II.3.1. The program for conversion ASCII velocity files from ProMAX to binary files and text SURFER files II.3.2. The program for conversion binary velocity files from # INVERSDH to text SURFER files II.3.3. The program for calculation RMSDV (Root Mean Square velocity error) II.3.4. The program for conversion ASCII files with picks from ProMAX to text FORTRAN format II.3.5. The program for 2D (spatial) smoothing of velocity fields II.3.6. The program for statics calculations II.3.7. The program for turning ray tracing II.4. Programs for graphical presentation III. RESULTS OF THE SOLUTION OF DIRECT AND INVERSE PROBLEM OF THE REFRACTION TOMOGRAPHY III.1. The ray approach and the wave approach to seismic wave propagation and their role in tomographic inversion III.2. Imaging of near surface velocity heterogeneities of the medium in wave pattern of acoustic modeling III.3. The solution of direct and inverse problem of the refraction tomography for selected models and statics estimation III.3.1. The two layer model with gradient medium over refractor (depth 176 m) III.3.2. The two layer model with gradient medium over refractor (depth 120 m) III.3.3. The model of gradient half-space III.3.4. The two layer model with constant velocities III.3.5. The three layer model with constant velocities III.3.6. The four layer model with constant velocities III.4. The analysis of the results of direct and inverse problem solution for model data III.5. The tomographic inversion of first breaks on field records CONCLUSIONS REFERENCES

5 - 5 - INTRODUCTION The first layer below the ground surface, known as the weathering layer (or Low Velocity Layer - LVL), is generally a thin layer made up of low velocity, unconsolidated material. The investigation of LVL may be undertaken for civil and mining engineering projects as well as in connection with structural investigations. In the application to civil engineering projects, the most common aim is the determination of the depth to the bedrock on dam sites and along high pressure tunnel routes. In mining, determination of the LVL thickness may be required in order to plan safely the underground operations or to determine the thickness of the overburden to be stripped if opencast operations are planned. During seismic structural investigations lateral variations in the thickness or velocity of the weathering layer can corrupt the continuity of events on the stacked section and, perhaps more seriously, can introduce apparent structure into deep reflectors. To avoid these effects we must define and introduce so called static corrections to the seismic records. Before this process the model of the weathering layer must be estimated. The recognition of the structure of near surface layer and its velocity distribution plays the main role in estimation of field static corrections specially in vibroseis seismics. Field statics and residual statics define total statics corrections. The accuracy of static corrections estimation has essential effect on results of many processing procedures. The most important of them are velocity analysis, stacking and migration. The errors in static corrections estimation are the source of serious structural interpretation errors as well as the source of decreasing of the effectiveness of many advanced seismic procedures: inversion, AVO which play main role in direct hydrocarbon prospecting and in stratygraphic interpretation. The main source of our knowledge about LVL are shot holes, uphole times in dynamite seismics, velocity measurements in deep shot holes, well to well measurements, interpretation of first breaks by means of refraction methods (GRM method, Delay Time method and others) and by means of tomographic inversion (head wave tomography, turning ray tomography, attenuation tomography). We can also use surface waves to LVL recognition. The refraction tomography - comprising head wave tomography and turning ray tomography - may be treated as one of the most important inversion method of LVL recognition in the case when we have no holes or wells to our disposal (Bohm 2006, Bridle 2006, De Amorim 1987, Ditmar 1999, Docherty 1992). Such a situation is commonly met

6 - 6 - when we have only to our disposal results of land vibrator seismics and we must obtain all the information from first breaks on seismic records. The effectiveness of tomographic inversion of first breaks depends on many parameters of seismogeological models of LVL, on many parameters of seismic acquisition and on many parameters of tomographic inversion. The main goal of the presented doctor thesis is the estimation of the effect of these parameters on results of tomographic inversion of head waves arrivals and on the accuracy of field statics estimation using model data simulating mainly standard reflection land records. The application of the model data for analysis of the tomographic inversion effectiveness is well known and commonly used procedure (Bridle 2006, De Amorim 1987, Ditmar 1999, Docherty 1992, Rajasekaran 1996, Scott 1991, Zhu 1992). Additionally the comparison will be carried out between results of tomographic inversion of head waves and the results of turning wave tomography on selected model data. Although the presented work has mainly the form of model study some results of field data processing will be presented too. The input model data were constructed using the advanced processes of seismic data processing system ProMAX (Interactive Velocity Editor, Finite Difference Modeling, Turning Ray Tracing) and using the original Fortran programs constructed in the Department of Geophysics (University of Science and Technology AGH Faculty of Geology, Geophysics & Environmental Protection). The tomographic inversion of turning wave arrivals was realized using ProMAX tomographic processes (Create Turning Ray Velocity, Turning Ray Tracing, Compute Residual Travel Times, Turning Ray Tomography). The tomographic inversion of head wave arrivals was realized using original Fortran programs constructed in the Department of Geophysics. All the original basic and auxiliary Fortran programs used in my work have been created with my contribution in the phase of their numerical testing and optimizing. It must be stressed that programs of head wave tomography are not actually available in seismic data processing systems used in Poland in main geophysical companies (system ProMAX and system OMEGA ). Only variants of turning wave tomography and reflection tomography are available. Access to the ProMAX system on Faculty of Geology, Geophysics & Environmental Protection was possible within Landmark University Grant Program sponsored by Landmark Graphics Corporation. The following chapters of my work include: analysis of different approaches to imaging the near surface velocity heterogeneities in seismic exploration, the theoretical

7 - 7 - principles of applied programs and description of input/output data, presentation of the seismogeological models used in the analysis, results of the solution of direct problem (defining of traveltimes and seismic ray trajectories) and inverse problem of refraction tomography (determination of velocity fields from traveltimes), results of field statics calculations.

8 - 8 - I. DIFFERENT APPROACHES TO IMAGING THE NEAR SURFACE VELOCITY HETEROGENEITIES IN SEISMIC EXPLORATION Shallow refraction seismics has long been used for the determination of the near surface layer structure. The most common goal of this investigation in seismic prospecting for gas and oil was the definition of the static corrections. The model of near surface layer was the result of refraction interpretation allowing to derive estimates of the thicknesses and velocities of the near-surface layers by analyzing the first breaks of head waves on the field records. Conventional analysis of first-break data makes use of intercept times and inverse slopes of the refracted-arrival segments of traveltime-distance graphs to interpret the depth and velocity structure of the shallow subsurface. Several methods have been proposed for the interpretation of refraction data, such as the intercept time method, the wavefrontreconstruction method (Thornburgh 1930), the plus-minus method (Hagedoorn 1959), the general reciprocal method (Palmer 1980, 1981), the delay time method (Barry 1967). All these methods are very useful tool of seismic interpretation and are still used to define starting model in more advanced modern interpretation techniques based on generalized linear inversion (Hampson and Russell, 1984) or on tomographic inversion. However, all these methods have certain drawbacks restricting their range of applications. First of all they were designed only for interpretation of refraction data and it was not simple to include other types of waves (for instance reflected waves). They cannot detect velocity inversions (a lowvelocity layer beneath a high-velocity) and cannot to resolve thin beds (known as the hidden layer problem). The interpretation of velocity increases with depth within a layer can be problematic with some implementations. Most of the refraction techniques were designed to compute static corrections for a constant velocity weathering layer of slowly changing thickness overlying a refractor of constant velocity. When these conditions do not exist, then unacceptable errors arise in the computed statics. Nowadays the CDP method (Common Depth Point method) is the dominating method of surface acquisition. When the dynamite seismics is applied we measure so called uphole time by using uphole geophone near each shot hole. This information may be used to estimate static corrections. Additionally we can realize special velocity surveys in deeper shot holes and use their results (traveltimes) to derive velocity model of LVL. Their is no such a possibility in the case of Vibroseis method. In this case we must practically obtain all the information about LVL only from reflection field records. This information is mainly inherent in first breaks of refraction arrivals.

9 - 9 - The generalized linear inversion of first breaks (Hampson and Russell, 1984) is from many years treated as one of the most effective tools of refraction statics estimation. In this approach, an input model is designed using standard refraction interpretation and theoretical first-break traveltimes are computed. The model is then perturbed iteratively until the computed and observed traveltimes match according to some squared-error criterion. Theoretical first break arrivals are calculated by ray tracing from each shot to each receiver. Theoretical times are compared to pick first break times and the error is then minimized using the GLI algorithm. As the error is minimized, the model is updated so that it matches the true earth more correctly. The computer program for computation statics using GLI algorithm is commonly known as GLI2D in two dimensional variant or GLI3D in three dimensional variant. There are still some limitations of the approach based on GLI algorithm. The interpretation model used is the layered one limiting the range of velocity models taken into consideration. The velocity in each layer cannot vary in vertical direction. The velocity in the following layers must increase with depth. The number of the layers must be established before interpretation. To overcome these limitations new solutions based on tomographic inversion have been proposed for determination of the LVL structure and static corrections (Bohm 2006, Bridle 2006, De Amorim 1987, Ditmar 1999, Docherty 1992). The static corrections based on tomographic approach are named tomostatics (Zhu 1992). Tomostatics has advantages over traditional refraction statics in regions where it is not easy to identify refractors and where we can meet velocity inversion. The tomographic method enables us to consider complex geological models with dipping or variously curved layers and with strong lateral velocity variations and rough topography. Model parametrization is much more flexible. The next advantage of this method is the possibility of jointly inverting the different kinds of waves generated within a seismic experiment (turning waves, head waves, reflected waves). Including different kind of waves increases the effective aperture in the data providing much more reliable solutions. The velocity fields resulting from tomographic inversion may be used for statics estimation as well as for determination of migration velocity field in shallow part of the medium. The tomographic inversions may be classified from different points of view. Using different types of waves we can consider turning wave tomography, head wave tomography and reflection tomography. Tomostatics is mainly based on turning wave tomography and head wave tomography. Taking into account the theoretical principles of tomographic inversion we can distinguish between ray tomography (traveltime tomography) and

10 diffraction tomography (Lo, Inderwiesen 1994). The first one is based on ray approach to wave propagation well known from theory of seismic modeling. Such a approach is commonly applied in all production applications of seismic tomography in seismic processing systems including system ProMAX and OMEGA used in Poland. The diffraction tomography and wave-equation tomography based on the wave approach found only limited applications in seismic prospecting (Nolet 1987, Pratt, Tura et al. 1994, Woodward 1992; Harris, Wang 1996). Taking into account the main purposes of seismic tomography we can distinguish between traveltime tomography and amplitude tomography (attenuation tomography). The main goal of the traveltime tomography is determination of the velocity distribution in the medium using propagation time of different waves (Scott, Fehler 1991, Michelena et al. 1993). The main goal of the amplitude tomography is to define the attenuation distribution using amplitudes (Brzostowski, McMechan 1992, Liao, McMechan 1997, Neep et al. 1996, Leggett et al. 1993) or spectral characteristics of the waves (Quan, Harris 1997, Chris et al. 2004). Among different types of tomography only the traveltime tomography in the variants of turning wave and head wave tomography have been widely applied to determination of the near surface velocity characteristics and statics calculations. The typical tomographic inversion of traveltimes based on ray approach consists of the following steps: - determination of the traveltimes from field records using the picking procedure, - construction of an initial velocity model using all available data, - defining the theoretical traveltimes for assumed acquisition parameters and assumed initial velocity field through the process of ray tracing (direct task of tomography), - calculations the differences between theoretical and measured (picked) traveltimes, - defining the corrections to an initial velocity field through the procedure of minimizing the errors between the observed arrivals on field records and those computed by ray theory through an initial model (inverse task of tomography), - refining an initial velocity model using calculated corrections. Effectiveness of the tomographic inversion of seismic data depends on many factors related to geological structure, acquisition geometry as well as to interpretation methods. The most important factors related to geological structure of medium are: - shape and size of velocity heterogeneities, - character of velocity variations (the occurrence of strong velocity gradients), - position of velocity heterogeneities.

11 The most important factors related to acquisition geometry are: - the number of shots and shot interval, - the number of receivers and receiver interval, - position of shots in relation to position of receivers and velocity heterogeneities, - dominant frequencies of seismic waves. The most important factors related to tomographic inversion are: - accuracy of traveltimes picking, - the step of velocity discretization of the medium, - the method of solving of the basic set of tomographic equations, - constraints applied to velocity distributions and the way of taking into account additional velocity information, - the methods of smoothing of velocity fields (averaging, convolutional smoothing - seismic quelling, regularization), - the range of taking into consideration different components of wave field, mainly different types of waves. Different aspects of tomographic inversion applied to definition the near surface layer characteristics and static corrections were analysed in many papers. The most important conclusions will be presented below. De Amorim et al. (1987) introduced the model of LVL consisting of an undulating earth surface with a planar refracting horizon between two media divided into blocks of constant velocity. Each block was of equal horizontal length and had an unknown constant velocity. The traveltimes were computed for the waves that are refracted at the bottom of the LVL between any source and receiver locations. These traveltimes were expressed in terms of the velocities in the blocks. Testing various other models for the LVL with dipping and curved refractor boundaries did not obtain improved field static corrections on the available data. The L2 norm was used during inversion and the tomographic set of equation had typical form: (Aˆ T Aˆ + λi) ˆ V = Aˆ T t where: I r - the unit matrix, A $ T - transposition of coefficient matrix A $, λ - arbitrary parameter, V - column vector of unknowns (values of velocity corrections in the nodes of computational grid), t - column vector of traveltime differences (measured between recorded and calculated - for assumed velocity distribution - traveltimes), A $ - matrix of coefficients defining relation between traveltimes and velocities.

12 Olsen K.B. (1989) used the procedure aimed to model a laterally varying distribution of velocities and the topography of n refracting interfaces. By the specification of n vertical grid lines of equal spacing, each layer of the model is divided into a number of cells. The upper limit of the model (the surface of the earth) is specified by known elevations. The cells situated below the lower refracting interface are considered half-infinite. For each cell, a constant velocity is determined in terms of the slowness. The depths to the base of the cells, which define the refracting interfaces, are among the inversion parameters. The ray tracing procedure is developed on the concept of minimal traveltime of the first arrivals. For each station, the raypath representing the shortest traveltime of all the possible direct and refracted waves from the shot is calculated. No refraction is assumed at the vertical cell boundaries. Lawton (1989) used differences in first-arrival traveltimes between adjacent records in multifold reflection surveys to compute the depth and velocity structure of near-surface layers. The traveltime differences as a function of source-receiver offset provide a direct indication of the number of refractors present, with each refractor being defined by an offset range with a constant time difference. For each refractor, the time-difference value at a common receiver from two shotpoints is used to partition the intercept time into the delay time at each shotpoint. This procedure is repeated until the delay times at all shotpoints and for all refractors have been computed. Refractor depths and velocities are evaluated from this suite of delay times. Zanzi and Carlini (1991) proposed a new method for refraction statics reducing the computational time without reducing accuracy. The first arrivals, common-offset organized, formed the data space. The method involves Fourier transforming any common-offset data vector with respect to the common mid-point. As a result, the data are decomposed in a number of subspaces, associated with the wave-numbers, which can be independently inverted to obtain any wavelength of the near-surface model. Docherty (1992) investigates the feasibility of computing the weathering model from the traveltimes of refracted first arrivals. The problem is formulated in terms of the difference in arrival time at adjacent receivers, resulting in a much sparser matrix for inversion. Lateral variations in both the weathering thickness and velocity are sought. In most cases, it is necessary to include a small number of constraints to obtain the true weathering model. Any roughness in the solution that is not required to fit the data is most effectively removed using a second difference smoothing technique. Two layers make up the model: a laterally inhomogeneous weathering layer and a uniform, high speed refractor. The weathering layer is divided into cells of constant velocity. Each cell is bounded above by the observation surface

13 and below by the refractor. Boundaries between adjacent cells are vertical. The base of weathering is described by a series of node points, joined by straight line segments. In this study a constant refractor velocity is assumed. Zhu et al. (1992) presented examples illustrated that turning ray tomography can image near-surface velocities more accurately than refraction statics methods. The medium to be imaged was discretized into a grid of small rectangular cells, each of which contains a single velocity. Sources and receivers are both located on the surface. The updated velocities were slightly smoothed (or damped) every few iterations. This was the Constrained Damped Simultaneous Iterative Reconstruction Technique (CDSIRT). It was confirmed that tomostatics is noticeably closer to the true statics where velocity inversions are significant. Generally, long spatial wavelength statics appear to be estimated better using tomostatics, although a tomostatics bias (bulk shift) exists with increasing depth. Due to damping and smoothing in the tomography algorithm, the output image of a linear inversion was remarkably robust to a wide range of reasonable initial models. Stefani (1995) used turning-ray tomography for estimating near-surface velocity structure in areas where conventional refraction statics techniques fail because of poor data or lack of smooth refractor/velocity structure. The method comprises nonlinear iterations of forward ray tracing through triangular cells linear in slowness squared, coupled with the LSQR linear inversion algorithm. Rajasekaran and McMechan (1996) performed the tomography on prestack time picks using the simultaneous iterative reconstructive technique (SIRT) algorithm with modifications to include reflected as well as turned rays. Traveltimes of head waves are well approximated by rays turned in a small velocity gradient below a high contrast reflector, and so are included automatically as a special case of turned rays. The reflections, which correspond to predominantly near vertical propagation, define horizontal changes in the model, but not the vertical changes. Conversely, the turned transmissions are better able to define the vertical changes. Increasing the effective aperture by combining reflection and transmission data and performing tomography on this composite data set produces a better image of the 2-D velocity distribution. Lanz et al. (1998) investigated the applicability of surface-based 2-D refraction tomography (turning ray tomography) for delineating the geometry of a landfill. The depth of the near surface model did not exceed 100 m. The velocity in this layer was defined by rapid increases from about 1000 m/s to 1500 m/s. Geophone and source spacing were set to 2 and 8 m, respectively. Sampling interval 0,25 ms was used. The results have demonstrated that the

14 tomographic refraction scheme may be an efficient means of studying the very shallow subsurface but complementary geological and other geophysical data are required to make interpretation reliable. Additonally the accurate first-break picking was critical because the traveltime anomalies associated with waste disposal sites may be only a few milliseconds. A two-stage picking strategy was proposed. In a first pass, the arrivals were determined with a commercial phase picker based on neural networks. After visual inspection, a crosscorrelation-based algorithm was used to refine these initial first-break picks. In the process of ray tracing fast finite-difference eikonal solver has been used. The inversion cell size was increasing from 1m to about 15 m. Zhang and Toksöz (1998) presented a nonlinear refraction traveltime tomography method that consists of a new version of the shortest path ray-tracing approach, a regularized nonlinear inversion method that inverts traveltime curves rather than traveltimes alone, and a Monte Carlo method for nonlinear uncertainty analysis of the final solution. Seismic raypaths were defined by calculating the shortest traveltime paths through a network consisting of nodes and representing the earth. They chose to solve an inverse problem that explicitly minimizes data misfit as well as model roughness. Ditmar et al (1999) developed a new algorithm for tomographic inversion of traveltimes of reflected and refracted seismic waves. In the case of a very inexact initial model, a layer-by-layer inversion strategy was recommended as a first inversion step. It was assumed that the model consists of several layers separated by interfaces represented by a set of points connected by straight segments. Velocity distribution in each layer was described by means of its own velocity grid, the layer being completely inside the grid. The velocity values were specified at gridnodes; bilinear interpolation was used in between them. Zhu (2002) summarized the results of tomostatics based on turning wave tomography. The main conclusions are: - if structures are relatively simple and the assumption for refraction statics is valid, tomostatics will not be better than refraction statics; - non-linear tomography is not sensitive to an initial velocity model but a good initial velocity model improves the convergence rate; - usually, the recording aperture (signed offsets) should be at least four times larger than the desired imaging depth; - if the first arrivals are contaminated by noise at near or far offsets, we should eliminate those offsets;

15 it is recommended to remove any previously applied elevation and velocity statics before tomostatics. Bohm et al. (2006) used a joint inversion of both first and refracted arrivals in order to obtain a well-resolved velocity field for the computation of static corrections. After the analysis of the diving waves, they inverted the traveltimes associated with the refracted events by using the velocity model obtained from the diving waves as the initial model. Also after inverting the two refracted arrivals separately they used the resulting output velocity field as a new initial model for jointly inverting again the direct arrivals and the traveltimes with the first and second refracted waves, in order to obtain a more accurate velocity field in depth. Bridle R., (2006) analysed the applications of refraction statics and tomostatics on test lines. For longer deeper anomalies with irregular raypaths, refraction statics and tomostatics were expected to provide major improvements; however, only marginal improvements were observed. In the test line considered the refraction statics provided the best section visually in terms of signal strength, sharpness and continuity, with a structure that seems geologically reasonable. The image provided by tomostatics was similar in structure, but was much noisier. However, only the tomostatics solution was able to focus some events in the most difficult area. Summaring the above presentation of tomographic publications we can state that the application of tomographic inversion to imaging near surface heterogeneities and to estimation static corrections is still the subject of many current investigations.

16 II. THE DESCRIPTION OF THE APPLIED PROGRAMS II.1. PROMAX S PROGRAMS FOR VELOCITY MODEL EDITING, SEISMIC MODELING, TURNING RAY TRACING, TURNING RAY TOMOGRAPHY, INTERACTIVE FIRST BREAK PICKING AND DATABASE OPERATIONS During the presented analysis several programs of ProMAX system have been used for the purposes of velocity model editing, seismic modeling, turning ray tracing and turning ray tomography. At the stage of constructing velocity fields the process named Interactive Velocity Editor was used. This tool is designed for building and editing velocity fields and is useful for creating a velocity and/or density field for finite difference modeling as well as for analyzing and adjusting the results of a tomographic velocity inversion. The main approach to editing a velocity field is to create a polygon and use the polygon to act on the velocity field. After closing a particular polygon we can define a velocity field using option Input/Set Velocity and then write the velocity field into the polygon using option Apply Velocity Edit. The example of velocity model (gradient model) defined in the work window of Interactive Velocity Editor is presented in fig Fig The example of velocity model (gradient model) defined in the work window of Interactive Velocity Editor The main parameters of the Interactive Velocity Editor process are: - the type of velocity functions to edit (interval velocity in depth was mainly used ), - CDP number on left and right edge of velocity field, - the spatial sampling of the output velocity field expressed in CDP s, - the minimum and the maximum depth of the input velocity field (in meters), - depth sample interval of new velocity field (in meters).

17 The presentation of work window of Interactive Velocity Editor is included in fig Fig The presentation of interactive work window of Interactive Velocity Editor The velocity field is the descrete one and is saved in the database as the velocity table in which the velocity in the node of assumed grid is the function of CDP s number and depth. The constructed velocity field may be exported using special ProMAX ASCII format including only nodes of the velocity grid which define velocity variations. Other nodes are omitted. The next applied ProMAX program was process named Finite Difference Modeling (FDM). This process provides an accurate, but compute intensive method of forward modeling the earth s response. A highly complex velocity and density field may be easily built using the Interactive Velocity Editor. FDM then divides the field into a very fine two dimensional grid. Within each rectangular grid point, velocity and density values are approximately constant, and a compressional wave can be accurately propagated through the grid point. FDM offers two modes of modeling: - Exploding Reflector being a poststack, zero-offset form of modeling, - Point Source simulating the firing of a shot into an array of receivers with finite offsets.

18 In the presented analysis the second mode of modeling was applied. FDM process is the type of acoustic modeling (only P waves are used) resulting in the set of seismic traces. The main parameters of the process are: - the name of the input velocity/density field constructed in Interactive Velocity Editor, - type of modeling mode (Exploding Reflector or Point Source), - the maximum depth of field for modeling (in meters), - length of time (in ms) to run modeling per shot, - the dominant frequency that will be contained in the wavelet produced by this model, - type of source wavelet (Zero/ Minimum Phase Ricker, Zero/Minimum Phase Ormsby, Zero/Minimum Phase Klauder, Gaussian Wavelet), - the grid adjustment factor changing the grid spacing and time steps, - CDP spacing in meters, - parameter of selection boundary condition at the surface (reflecting or absorbing multiples boundary), - CDP of left and right edge of modeling field, - number of shots, - CDP increment between shots, - depth of the first shot (the same for all), - number of receiver stations, - CDP of the first receiver, - distance (in CDPs) between shot and first group, - receiver location increment (in CDPs). The parametrization window of Finite Difference Modeling is presented in fig The seismic record calculated with FDM using the velocity field from fig. 2.1 and parameters from the fig. 2.3 are presented using ProMAX procedure Trace Display in fig The direct task of turning ray tomography in ProMAX system was solved using the process named Turning Ray Tracing. This process produces the predicted travel times of turning wave for a starting model which are compared with the picked travel times. It is also used to produce the ray paths which are used in the tomographic inversion. In according with the Promax manual process Turning Ray Tracing uses a Langan & Lerche style ray tracer, which is a shooting method that traces rays through a gridded velocity field using triangles. Since this ray tracer uses a shooting approach to find the ray from a source to a receiver, it cannot trace rays into a shadow zone. The shooting approach works by first shooting a fan of rays at all angles. Then the two rays that span each receiver are used to iterate to a receiver.

19 Fig The parametrization window of Finite Difference Modeling Fig The seismic record calculated with FDM using the velocity field from fig.2.1 and parameters from the fig. 2.2 are presented using ProMAX procedure Trace Display Ray tracers can fail to find a receiver for several reasons (to small width od the area with relation to the position of first/last receiver, too small depth od the area, shadow zones). The main parameters of the process are: - the name of an interval vs. depth file from the database, - option of creating a ray path file (needed in Turning Ray Tomography), - the name of output ray path file to store in the RAYS database; although the ray paths are

20 stored in the table database, they are not tables and cannot be viewed/exported/ or edited using the menu features, - the number of the first/last shot at which to start/end tracing rays, - the shot increment for ray tracing, - the nmber of arrivals for each receiver to store in the database, this value needs to be greater than 1 to save the ray information when multiple arrivals reach a receiver, - the order the arrivals are written to the database; the options are Minimum travel time, Maximum travel time, Minimum amplitude, and Maximum amplitude; generally, a sort of Minimum Travel Time First is used; - CDP spacing of the velocity field in m, the x axis of the velocity field is defined by CDP number, - horizontal cell size in meters for ray tracing, values on the order of m are recommended, - vertical cell size, it is recommended to make this the same as the horizontal cell size, or a little smaller, - maximum depth of velocity field to use for ray tracing, this depth should be large enough to allow room for rays reaching the last receiver and beyond, - buffer for ray tracing field beyond first and last station (enter the distance in CDPs to extend the velocity field to the left and right of the first and last source or receiver positions, this buffering is necessary for the fan shooting to send a ray, - option of storing ray tracing information in the database under the default entry names, - a robustness factor (a value greater than 1.0 will reduce program error, but it will increase computation time, a reasonable range for this value is The parameterization window of Turning Ray Tracing is presented in fig The results of Turning Ray Tracing process are saved in ProMAX Database and may by displayed and edited using Database procedures. Example of such a display of calculated traveltimes are presented in fig. 2.6 for the case of gradient model without anomaly (red graph) and with low velocity anomaly (blue graph).

21 Fig The parameterization window of Turning Ray Tracing process Fig Example of a display of calculated traveltimes for the case of gradient model without anomaly (red graph) and with low velocity anomaly (blue graph). If the depth of velocity model is too small the achieving receivers on far offsets may be impossible. In this case traveltimes are not estimated and in Database on the positions of distant receivers (stations) we have constant time extrapolated from last achieved receiver. This case is illustrated in fig. 2.7 (horizontal part of the left branch of the traveltime graph) and confirmed in fig. 2.8 by behaving of ray trajectories determied with the help of Fortran programs (#RAYEDT and #RAYPLTT).

22 Fig Example of a display of calculated traveltimes for the case of gradient model without anomaly when the model depth is too small for assumed velocity gradient; vertical axis traveltimes in ms, horizontal axis channel number The presented problem of turning ray tracing by means of Turning Ray Tracing procedure is very important in the case of low velocity layer which practically has small thickness and the problem of achieving distant receivers arises. Fig Example of a display of calculated ray trajectories for the case of gradient model without anomaly when the model depth is too small for assumed velocity gradient The next two ProMAX programs were used before tomographic inversion. The first named Create Turning Ray Velocity produces an approximate interval velocity model for the ray tracing from the first break picks in depth using flat layer theory of refracted arrivals. This approach is sometimes known as the slope/intercept method for refractor inversion. The travel time versus offset points are converted to refractor velocity/depth under the assumption that

23 velocity is monotonically increasing in depth. Then the process Create Turning Ray Velocity is used in the case of layered model when the velocity increases with depth and all picks of head waves arrivals are available. In practice it is difficult to have all these conditions satisfied. The second ProMAX program named Compute Residual Travel Times subtracts the predicted travel time database values from the picked travel time database values and stores the result in a new database value. These subtracted values are the residual travel times used for tomographic inversion. The picking of the first breaks is realized using interactive procedure available from the work window of Trace Display procedure. The picker is activated from the mouse button and is the type of the crosscorelation picker. The resulting picks are saved to database file. The last and the most important in my analysis ProMAX program is named Turning Ray Tomography. This is the main program of the Turning Ray Tomography package that performs the velocity inversion. It takes the travel time deviations between picked travel times and those predicted for a starting model to adjust the velocity along the ray paths of the starting model to reduce the travel time deviations. Following manual we can say that this process is an inversion and should not be considered automatic. Inversions are very error prone. Artifacts can cause problems except in the best of situations. Inversion should be treated as a mathematical aid that assists the interpreter, but it is the interpreter making the final decision. The main parameters of Turning Ray Tomography comprise: - minimum eigenvalue to invert, - horizontal smoothing (in meters), - vertical smoothing (in meters), - damping factor (in seismic ray number), - maximum residual travel time to use (in ms), - norm (L2 or L1) used in the process of minimization of the objective funtion, - defining model weighting input to the inversion. The parameterization window of the Turning Ray Tomography is presented in fig The problem of selection the optimal values of the input parameters of Turning Ray Tomography process is essentially important in the case of near surface thin layer where we have small distances between receivers and medium heterogeneties. To select the proper values of inversion parameters in the analyzed case the analysis has been accomplished

24 Fig The parameterization window of the Turning Ray Tomography (Kasina, Zawam 2006) on synthetic records generated with the finite difference method in the acoustic variant. The construction of input data to tomographic inversion Several gradient models of near surface medium with velocity anomalies have been used in the analysis of tomographic inversion effectiveness. The first model was the gradient one with low velocity zone (Fig. 2.10a). The velocity was increasing from 800 m/s near the surface to 2900 m/s for the maximum depth 300 m. The zone with low velocity (800 m/s) was located in the range of depth m and its width was 104 CDP (for CDP interval equal 5 m). The velocity model used in the modeling process contained 1200 CDPs. In the second model (fig. 2.10b) with high velocity anomaly the constant velocity layer was embeded in the gradient model with velocity 2500 m/s on depth 160 m and the velocity in the anomaly zone was changed from 800 m/s to 1500 m/s. For such a model we can observe the turning wave in the first gradient layer and the head wave generated on the boundary between first and second layer. To model the full wave pattern of records with the finite difference method in acoustic variant the procedure Finite Difference Modeling (FDM) was applied in the processing system ProMAX. The surface point source was assumed with source signal in the version of minimum-phase Ricker signal with pick frequency 40 Hz. In the modeling process the traces of 10 split-spreads have been generated. Each spread contained 601 receivers with receiver interval equal 10 m (2 CDPs). The shot interval was equal 100 m (20 CDPs). The modeling

25 has been performed fo the following effective dimensions of the grid: x = z = 2.66 m and for the time interval equal t = 0.53 ms. These values of modeling parameters let us to avoid numerical errors and to secure calculation stability. The selected synthetic record calculated for the first velocity model (model with low velocity anomaly) is showed in Fig The selected synthetic record calculated for the second velocity model (model with high velocity anomaly) is showed in Fig Fig. 2.10a. Velocity model of the gradient medium with low velocity anomaly (v = 800 m/s); horizontal axis CDP s number, vertical axis depth in meters Fig. 2.10b. Velocity model of the gradient medium with high velocity anomaly (v = 1500 m/s); horizontal axis CDP s number, vertical axis depth in meters

26 Fig Theoretical wave pattern of selected record obtained with FDM modeling for the first velocity model (from Fig. 1) with low velocity anomaly with results of first break picking marked (red); horizontal axis offset in meters Fig Theoretical wave pattern of selected record obtained with FDM modeling for the second velocity model with high velocity anomaly with results of the first break picking marked (red); horizontal axis offset in meters The nest step of preparation input data to tomographic inversion was the defining the traveltimes of turning ray wave for the starting velocity model. The starting models were created from original velocity models (fig. 2.10a, 2.10b) by removing the anomalous zone. Additionally, one starting gradient model in the case of second velocity model has been

27 created using interpretation of first breaks on model records without velocity anomaly by means of Create Turning Ray Velocity procedure available in ProMAX system. Fig The traveltimes graphs of turning wave (defined with picking first breaks blue) in the medium with high velocity anomaly and calculated traveltimes (red) in the medium without velocity anomaly; horizontal axis trace number, vertical axis traveltimes in ms The traveltimes for starting velocity model have been defined using procedure Turning Ray Tracing in ProMAX system or have been identified with first breaks traveltimes picked on the records without velocity anomaly. The example of the graph containing first breaks traveltimes from records with high velocity anomaly and the traveltimes calculated for the second starting velocity model by means of Turning Ray Tracing is illustrated in Fig The last step of preparation input data to tomographic inversion was to define time differences between picked and calculated (for starting velocity models) traveltimes using procedure Compute Residual Travel Times available in ProMAX system. The tomographic inversion in discussed variant of turning waves is realized in ProMAX system with Turning Ray Tomography procedure. According to the well known ray tomography theory during the consecutive tomographic inversions, in which the resulted velocity field of previous inversion is after process of smoothing and adding constraints the input velocity field to the next inversion, we obtain for correct selection inversion parameters the reducing of the differences between picked and calculated traveltimes. These differences are named residual traveltimes and their reduction is the basic criterion of evaluation the inversion effectiveness. The first step before inversion is to select the proper

28 parameters of inversion. The basic principles of parameter values selection are presented below in accordance with the instructions of ProMAX manual: - the value of ray damping should be equal a fourth of the average ray density; - if the residual travel times are not improved, then we use a smaller eigenvalue or less smoothing, - the L1 norm is useful when your travel time picks are noisy or have many bad picks, - variable model weighting can be used to disallow velocity variations in certain parts of the model and encourage them in others; we use the values of weights from the - a smaller value of eigenvalue inverts more subtle features, even features with some imposed smoothing, but increases noise effects; this value should be about 0,05 0,2, - the scale of velocity field smoothing (parameter horizontal smoothing) should be decreased in consecutive iterations. The analysis of the tomographic inversion effectiveness The main goal of realized calculations was the evaluation the effect of selection of tomographic inversion parameters on the accuracy of reconstruction of near surface velocity fields. The precision of such reconstruction has been evaluated by means of the analysis of the resulted velocity fields, the difference between resulted and assumed velocity fields as well as the differences between picked and theoretically calculated traveltimes. These time differences are named residual traveltimes. Three consecutive iterations of tomographic inversion have been accomplished for the selected parameters values and then the value of selected parameter has been changed. Three iterations of tomographic inversion have been accomplished for established smoothing parameters of horizontal smoothing (600 m) and vertical smoothing (75 m) for established minimum eigenvalue 0,2. The resulted velocity field is showed in Fig The anomalous velocity zone which assumed position is marked by white rectangle is difficult to interpret. The error of velocity estimation in this zone reaches values m. The graph of residual traveltimes is presented in Fig These times have been reduced from the starting values ( 4) ms 24 ms to the values 0 ms 13 ms after third inversion. The differences between residual traveltimes after second and third inversion were very small.

29 Fig The velocity field after 3 tomographic inversions without changing the parameter of horizontal (600 m) and vertical (75 m) smoothing; horizontal axis CDP number, vertical axis depth in meters Introducing variation of horizontal smoothing parameter in consecutive inversions from value 600 m, through value 300 m to the value 150 m after third inversion improved not so much reconstruction of anomalous zone (Fig. 2.16) at the cost of some deformation of the deepest part of velocity field below anomaly. We can observe essential decreasing the residual traveltimes for the largest offsets to the value about 2 ms after third inversion (Fig. 2.17). In the next step the calculations have been accomplished for established horizontal smoothing (600 m) and varying value of vertical smoothing through values 75, 50 and 25 in consecutive inversions. We observe decreasing the residual traveltimes in the range of intermediate offsets. The application of vertical smoothing 25 m and varying horizontal smoothing parameter in consecutive inversions through values 600, 300 and 150 m brought the best reconstruction of anomalous zone (Fig. 2.18) and caused the essential reduction of residual traveltimes for small, intermediate and large offsets after third inversion (Fig. 2.19).

30 Fig The graphs of residual times before first inversion (orange), after first inversion (green), after second inversion (red), after third inversion (blue); parameter of horizontal smoothing 600 m, parameter of vertical smoothing 75 m; vertical axis residual time in ms Fig The velocity field after 3 tomographic inversions for established value of vertical smoothing (75 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis CDP number, vertical axis depth in meters

31 Fig The graphs of residual traveltimes calculated for established value of vertical smoothing (75 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis - residual traveltimes in ms Beside the smoothing parameters the effect of other parameters have been analyzed too. Decreasing the minimum eigenvalue from value 0,2 to value 0,1 and established remaning parameters (horizontal smoothing 600 m, vertical smoothing 75 m) caused some lowering residual traveltimes for the largest offsets. The effect of changing damping parameter marked only for intermediate and largest offsets. Among four values of this parameter (5, 10, 15, 20) for the value 5,0 we can observe noticeable decreasing of residual traveltimes. Application the norm L1 instead of the norm L2 didn t introduce as one could expect for data with good quality of first break picks essential changes. In the above analysis it was assumed that the starting velocity model is the exact gradient model of the medium without anomaly. Two starting gradient models have been additionally considered to evaluate the effect of error of defining starting model: one with vertical gradient increased by 10% (gradient value 7,7 m/s per meter) and the second one with gradient decreased by 10% (gradient value 6,3 m/s per meter). Differences of residual traveltimes for these two cases were essential only after first inversion. After second and third inversion the residual traveltimes practically were the same. Lowering the value of parameter named Maximum travel time residual to use from 10 to 5 after second and third

32 inversion didn t decreased the maximum values of residual traveltimes in the zone of intermediate offsets. Fig The velocity field after 3 tomographic inversions for established value of vertical smoothing (25 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis CDP number, vertical axis depth in meters In the above analysis it was assumed that the starting velocity model is the exact gradient model of the medium without anomaly. Two starting gradient models have been additionally considered to evaluate the effect of error of defining starting model: one with vertical gradient increased by 10% (gradient value 7,7 m/s per meter) and the second one with gradient decreased by 10% (gradient value 6,3 m/s per meter). Differences of residual traveltimes for these two cases were essential only after first inversion. After second and third inversion the residual traveltimes practically were the same. Lowering the value of parameter named Maximum travel time residual to use from 10 to 5 after second and third inversion didn t decreased the maximum values of residual traveltimes in the zone of intermediate offsets. Summaring the analysis results in the considered case of gradient model with low velocity anomaly we can state that: - tomographic inversion let to locate the low velocity anomaly; the accuracy of positioning is greater in the horizontal direction than in the vertical direction; - the velocities in anomalous zone are reconstructed for considered iteration number with the error about 19-25% (the velocities are generally too high); - the greatest lowering of the residual traveltimes ca be observed after first inversion;

33 when the starting model is taken with error the model with lowered gradient is much better; - using the gradual lowering of horizontal smoothing after consecutive inversions with lower value of vertical smoothing and lowered minimum eigenvalue made it possible to obtain reduction of residual traveltimes; this reduction appeared to be offset dependent. Fig The graphs of residual traveltimes calculated for established value of vertical smoothing (25 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis residual traveltimes in ms Summaring the analysis results in the considered case of gradient model with low velocity anomaly we can state that: - tomographic inversion let to locate the low velocity anomaly; the accuracy of positioning is greater in the horizontal direction than in the vertical direction; - the velocities in anomalous zone are reconstructed for considered iteration number with the error about 19-25% (the velocities are generally too high); - the greatest lowering of the residual traveltimes ca be observed after first inversion; - when the starting model is taken with error the model with lowered gradient is much better; - using the gradual lowering of horizontal smoothing after consecutive inversions with lower value of vertical smoothing and lowered minimum eigenvalue made it possible to obtain reduction of residual traveltimes; this reduction appeared to be offset dependent. The final calculations have been accomplished to evaluate the effect of the model parameters on tomographic inversion effectiveness. The second model of the medium with gradient layer over constant velocity basement (2500 m/s) has been considered. The high

34 velocity anomaly (1500 m/s) was embeded in the gradient layer. In the starting model velocity anomaly has been removed from the gradient layer. The calculations of the theoretical traveltimes have been done using two ways. One way was to calculate traveltimes with the procedure Turning Ray Tracing. The second way was to pick first breaks on the synthetic records obtained with Finite Difference Modeling procedure to create gradient medium using procedure Create Turning Ray Velocity and then to calculate traveltimes with the procedure Turning Ray Tracing. The calculated traveltimes have been used to define residual traveltimes. The inversion has been accomplished for the following parameters: minimum eigenvalue 0.1, damping factor 5, vertical smoothing 25 m, horizontal smoothing changing from 600 m through 300 m to 150 m in the consecutive inversions. The inversions have been realized with and without velocity weights. The results of calculations for the case of inversion without velocity weights are showed in Figs The location of velocity anomaly after third inversion is very good in horizontal and vertical direction (Fig. 2.20). The errors of inversion presented in Fig in the form of map of differences between assumed and resulted velocities have generally small values with the exception of central part of anomaly (error values about 200 m/s) and the deepest part of the map in its central part (error values about 300 m/s). The residual traveltimes are significantly reduced after third inversion from starting values about 20 ms to values about ± 4 ms (Fig. 2.22). If we look at the ray density map (Fig. 2.23) after first inversion we can see that the significant velocity errors occur even in the area of higher ray density. If we treat this area as well illuminated and if we select it as the area with great velocity weights (that means as the area in which significant velocity changes are allowed) we will obtain after third inversion the velocity field showed in Fig We can observe some improvement of velocity field reconstruction appearing as better smoothing its values in deeper part of the velocity field without significant changes of residual traveltimes (Fig. 2.22). Applications of the picked times in computations of theoretical traveltimes practically didn t change the results of inversion. Conclusions The realized model computations and the analysis of tomographic inversion results made it possible to formulate the following conclusions: - the procedures of tomographic inversion realized in seismic data processing ProMAX in the variant of turning ray tomography may be efficiently used to reconstruction of near surface gradient velocity fields with anomalous zones basing on the picked first breaks of reflection

35 seismics records, - the effectiveness of tomographic inversion appeared to be much better in the case of high velocity anomalous zones embeded in the gradient medium, - the selection of the proper values of tomographic inversion parameters made it possible to improve the quality of velocity field reconstructions and to minimize the values of residual traveltimes, - the horizontal and vertical smoothing parameters appeared to be the most meaning ones among others, - we can observe a little better velocity reconstruction in horizontal direction then in vertical direction, - the errors of velocity anomaly reconstruction can achieve values of about 25% in the case of low velocity anomaly and about 15% in the case of high velocity anomaly, - we can observe significant reduction of residual traveltimes after first inversion in all considered cases. Fig The velocity field after 3 tomographic inversions in the case of gradient model with high velocity anomaly for established value of vertical smoothing (25 m) and varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; horizontal axis CDP number, vertical axis depth in meters

36 Fig The map of differences between assumed and resulted velocities after 3 inversions in the case of of gradient model with high velocity anomaly; the graph on right side is defined on the vertical line crossing the central part of velocity anomaly Fig The graphs of residual traveltimes in the case of of gradient model with high velocity anomaly calculated for established value of vertical smoothing (25 m) after first inversion (green), after second inversion (red), afer third inversion (blue) for varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions; vertical axis residual traveltimes in ms

37 Fig The map of ray density in the case of of gradient model with high velocity anomaly with marked area used for velocity weighting; horizontal axis CDP s number, vertical axis depth in meters Fig The velocity field after 3 tomographic inversions in the case of gradient model with high velocity anomaly for established value of vertical smoothing (25 m),varying horizontal smoothing through the values 600, 300 and 150 m in consecutive inversions and application of velocity weighting; horizontal axis CDP number, vertical axis depth in meters

38 II.2. FORTRAN PROGRAMS FOR DIRECT AND INVERSE PROBLEM OF THE HEAD WAVE TOMOGRAPHY Two original Fortran programs have been tested and used to solve direct and inverse task of head wave tomography. The first one named #RAYEDH was appropriated for defining head wave trajectories and traveltimes and the second one named #INVERSDH was appropriated for reconstruction velocity fields in the process of head wave traveltimes tomographic inversion. These two programs work for discrete velocity fields and can be use to solve direct and inverse task of curved ray tomography. II.2.1. The description of the program for direct problem of the head wave tomography The algorithm of #RAYEDH is based on the following assumptions defining the interpretation model: - we consider 2D velocity model of the medium in the plane (x,z) consisting of the two layers separated by linear refractor, - the velocities of the first layer are defined in the nodes of rectangular regular grid, - the velocity below the refractor is assumed to be constant, - the position of the flat refractor is defined by coordinates (x,z) of two its points, - the positions of shot points are defined by (x,z) coordinates, - the positions of receiver points are defined by (x,z) coordinates where receiver depth z is constant for all receivers, - the curvilinear trajectory of seismic ray propagating from the source is defined by Fermat principle. The assumption of constant receiver depth is some simplification but my analysis is mainly directed towards estimation of the effect of velocity heterogeneities in the first layer not topography. The scheme of the seismic ray trajectory against a background of assumed grid is presented in fig

39 Fig The scheme of the seismic ray trajectory against a background of assumed grid and positions of source (PW) and receiver (PO) The consecutive points of seismic ray trajectory are determined through solution of the following set of differential equations resulting from Fermat s principle (Kasina, 1988): 1 v v dα α = sin cos α ds dx = ds cosα v x z dz = ds sinα dt = ds v where: dα - angle between tangents to a ray trajectory defined in two points located at distance ds; v - velocity function; α - angle between x-axis and ray trajectory. That set of equations has been solved using numerical Runge-Kutta method of fourth order. Interpolation of velocity function and its derivatives has been performed using linear or cubic interpolation and velocity values defined in the nodes of rectangular grid. Before interpolation the estimation of the first and second spatial derivatives of the velocities in the nodes of assumed grid is realized. If we introduce the velocity grid consisting of I columns and J rows and the dimensions of the cell x and z then the coordinates of the nodes are defined by the relations: x = i x, z = j z, i = 0, 1, 2,..., I j = 0, 1, 2,..., J If we introduce the following notation for velocities in the nodes: v(x,z) = V(i x, j z) = V i,j

40 then we can define the spatial derivatives of velocities using central differences: v V ( x, z) = V ' i+ 1, j i 1, j x v v V ( x, z) = 2 x V ' i, j+ 1 i, j 1 z 2 z V V V + V ( x, z) = 4 x z '' i+ 1, j+ 1 i 1, j+ 1 i+ 1, j 1 i 1, j 1 xz where: v '' xz 2 v ' v ' v =, v z =, v x =. x z z x The execution of the #RAYEDH program consists of several steps. First, the critical angle is estimated for assumed velocities on both sides of refractor under the shot. Then for assumed range of starting angles from the shot the proper starting angle is estimated for which incidence angle of the ray on refractor is equal the critical angle with assumed accuracy. The trajectory of the ray with critical incidence angle is defined and its intersection with refractor. Next the coordinates of the following points of the ray trajectory along refractor are defined for assumed propagation interval. For each point on refractor the emergence angle of head wave is estimated. Then the trajectory of head wave ray is defined from refractor to level of receivers with assumed step of propagation along ray. The coordinates of intersections of all rays with receiver level is estimated. For the variant of established positions of receivers the trajectories of rays hitting receivers with assumed accuracy are established. When the positions of all rays is defined the coordinates of trajectory points are saved and the traveltimes are calculated from the source to each receiver in accordance with assumed velocity field. The example of ray trajectories against a background of assumed seismogeological model and positions of source/receivers is fig Program #RAYEDH includes the master segment and the following subroutines: DIFR, INTRP, TRAF, TRAS, INT1, INT2, GRAF. In the master program the input data are red, the parameters used in the subroutines are defined, the results of calculations are printed and saved. The subroutine DIFR is appropriated for the estimation of spatial derivatives of the velocity field. The subroutines INTRP, INT1, INT2 are used to linear or cubic interpolation of velocities and their spatial derivatives. The subroutine TRAS is used to define the seismic ray

41 Fig The example of ray trajectories against a background of assumed seismogeological model and positions of source/receivers trajectories through solution of the set of differential equations resulting from Fermat s principle. The subroutine TRAF is used to define the starting angles of the rays propagating from the shot to assumed receivers. The subroutine GRAF is used for simplified presentation of input and output small velocity fields by means of character map. The input data for the case of two shot points are red in the following sequence: READ(1,*) NAME2 READ(1,*) IPRINT, NRMOD, PLOT, ITP, IPRNT READ(1,*) ITAPE IF(ITAPE.NE.0) READ(1,*) NAME7 IF(IPLOT.NE.0) READ(1,*) NAME3 IF(ITAPE.NE.0) READ(7) LW, T, DX, DZ, LWT, (V(I), I= 1, LWT) IF(ITAPE.EQ.0) READ(1,*) LW, LT, DX, DZ READ(1,*) DS, DXMAX, DZMAX, DALFOP, DF1, INTL READ(1,*) WXPS1, WZPS1, WXPS2, WZPS2 READ(1,*) LPG, WZG, WXG(1), DXG READ(1,*) X1RF, Z1RF, X2RF, Z2RF, V2, V1S1, V1S2, DSRF, SRFMAX, DI READ(1,*) ALFOS1, ALFGS1, ALFOS2, ALFGS2 IF(ITAPE.NE.0) READ(1,*) NRR, (NR1(I), NR2(I),VN(I), I=1, NRR) where: NAZWA2 the name of output text file for printing the results of calculations

42 NAZWA3 the name of output binary file used for plotting rays NAZWA7 the name of input binary file with velocities IPRINT - indicator of printouts: IPRINT=0 - variant without control printouts IPRINT=1 - variant with control printouts NRMOD - model number IPLOT - indicator of saving data for plotting IPLOT=0 variant without saving data for plotting IPLOT=1 variant with saving data for plotting ITP - indicator of calculation option ITP=0 calculation for assumed receivers ITP=1 calculation for assumed values of "s" parameter (DSRF) along refractor IPRNT - indicator of additional control printouts (ray points) ITP=1 and IPRNT=1 - all control printouts without ray points ITP=1 and IPRNT=2 - all control printouts with ray points ITP=0 and IPRNT=1 - limited control printouts without ray points ITAPE - indicator of input velocity data ITAPE=0 - reading from text file ITAPE=1 - reading from binary file LW - columns number of velocity matrix LT - rows number of velocity matrix DX - the dimension of the cell in x direction (in meters) DZ - the dimension of the cell in z direction (in meters) LWT number of grid nodes DS ray step (in m) DXMAX, DZMAX assumed error of hitting receiver (in m) DALFOP increment of starting ray angle from the source (in deg) DF1 - minimal increment of starting ray angle from the source (very small value) INTL - indicator of interpolation type INTL=0 - cubic velocity interpolation INTL=1 - linear velocity interpolation WXPS1,WZPS1,WXPS2,WZPS2 coordinates of shot points 1 and 2 (in m) LPG - number of receivers WZG - coordinates z of receivers (in m)

43 WXG - coordinates x of receivers (in m) DXG receiver interval (in m) X1RF, Z1RF, X2RF, Z2RF coordinates of two refractors points (in m) V2 head wave velocity (in km/s) V1S1,V1S2 overburden velocity near refractor under SP. 1 and SP. 2 DSRF - assumed propagation step along refractor (in m) SRFMAX - assumed maximum propagation distance along refractor (in m) DI - assumed accuracy of reaching critical angle (in deg) (very small value) ALFOS1, ALFGS1, ALFOS2, ALFGS2 minimum and maximum starting angle from the SP.1 and SP. 2; for SP. 1 (source on the left of the spread)) angles are decreasing from ALFOS1 to ALFGS1; for SP. 2 (source on the right) angles are increasing from ALFOS2 to ALFGS2 VN(I) - input velocities red from text file in km/s and defined in grid nodes NR1(I), NR2(I) - pair of grid nodes between which velocity is equal VN(I) NRR - number of pairs (NR1,NR2) The main output data for the case of two shot points are written excluding the writing of input parameters - in the following sequence: IF(IPLOT.NE.0) WRITE(3) NRM IF(IPLOT.NE.0) WRITE(3) IPS, IG, (I, XA(I), ZA(I), VA(I), I=1, NRM) WRITE(2,140) IPS, IG, (I, XA(I), ZA(I), VA(I), I=1, NRM) WRITE(2,150) ((J, I, T(I,J), I=1, LPG), J=1, LPS) IF(IPLOT.NE.0) WRITE(3)((J, I, T(I,J), I=1, LPG), J=1, LPS) where: NRM the number of ray points for selected shot IPS and selected receiver IG XA(I), ZA(I), VA(I) the coordinates of I-th point of the ray and the value of velocity in that point; these values are saved in binary file (NAME3 on unit 3) and in text file (NAME2 on unit 2) LPS number of shot points LPG number of receivers T(I, J) the values of calculated traveltimes of head wave

44 The main difference in input data for the case of many shots and split-spreads refers reading the parameters of aquisition: READ(1,*) LPS, (WZPS(I), I=1,LPS), (WZPS(I), I= 1,LPS) READ(1,*) LPG, WZG, WXG(1,1), WXG(LPG/2+1,1), DPS, DXG where: WXPS(I), WZPS(I) coordinates (x,z) of I-th shot point (in m) WXG(I,J) - coordinate (x) of I-th receiver for J-th shot (in m) WXG(1,1) coordinate (x) of the first receiver of first spread (first shot) WXG(LPG/2+1,1) coordinate (x) of the first receiver of right half of the first split-spread WZG coordinate (z) of receivers (in m) DPS shot interval (in m) DXG receiver interval (in m) The example of input and output data are presented below. Example of the input data of #RAYEDH for the case of 2 shots placed outside of the spread m200_2w_1200_1800z100_200a.out m200_2w_1200_1800z100_200a.vel m200_2w_1200_1800z100_200a.plt E E

45 Example of the output data of #RAYEDH for the case of 2 shots placed outside of the spread ***HEAD WAVES -DIRECT TASK (KINEMATICS)*** INPUT DATA MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= Z1RF= X2RF= Z2RF= HEAD WAVE VELOCITY (in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-2(in km/s)= ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 5.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)= ASSUMED ACCURANCY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09 FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= WZPS1= 0.00 WXPS2= WZPS2= 0.00 NUMBER OF RECEIVERS = 70 COORDINATES (z) OF RECEIVERS (in m)= 0.00 COORDINATES (x) OF RECEIVERS (in m))= CALCULATION PARAMETERS: MODEL NR= 1 IPLOT= 1 ITP= 0 RAY STEP DS= 0.80(m) ERROR OF HITTING RECEIVER (in m) - DXMAX DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1= ALFGS1= ALFOS2= ALFGS2= INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.1E+01 DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) THE NUMBER OF UNKNOWS (GRID NODES)

46 CALCULATION RESULTS Example of the output data of #RAYEDH for the case of 2 shots placed outside of the spread (c.d.) REFRACTOR DIP = CRITICAL ANGLE UNDER SP.-1 = CRITICAL ANGLE UNDER SP.-2 = XKR1= ZKR1= AWYJ1= TRAY1= CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP NO SOLUTION FOR RECEIVER NO. 38 XKR2= ZKR2= AWYJ2= TRAY2= CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP NO SOLUTION FOR RECEIVER NO. 67 NO SOLUTION FOR RECEIVER NO. 68 NO SOLUTION FOR RECEIVER NO. 69 NO SOLUTION FOR RECEIVER NO. 70 CALCULATED TRAVELTIMES SHOT GEOPHONE TIME(S)

47 II.2.2. The description of the program for inverse problem of the head wave tomography The algorithm of the tomographic inversion of applied head wave tomography is based (Kasina 2001) on the same interpretation model as used in the program # RAYEDH. Let s consider the seismic ray, propagating from the source to the assumed receiver. The traveltime of the ray may be expressed using relation: t r 1 = s v( P ) P k k... (2.1) where v(p k ) is the propagation velocity along k-th segment of ray trajectory of length s and position of its center P k. The velocity v(p k ) may be expressed as linear combination of velocities in the neighbouring grid nodes. When m is the number of these nodes then: v( P ) = c V... (2.2) k m m m where V m is the velocity in m-th grid node. If we use cubic interpolation m has the value 16. Let s define now the result of a subtraction: ' r r r t = t t... (2.3) where t r is the recorded time and t r is the traveltime calculated during ray tracing for assumed initial velocity field. If we treat this value as the error of defining traveltime (caused by the errors of selection velocity in grid nodes) then we obtain: and t r v Pk = s ( ) 2 v ( P ) P k k... (2.4) v( P ) = c V... (2.5) k m m m where v(p k ) is the error of velocity estimation in the point P k, and V m is the velocity error in m-th neighbouring node. Now the relation (2.4) has the form: v( P t s v P v P 1 ) ( 2 ) ( k ) r = = v ( P ) v ( P ) v ( P ) 1 2 c m Vm c n Vn c r Vr m n r = s v ( P1 ) v ( P2 ) v ( Pk ) k

48 Finally, for the assumed r-th seismic ray we obtain the equation: t = a, V... (2.6) r r m m m where a r,m defines the sum of coefficients relating to the same velocity increment V m. If we consider R seismic rays and by M we denote the number of used velocities in grid nodes then we obtain the set of equation: M a r, m Vm = t r, r = 1, 2,..., R m= 1... (2.7) In the above set of equation we treat the increments V m as the unknowns which must be added to the values of starting velocities to obtain the better estimation of the recorded traveltimes using modeled traveltimes. To avoid too big irregularities of the velocity in the solution we introduce the constraints imposed on estimated velocity increments V m. Finally, we minimize using norm L2 - the quantity: R M 2 S = a r, m Vm t r + λ ( Vm Vi )... (2.8) r= 1 m= 1 2 where i denotes indexes of neighbouring nodes in relation to node with index m, λ denotes the coefficient of constraints. If we introduce the matrix notation: $ $ $ A r, m, V m, t ( R, M) ( M, 1) ( R, 1) = a = V = t then the equation (2.7) may be written in the form: A$ V$ = $ t... (2.9) The condition of minimizing the quantity S may written in the form: S V As a result we obtain: k i r = 0, k = 1, 2,..., M... (2.10) T ( ) $ $ $ $ T A A + λ Ω V = A $ t... (2.11) where Ω $ is the matrix of constraints, A $ T is the transposition of matrix A $. The solution of the set of equation (2.11) gives us the column vector V $ of the velocity corrections. If we solve the set of equations using iterative method we obtain the consecutive approximations of the solution. Adding defined corrections V $ to the velocities

49 (assumed or resulting from previous iteration) in the grid nodes we obtain the consecutive approximations of the desired velocity distribution. The execution of the #INVERDH program consists of the following steps: - for assumed starting velocity distribution - defining the initial model - we perform ray tracing and traveltime calculations for established positions of sources and receivers in accordance with the algorithm of direct task of tomography, - in the process of ray tracing we generate the coefficient matrix A of equation set (2.9), - the matrix A is modified using algorithm of convolutional quelling, - generation of constraint matrix Ω, - solving the set of equation (2.11) for λ = 0, - we estimate (using the solution (2.11) for λ = 0) or we assume the value of observation error ε of the traveltime, - we solve the set of equation (2.11) for different, assumed values of constraint coefficient λ k and then we estimate the optimal value λ opt, - we solve the set of equation (2.11) for λ opt, - we modify resulting velocity matrix using once more operator of convolutional quelling, - we add the obtained velocity corrections V $ to the values of initial velocity field, - we use (or not) 2D smoother (average type) to resulting velocity field, - we treat modified velocities as input data to the next iteration step, - we continue calculations until we exceed the assumed number of iterations or at the moment when the maximum differences between velocities of consecutive iterations will be lower then assumed value. The program #INVERDH consists of master segment and the following subroutines: TRAS, DIFR, INT1, INT2, INTRP, TRAF, GRAF, PLUS, OPTY, SORT, TRANS, GRADSPR, COEF, OMEGA,SPLOT. Subroutines TRAS, DIFR, INT1, INT2, INTRP, TRAF, GRAF are the same as in the program #RAYEDH. Subroutine OMEGA is used to generate the matrix of constraints. Only nonzero elements (INTEGER type) of the matrix are calculated and saved with their indexes. The range of constraints between velocity value V m (or the value of its increment V m ) in selected grid node and values of neighbouring nodes is the matter of choice. If we limit this range to 4 neighbouring nodes we minimize the quantity:

50 where: m4 B = ( Vm Vm ) k mk = m1 2 V m = V i,j m 1 = (j - 2) W + i m 2 = (j - 1) W + i - 1 m 3 = (j - 1) W + i + 1 m 4 = j W + i Therefore: B V m m 4 = 2 ( V V ) = 0 m k = m1 m mk and we can write the relation defining the way of $ Ω matrix generation (Kasina 2001): M m= 1 { λ } M m4 Ω $ V $ = λ ( V V ) m= 1m= 1 m where M is the number of grid nodes, Ω $ is the matrix with dimensions M M, V $ is the matrix with dimensions M 1. In the considered case the values of Ω $ matrix elements are INTEGERS in the range (-1, 4). Subroutine COEF is used to generate nonzero elements of the coefficient matrix A. These coefficients are modified during introducing new seismic rays. Subroutine TRANS defines the transposition of matrix A and realizes multiplication matrix A T by A. Only nonzero elements of matrices are taken into account. Also the right side of equation (2.11) is calculated. Subroutine GRADSPR is used to solve the set of equation by means of conjugate gradient method. During iterations the actual value of the sum of squared residuals GAMMA2 is defined. Application of iterative method to solve the set of tomographic equation is necessary because the coefficient matrix is the singular one. Additionally, this matrix is sparse one. So we can save the computational time using only nonzero elements of matrices in the process of iterative solution. For solution our set of equation we can use also technique based on Singular Value Decomposition (Michelena 1993), but for large set of equations met in practice iterative methods (conjugate gradient method, Gauss-Seidel method) are much more effective (Young 1971, Kasina 1994, Kasina 1999). Subroutine PLUS i used to add matrix A T A and λ Ω $. Only nonzero elements are taken into account. Subroutine SORT is used for sorting the elements of resulting matrix. Subroutine OPTY is used to define optimal value of constraint coefficient λ opt. i

51 Subroutine SPLT is used to realize convolutional quelling. This is a technique improving imaging of seismic tomography data. The results were found to be superior to a simple, least-squares solution because convolutional quelling suppresses side bands in the resolving function that lead to imaging artifacts (Meyerholtz et al. 1989, Kasina 2001). The operator of convolutional quelling is a matrix with a special, banded structure. After convolution with a smoother (in the case illustrated, the smoother uses the eight nearest neighbors), the raypath is effectively broadened into a band with a width controlled by the width of the smoothing function. Convolutional quelling with a smoother is equivalent to specifying that the velocities in neighboring cells are to be highly correlated. Fig Graphic illustration of the process of convolutional quelling. The upper corner shows a perspective picture of the smoother used in the convolutional quelling. The left diagonal line of boxes illustrates pixels crossed by a ray associated with a given datum (nonzero values of the matrix A). After quelling, the raypath is broadened by the width of the smoother. The result for the eight nearest-neighbor example is illustrated on the right. For this example, the boxes with the solid boundaries would have larger amplitudes than those with the dashed boundaries (after Meyerholtz et al. 1989). In the presented program nine-points convolutional operator was used with the matrix structure presented below a a a a b a a a a with symbolic notation q=b/a used in the description of input data.

52 If we look at the comparison of popular methods of tomographic inversion (Scott and Fehler 1991) it will possible to classify the algorithm of inversion used in #INVERSDH. That comparison have been carried out using synthetic crosswell data and was based on the misfit with the true model, solution stability under different sets of noise of the same level, and resolution-covariance relationships. In the mentioned below relations G is the coefficient matrix of tomographic set of equations. The basic solution named DLS (Damped Least Squares) is obtained by minimizing a combination of data residual and solution length terms where the solution is really a perturbation from some starting model: G G G 1 T 2 g = ( + ε I ) 1 G T where I is the identity matrix. The damping parameter ε regulates the balance between data residual and solution length terms. Damping reduces the total length of the solution perturbation vector, but puts no constraint on the similarity of adjacent elements of the solution. Thus, DLS images tend to be rough when data are noisy or inconsistent. Damped least squares with a spatiul average smoother worked reasonably well in the smooth model case, but performed poorly when applied to the high-gradient model data. DLS plus averaging smoother means that we apply convolution with a spatial averaging operator to smooth DLS images: G S G G 1 T 2 g = ( + ε I ) 1 G T where S represents the smoothing operator. The smoother can have any shape. A given pixel value is replaced with a weighted average of the surrounding pixels and itself. Smoothing is applied without regard to how well constrained each model parameter may be. Iteratively reweighted least squares (IRLS) is an iterative scheme based on least squares that estimates the general L p solution (Scales et al. 1988). The L p solution is defined as that which minimizes the sum of absolute values of the data residuals, each raised to the power p. The L 1, L 2 (least squares) norms are most commonly used in geophysical inversion. The normal equations obtained by minimizing the L p norm are: G T R G m = G T R d where m is model data, d is observed data, R is a diagonal matrix representing reweighting with elements dt i p-2 and dt i are the data residuals. These equations must be solved iteratively if p 2. For p < 2, a threshold must be set for very small dt i.

53 The next solution named convolutional quelling (Meyerholtz et al. 1989) may be described as a ray broadening. The basic equation has the form: 1 T T 2 G g = W ( W G G W + ε I ) W G where the smoothing operation (operator W) distributes the path lengths to neighboring pixels which broadens the raypath. Convolutional quelling proved the best of the damped leastsquares methods, but had resolution limited by the dimension and shape of the smoothing function. Convolutional quelling performed the best of the damped least squares based methods in the misfit and stability tests, and is a very low error technique. We can treat this approach as some step towards solutions of wave approach type based on so called wavepaths and Fresnel volumes (Vasco et al. 1995). A slightly different class of methods falls under the category of regularization (Scales et al., 1990). These techniques are widely used and involve the minimization of a measure of model roughness which can be some combination of first and second spatial derivatives in various directions. The solution has the form: G G G T T g = ( + λ D T D) where D represents a first or second difference operator, and λ regulates the balance between the data residual and solution roughness. A first difference regularization method performed the best in terms of rms misfit from the true model, and solution stability under different realizations of noise. 1 From the analysis in the Scott and Fehler s paper (1991) it can be concluded that convolutional quelling is very effective technique, only a little weaker then the best one - first difference regularization. Taking into account the presented comparison of different tomographic inversions we can classify the algorithm used in #INVERSDH (comprising matrix of constraints, convolutional quelling and 2D smoother applied to resulting velocity fields) as very close to the solution based on first difference regularization. The input data to #INVERSDH for the case of two shot points are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA3 READ(1,*) NAZWA4 READ(1,*) ICASE, IPRINT, LMAX, ITAPE, IPRNT, IPLOT, LOPT, LATA READ(1,*) LBIEL,LSKAL,BL,SK,LOPT1,ATAMIN,LCON,LSPLT IF(IPLOT.NE.0) READ(1,*) NAZWA8 IF(ITAPE.NE.0) READ(1,*) NAZWA7 G T T

54 IF(LCON.NE.0) READ(1,*) NAZW10 READ(1,*) LW, LT, DX, DZ, ILW READ(1,*) WXPS1, WZPS1, WXPS2, WZPS2 READ(1,*) LPG, WZG, (WXG(I), I=1, LPG) READ(1,*) X1RF, Z1RF, X2RF, Z2RF, V2, V1S1, V1S2, DSRF, SRFMAX, DI READ(1,*) DS, DXMAX, DZMAX, DALFOP, DF1, INTL READ(1,*) LIT, ERES READ(1,*) ALFOS1, ALFGS1, ALFOS2, ALFGS2 READ(1,*) LEPS, DLMBD, DXMAXV IF(LEPS.NE.0) READ(1,*) EPS IF(LSPLT.EQ.1) READ(1,*) ((SS(I,J),J= 1,3), I=1,3) READ(1,*) ((J, I, TR(I,J),I=1, LPG), J=1, LPS) where: NAZWA2 - the name output data text file for printer NAZWA3 - the name of binary file *.ATA (created as product of matrix A and A T ) NAZWA4 - the name of binary file *.ATB with matrix ATB (created as product of matrix A T and B) NAZWA7 - the name of input binary file with velocities NAZWA8 - the name of output data binary file (with velocities) for plotter NAZW10 - the name of input data file with invariable velocities ICASE indicator of calculation variant ICASE.LE.0 we start with velocities defined as constant (V1S1), ICASE.GT.0 - we start with assumed velocity field from text file or binary file (for data from ProMAX ICASE.EQ.1) or from previous iteration (ICASE.EQ.2) ITAPE - indicator of reading velocities V(I) for ICASE.GT.0 from text file (ITAPE=0) or from binary file (ITAPE=1) LMAX - maximum number of iteration IPRINT indicator of printouts IPRINT = 1 - full control prints are available with (IPRNT=2) or without (IPRNT.NE.2) ray points IPRNT = 0 - only basic control prints are available (without ray points) IPLOT - indicator of saving velocity field IPLOT= 1 - saving resulting velocity field in binary file IPLOT= 0 - without saving resulting velocity field in binary file

55 LOPT indicator of constraints LOPT= 0 variant without constraints LOPT= 1 variant with constraints (LAMBDA>0.) LATA indicator of matrices (ATA, ATB) generation LATA= 0 - first iteration, we save resulting matrices ATA, ATB on disk LATA= 1 - we don t generate but read resulting matrices from previous iteration LCON indicator of introducing selected non variable velocities LCON=1 - we introduce from binary file non variable velocities LCON=0 - we don t introduce from binary file non variable velocities ATAMIN - the value of element of matrix ATA below which we are zeroing value LSPLT - indicator of convolutional quelling LSPLT= 1 - we introduce convolutional quelling LSPLT= 0 - we don t introduce convolutional quelling LBIEL indicator of "whitening" the coefficient matrix LBIEL = 1 calculations with "whitening" the coefficient matrix by adding small value BL to the diagonal elements LBIEL = 0 calculations without "whitening LSKAL indicator of matric scaling LSKAL= 1 - we introduce matrix scaling (multiplication matrix ATA and column matrix DT by value SK) LSKAL= 0 variant without scaling LOPT1 indicator of optimization of coefficient LAMBDA LOPT1 = 1 - we assume the value DLMBD of constant LAMBDA (for LOPT=1) LOPT1 = 0 - we assume increment DLMBD of constant LAMBDA (for LOPT=1), optimal LAMBDA is evaluated LW - number of columns of velocity grid LT - number of rows of velocity grid DX, DZ - dimensions of grid cell in meters ILW= LPS x LPG number of observations LPG - number of receivers WZG - coordinates z of receivers (in m) WXG(I) coordinates x of receivers (in m) X1RF, Z1RF, X2RF, Z2RF coordinates of two refractors points (in m) V2 head wave velocity (in km/s)

56 V1S1,V1S2 overburden velocity near refractor under SP. 1 and SP. 2 DSRF - assumed propagation step along refractor (in m) SRFMAX - assumed maximum propagation distance along refractor (in m) DI - assumed accuracy of reaching critical angle (in deg) (very small value) DS ray step (in m) DXMAX, DZMAX assumed error of hitting receiver (in m) DALFOP increment of starting ray angle from the source (in deg) DF1 - minimal increment of starting ray angle from the source (very small value) INTL - indicator of interpolation type INTL=0 - cubic velocity interpolation INTL=1 - linear velocity interpolation LIT - maximum number of iterations of conjugate gradient method ERES - value of sum of squared residuals below which we stop calculation ALFOS1, ALFGS1, ALFOS2, ALFGS2 minimum and maximum starting angle from the SP.1 and SP. 2; for SP. 1 (source on the left of the spread)) angles are decreasing from ALFOS1 to ALFGS1; for SP. 2 (source on the right) angles are increasing from ALFOS2 to ALFGS2 LEPS indicator of estimation the observation error LEPS= 0 - we estimate the observation error LEPS=1 - we read the known observation error DLMBD - increment of constant LAMBDA (for LOPT1=0) or assumed value of constant LAMBDA (for LOPT1=1) DXMAXV - assumed value of maximum difference of consecutive velocity solutions below which we stop calculations EPS - assumed observation error SS(I, J) - matrix of convolutional quelling TR(I, J) - recorded traveltimes in seconds Examples of input and output data of #INVERSDH are presented below.

57 Example of the input data of #INVERSDH for the case of 2 shots placed outside of the spread INVm200_2w_1200_1800z100_200a.OUT INVKLINR.ATA INVKLINR.ATB E-0 0.E E INVm200_2w_1200_1800z100_200a.vel E E E E

58 Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread ***HEAD WAVE TOMOGRAPHY - INVERSE TASK*** INPUT DATA NUMBER OF ITERATIONS= 1 CALCULATION PARAMETERS- ICASE=0 IPRINT=0 ITAPE=0 IPRNT=0 IPLOT=1 LOPT=1 LBIEL=0 BL= E+00 LSKAL=0 SK= E+00 ATAMIN= 0.1E-59 PARAMETER OF INVARIABLE VELOCITIES=0 PARAMETER OF CONVOLUTIONAL SMOOTHING= 1 MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= Z1RF= X2RF= Z2RF= HEAD WAVE VELOCITY (in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP1(in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP2(in km/s)= ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 5.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)= ASSUMED ACCURACY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09 FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= WZPS1= 0.00 WXPS2= WZPS2= 0.00 NUMBER OF RECEIVERS = 70 COORDINATE (z) OF RECEIVERS (in m)= 0.00 COORDINATES (x) OF RECEIVERS (in m)= CALCULATION PARAMETERS: RAY STEP DS= 0.50(M) ERROR OF HITTING RECEIVER (in m) - DXMAX DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1= ALFGS1= ALFOS2= ALFGS2= INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.1E+01

59 Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.) DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) PARAMETERS OF CONJUGATE GRADIENT METHOD- MAXIMUM NUMBER OF ITERATIONS- 30 MINIMUM VALUE OF SUM OF SQUARED RESIDUALS E-06 OPTIMIZATION PARAMETERS- ASSUMED VALUE OF CONSTANT LAMBDA= E+02 NUMBER OF UNKNOWS(GRID NODES) MATRIX OF CONVOLUTIONAL SMOOTHING OPERATOR OBSERVED TIMES - TRIO (SP NUM,GEOPH,TIME(MS)) CALCULATION RESULTS REFRACTOR DIP = CRITICAL ANGLE UNDER SP.-1 = CRITICAL ANGLE UNDER SP.-2 = XKR1= ZKR1= AWYJ1= TRAY1= XKR2= ZKR2= AWYJ2= TRAY2= CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP

60 Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.) H1= H2= DEAD ZONE - X < AND X > GEOPH NO- 65 WITH COORD IN DEAD ZONE OF SP NUM 2 WITH COORD GEOPH NO- 66 WITH COORD IN DEAD ZONE OF SP NUM 2 WITH COORD XKR2= ZKR2= AWYJ2= TRAY2= CRITICAL EMERGING ANGLE OF RAY FROM REFRACTOR UNDER SP RMSDT= NUMBER OF ROWS OF SET EQUATION AX=B - 64 NUMBER OF ZERO VALUE COLUMNS OF COEFFICIENT MATRIX = NUMBER OF NONZERO ELEMENTS OF MATRIX OMEGA= GAMMA2= E+09 GAMMA2= E+06 GAMMA2= E+05 GAMMA2= E+04 GAMMA2= E+05 GAMMA2= E+05 GAMMA2= E+04 GAMMA2= E+03 GAMMA2= E+03 GAMMA2= E+03 GAMMA2= E+04 GAMMA2= E+03 GAMMA2= E+03 GAMMA2= E+02 GAMMA2= E+03 GAMMA2= E+03 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+03 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+02 GAMMA2= E+01 NUMBER OF ITERATIONS IN CONIUGATE GRADIENT METHOD- 30 FINAL SUM OF SQUARED RESIDUALS E+01

61 Example of the output data of #INVERSDH for the case of 2 shots placed outside of the spread (c.d.) ESTIMATED VELOCITIES FOR ITERATION NR - 1 DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 484 ROWS NUMBER- 151 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) END OF CALCULATIONS FOR ITERATION NR- 1 NUMBER OF ITERATIONS= 1 The effectiveness of the #INVERSDH program was tested on many models. The results of selected test are presented below. The assumed velocity model is presented in fig To secure relatively good illumination of the object (velocity anomaly) the dimensions of the anomaly are small. The ray trajectories (from # RAYEDH) on the background of the seismogeological model and positions of shots/receivers are presented in fig The results of inversion (from # INVERSDH) are presented in figs

62 Fig The assumed velocity model used for testing #INVERSDH; velocity in the first layer m/s, velocity in the second layer m/s, velocity of anomaly m/s Fig The ray trajectories (from # RAYEDH) on the background of the seismogeological model and positions of 2 shots (black circles) and receivers (circles)

63 Fig The results of inversion (from # INVERSDH) presented as 2D and 3D velocity fields using #SURFER Fig The result of inversion (from # INVERSDH) presented as 2D velocity field using #SURFER

64 II.3. AUXILIARY FORTRAN PROGRAMS II.3.1. The program for conversion ASCII velocity files from ProMAX to binary files and text SURFER files The velocity fields used as input data to program #RAYEDH were edited in ProMAX by means of the process named Interactive Velocity Editor and exported in the Promax ASCII format. It was necessary to do conversion of these ASCII velocity fields to binary files used as input data in #RAYEDH for the purpose of modeling. These binary files were also used as input data (initial models) in #INVERSDH. Additionaly it was necessary to do conversion of these ASCII files to text files used as input data (so called *.dat files) in SURFER program. The original Fortran program was named PROMXOUTR. The input data to # PROMXOUTR program are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA3 READ(1,*) NAZWA4 READ(1,*) NAZWA7 READ(1,*) ILCDP, DXCDP, CDPMAX, DH, HMAX, X0, Z0 READ(1,*) IPRINT where: NAZWA2 - the name of output ASCII file with input velocities from ASCII file of ProMAX at all nodes and output ASCII file (x,z,v) for plotting in #SURFER NAZWA3 - the name of output ASCII file from ProMAX process Interactive Vel Editor NAZWA4 - the name of *.DAT file used as input data for SURFER NAZWA7 - the name of output binary file with full model velocities at all nodes DXCP - interval CDP in meters DH - interval of depth in meters HMAX - maximum depth in meters ILCDP - number of CDP points (column number of velocity matrix) X0, Z0 - coordinates of new coordinate origin IPRINT - index of printing input velocities at all nodes IPRINT.EQ.0 - variant without printing velocity fields IPRINT.NE.0 - variant with printing velocity fields The examples of input and output data of # PROMXOUTR program are presented below. Example of input data of # PROMXOUTR Anom_Dip.out Anomaly_Dip Anom_Dip.vel

65 Example of Promax ASCII velocity file named Anomaly_Dip ProMAX Data Export CDP X Coor Y Coor DEPTH VEL_INT

66 *** DANE WEJSCIOWE *** Example of output data of # PROMXOUTR CDPs number = 981 CDPs interval in meters = Maximum CDP in ProMAX file = Depth interval in meters = 1.0 Maximum depth in meters = Coordinates of new origin: X0= 10.0 Z0= 2.0 INPUT VELOCITIES- DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 981 ROWS NUMBER- 301 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) *** 2D VELOCITY DISTRIBUTION *** COORDINATES OF NEW ORIGIN: X0= 10.0 Z0= 2.0 VELOCITIES IN ASCII FILE - TRIOS (X,Z,V):

67 The examples of velocity field prepared in Interactive Velocity Editor is shown in fig The same field after exporting in ASCII format and processing in # PROMXOUTR is displayed in fig using SURFER program. Fig The window of Interactive Velocity Editor, velocity field prepared to export in ASCII format Fig The velocity field presented in SURFER, input data were prepared in # PROMXOUTR

68 II.3.2. The program for conversion binary velocity files from # INVERSDH to text SURFER files The ouput velocity fields from #INVERSDH have the form of binary files. The Fortran program named #VEL2D has been prepared for conversion binary velocity files to text SURFER files. The input data are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA3 READ(1,*) NAZWA7 READ(1,*) IPRINT, IPRNT where: NAZWA2 - name of output ASCII file (x,z,v) for plotting in #SURFER NAZWA7 - name of input binary file with velocities (results from INVERSHE.EXE) NAZWA3 - name of output ASCII file (x,z,v) for plotting in #SURFER: *.dat file IPRINT - index of printing velocity or velocity corrections in file NAZWA2 IPRINT=1 IPRNT=0 - printing velocities, IPRINT=1 IPRNT=1 - printing corrections, IPRINT= 0 - we save only *.dat file II.3.3. The program for calculation RMSDV To create a possibility of fast evaluation differences beetwen assumed velocity model and velocity field resulting from tomographic inversion the Fortran program was prepared. The program named RMSDVEL calculates the RMS error in accordance of formula: where: RMSE = R, i= 1 ( Vi est Vi,mod ) R 2 1/ 2 V i,est the value of velocity estimated in the i th node, V i,mod the known value of model velocity in the i th node, R the number of nodes in velocity grid

69 The input data are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA7 READ(1,*) NAZWA3 READ(1,*) V2,V1S1,HMAX,LPROMX where: NAZWA2 the name of output text file for printer NAZWA7 the name of input binary file with model velocities (output from PROMAXOUT) NAZWA3 the name of input binary file with tomoinvers velocities (output from INVERSDH) V2 - head wave velocity (in km/s) V1S1 - overburden velocity near refractor under shot point (in km/s) HMAX max depth of calculations in m LPROMX - indicator of input data type LPROMX =1 for data from ProMAX programs LPROMX =0 for data from FORTRAN programs Examples of input and output data from # RMSDVEL are presented below. Examples of input data from # RMSDVEL RSMDVEL.OUT PROMVEL-Velocity-every-one.OUt PROMVEL-Velocity-every-fifth.OUt Examples of output data from # RMSDVEL *** CALCULATION OF RMS VELOCITY ERROR *** MODEL VELOCITY DATA FILE NAME - PROMVEL-Velocity-every-one.OUt TOMOINVERS VELOCITY FILE NAME - PROMVEL-Velocity-every-fifth.OUt INPUT DATA MEDIUM PARAMETERS : HEAD WAVE VELOCITY (in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= MAX DEPTH (in m)= DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 125 ROWS NUMBER- 26 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) THE NUMBER OF GRID NODES THE NUMBER OF GRID NODES USED RMS VELOCITY ERROR (m/s) = 23.

70 II.3.4. The program for conversion ASCII files with picks from ProMAX to FORTRAN text format One of the essential problems of tomographic inversion is defining the traveltimes in the process of picking. The comparison between picks obtained from model data generated using Promax process named Finite Difference Modeling and traveltimes obtained using ray approach (FORTRAN program #RAYEDH) needed exporting ProMAX picks and conversion ProMAX ASCII files to FORTRAN text format. The FORTRAN program named #PICKSOUT has been prepared. The input data of the # PICKSOUT program are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA3 READ(1,*) NAZWA4 READ(1,*) LPS,LCHAN where: NAZWA2 the name of output temporary ASCII file with times NAZWA3 the name of input ASCII file (result of picking) from ProMAX NAZWA4 the name of output ASCII file with times of picks LPS the number of shots LCHAN the number of channels The example of input and output data of #PICKSOUT are presented below. The example of input data of #PICKSOUT PICKSOUT.OUT Picks PICKSOUT2.OUT 1 102

71 The example of output ASCII data with picks from ProMAX ASCII database file write for Area="kasina", Line="tomography2005" Value list for Database Order TRC Test_05_100_1_sh1 >TRCGEOMETRY TRC < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < < *END

72 The example of output data of # PICKSOUT CALCULATED TRAVELTIMES SHOT GEOPHONE TIME(S)

73 II.3.5. Program for 2D (spatial) smoothing of velocity fields The smoothing of velocity fields resulting from tomographic inversion is commonly used procedure after each tomographic inversion and before application the velocity fields to statics calculations. The FORTRAN program named #VELAVE has been prepared to smooth velocity fields from binary files using 2D spatial rectangular operator. If we introduce the following notation for velocities in the nodes: v(x,z) = V(i x, j z) = V i,j i = 0, 1, 2,..., I; j = 0, 1, 2,..., J then for radius of velocity smoothing ( LAVEX, LAVEZ) equal 2 in both directions we use the following formula to calculated the velocity V s i,j after smoothing: V s i, j V = i, j + V i 1, j + Vi 2, j + Vi+ 1, j + Vi+ 2, j + Vi, j 1 + Vi, j 2 + Vi, j V i, j+ 2 The input data of #VELAVE are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) NAZWA7 READ(1,*) NAZWA3 READ(1,*) IPRINT, LPROMAX READ(1,*) LAVEX, LAVEZ where: NAZWA2 the name of output ASCII file (x,z,v) (with averaged velocities) NAZWA3 the name of output binary file with averaged velocities NAZWA7 the name of input binary file with velocities IPRINT - index of printing averaged velocity in file NAZWA2 IPRINT=1 - printing averaged velocities IPRINT=0 - no printing averaged velocities LPROMAX=0 velocities (in m/s) from PROMAX =1 velocities (in km/s) from FORTRAN LAVEX - radius of velocity smoothing in X direction defined in DX (increment of X variable in m), LAVEX=2 means radius = 2*DX LAVEZ - radius of velocity smoothing in Z direction The example of input and output data of #VELAVE are presented below. The example of input data of #VELAVE Velave.out INVanomaly.vel INVanom_ave2_2.vel

74 The example of output data of #VELAVE Radius of velocity smoothing in X direction= 2 Radius of velocity smoothing in Z direction= 2 INPUT VELOCITIES (in km/s)- DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) AVERAGED OUTPUT VELOCITY- DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) II.3.6. Program for statics calculations The estimation of static corrections based on the near surface velocity fields resulting from tomographic inversion was one of the main goal of presented dissertation. The

75 FORTRAN program was prepared for statics calculations using as input data binary velocity files obtained during inversion as well as the velocity fields of considered models. The statics is calculated assuming vertical propagation of the rays from the shot/receiver to the datum identified with refractor. sequence: The input data of #STATICS are red - for the variant of two shots - in the following READ(1,*) NAZWA2 READ(1,*) IPRINT,NRMOD,IPLOT,ITP,IPRNT READ(1,*) ITAPE IF(ITAPE.NE.0) READ(1,*) NAME7 READ(1,*) LMOD,LPROMAX IF(ITAPE.NE.0) READ(7) LW,LT,DX,DZ,LWT,(V(I),I=1,LWT) IF(ITAPE.EQ.0) READ(1,*) LW,LT,DX,DZ READ(1,*) DS,DXMAX,DZMAX,DALFOP,DF1,INTL READ(1,*) WXPS1,WZPS1,WXPS2,WZPS2 READ(1,*) LPG,WZG,WXG(1),DXG READ(1,*) X1RF,Z1RF,X2RF,Z2RF,V2,V1S1,V1S2,DSRF,SRFMAX,DI READ(1,*) ALFOS1,ALFGS1,ALFOS2,ALFGS2 IF(ITAPE.NE.0)READ(1,*) NRR,(NR1(I),NR2(I),VN(I),I=1,NRR) where the only new parameters with comparison with input data of #RAYEDH are: LMOD indicator of the type of input velocity field LMOD=0 - we read output velocity field from #INVERSDH LMOD=1 - we read input velocity model from Promax LPROMAX indicator of the source of the velocity field from inversion LPROMAX = 0 - we read data from FORTRAN program LPROMAX = 1 - we read data from PROMAX ALFOS1=ALFGS1=ALFOS2=ALFGS2=90. The examples of input and outpu data of #STATICS are presented below. Example of input data of #STATICS statanom2_model.out PROMVEL.OUT E E

76 Example of output data from #STATICS *** STATICS FOR THE ASSUMED VELOCITY MODEL *** INPUT DATA MEDIUM PARAMETERS : COORDINATES OF TWO REFRACTOR POINTS (in m) - PAIRS(X,Z): X1RF= Z1RF= X2RF= Z2RF= HEAD WAVE VELOCITY (in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-1(in km/s)= OVERBURDEN VELOCITY NEAR REFRACTOR UNDER SP.-2(in km/s)= ASSUMED PROPAGATION STEP ALONG REFRACTOR (in m)= 1.00 ASSUMED MAX PROPAGATION DISTANCE ALONG REFRACTOR (in m)= ASSUMED ACCURANCY OF REACHING CRITICAL ANGLE (in deg)= 0.1E-09 FIELD PARAMETERS: COORDINATES(x,z) of SP (in m)- WXPS1= WZPS1= 4.00 WXPS2= WZPS2= 4.00 NUMBER OF RECEIVERS = 70 COORDINATES (z) OF RECEIVERS (in m)= 4.00 COORDINATES (x) OF RECEIVERS (in m))= CALCULATION PARAMETERS: MODEL NR= 1 IPLOT= 0 ITP= 0 RAY STEP DS= 0.00(m) ERROR OF HITTING RECEIVER (in m) - DXMAX DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION RANGES OF OUTPUT RAY ANGLES FROM THE SOURCE (in deg)- ALFOS1 and ALFGS1 for right part of the spread ALFOS2 and ALFGS2 for the left part of the spread ALFOS1= ALFGS1= ALFOS2= ALFGS2= DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 213 ROWS NUMBER- 51 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m)

77 Example of output data from #STATICS (c.d.) CALCULATION RESULTS REFRACTOR DIP = CALCULATED STATICS FOR ASSUMED VELOCITY MODEL SHOT GEOPHONE STATION(M) STAT_ALL(S) CALCULATED RECEIVER STATICS FOR ASSUMED VELOCITY MODEL SHOT GEOPHONE STATION(M) STAT_RCV(S) CALCULATED SHOT STATICS FOR ASSUMED VELOCITY MODEL SHOT TIME(S) II.3.7. The program for Turning Ray Tracing In the Promax system we have the possibility to calculate the traveltimes of turning waves and ray trajectories using process Turning Ray Tracing. Unfortunately we have no any

78 tools for displaying or exporting ray trajectories. Therefore the new original FORTRAN program has been constructed for turning ray tracing (#RAYEDT) and another for ray trajectories displaying (#RAYPLTT). The ray tracing in gradient medium is realized using the same algorithm as in RAYEDH program. Two variants of calculations are possible. In the first case we can calculate the ray trajectories for assumed range of starting angles from the shot (fig. 2.34). In the second case we calculate the ray trajectories from the source to the selected receivers (fig. 2.35). The velocity field is prepared in the ProMAX system using process Interactive Velocity Editor. Fig The ray trajectories for assumed range of starting angles from the shot

79 Fig The ray trajectories of turning wave for assumed positions of receivers The input data of #RAYEDT are red in the following sequence: READ(1,*) NAZWA2 READ(1,*) IPRINT, NRMOD, IPLOT, ITP, IPRNT READ(1,*) ITAPE IF(ITAPE.NE.0) READ(1,*) NAME7 IF(IPLOT.NE.0) READ(1,*) NAZWA3 IF(ITAPE.EQ.0) READ(1,*) LW, LT, DX, DZ READ(1,*) DS, DXMAX, DZMAX, DALFO, DF1, INTL READ(1,*) LPS, (WXPS(I),I=1, LPS), (WZPS(I), I=1, LPS) READ(1,*) LPG, WZG,WXG(1,1), DPS, DGEOF READ(1,*) ALFO, ALFG IF(ITAPE.eq.0) READ(1,*) NRR, (NR1(I), NR2(I),VN(I), I=1, NRR) where: NAZWA2 the name of output text file for printing the results of calculations NAZWA3 the name of output binary file used for plotting rays NAZWA7 the name of input binary file with velocities IPRINT - indicator of printouts: IPRINT=0 - variant without control printouts IPRINT=1 - variant with control printouts NRMOD - model number IPLOT - indicator of saving data for plotting

80 IPLOT=0 variant without saving data for plotting IPLOT=1 variant with saving data for plotting ITP - indicator of calculation option ITP=0 calculation for assumed receivers ITP=1 calculation for assumed values of "s" parameter (DSRF) along refractor IPRNT - indicator of additional control printouts (ray points) ITP=1 and IPRNT=1 - all control printouts without ray points ITP=1 and IPRNT=2 - all control printouts with ray points ITP=0 and IPRNT=1 - limited control printouts without ray points ITAPE - indicator of input velocity data ITAPE=0 - reading from text file ITAPE=1 - reading from binary file LW - columns number of velocity matrix LT - rows number of velocity matrix DX - the dimension of the cell in x direction (in meters) DZ - the dimension of the cell in z direction (in meters) LWT number of grid nodes DS ray step (in m) DXMAX, DZMAX assumed error of hitting receiver (in m) DALFOP increment of starting ray angle from the source (in deg) DF1 - minimal increment of starting ray angle from the source (very small value) INTL - indicator of interpolation type INTL=0 - cubic velocity interpolation INTL=1 - linear velocity interpolation LPS the number of shots WXPS(I), WZPS(I) coordinates (x,z) of I-h shot LPG - number of receivers WZG - coordinates z of receivers (in m) WXG(I,J) - coordinates x of receivers (in m) DPS shot interval (in m) DGEOF receiver interval (in m) ALFO, ALFG - maximum and minimum starting angle from the shot (ALFO > ALFG, ALFO is drceasing, 180>ALGO,ALFG>0) NRR - number of pairs (NR1(I),NR2(I))

81 VN(I) - input velocities red from text file in km/s and defined in grid nodes NR1(I), NR2(I) - pair of grid nodes between which velocity is equal VN(I) Examples of input and output data of #RAYEDT program are presented below. Example of input data of #RAYEDT Anomaly_gradr_ps3.OUT m-grad.vel Anomaly_gradr_ps3.PLT E

82 Example of output data of #RAYEDT ***TURNING WAVES -DIRECT TASK (KINEMATICS)*** INPUT DATA SP NUMBER COOR. XPS SP NUMBER COOR. ZPS COORDINATES (z) OF RECEIVERS (in m)= 0.00 NUMBER OF RECEIVERS = 24 COORDINATE WXG(1,1) = SHOT INTERVAL = RECEIVER INTERVAL = SHOT NUMBER= 1 COORDINATES (x) OF RECEIVERS (in m))= MODEL NR= 1 IPLOT= 1 ITP= 0 RAY STEP DS= 0.50(m) ERROR OF HITTING RECEIVER (in m) - DXMAX DZMAX= 0.05 LINEAR VELOCITY INTERPOLATION ALFO= ALFG= INCREMENT OF SOURCE OUTPUT ANGLE(in deg) - DALFOP= 0.5E+00 DIMENSIONS OF COORDINATE GRID - COLUMNS NUMBER- 125 ROWS NUMBER- 26 INCREMENT OF X VARIABLE (in m) INCREMENT OF Z VARIABLE (in m) THE NUMBER OF UNKNOWS (GRID NODES)- 3250

83 Example of output data of #RAYEDT (c.d.) NO SOLUTION FOR RECEIVER NO. 10 CALCULATED TRAVELTIMES SHOT GEOPHONE TIME(S)

84 II.4. PROGRAMS FOR GRAPHICAL PRESENTATION Several programs were used for graphical presentation the input data (seismogeological models) and output data (velocity fields from inversion, ray trajectories, hodographs, statics). In the ProMAX system the following tools were applied for graphical presentation: - Interactive Velocity Editor for creating velocity models and displaying the resulting velocity fields from Turning Ray Tomography, - Velocity Viewer/Point Editor for displaying the resulting velocity fields from Turning Ray Tomography and differences between velocity fields, - DATABASE for displaying results of picking, traveltimes from Turning Ray Tracing. The velocity fields resulting from inversion realized by means of #INVERSDH were presented using SURFER program. The hodographs and statics distributions were presented using GRAPHER. The ray trajectories of head waves resulting from # RAYEDH were presented using original FORTRAN program named #RAYPLTH. That program was constructed using Lahey/Fujitsu Fortran 95 (LF95) with application the tool Winteracter Starter Kit (WiSK) and procedures of High Resolution Graphics. The input data to the #RAYPLTH were red using the following sequence: READ(2,*) NAZWA3 READ(2,*) TEXT READ(2,*) LW,LT,DX,DZ,ITP READ(2,*) ILMOD DO 99 I=1,ILMOD READ(2,*) LMOD(I),(BX(K,I),BZ(K,I),K=1,LMOD(I)) 99 CONTINUE READ(2,*) DS,DXMAX,DZMAX,DSRF,DF1,INTL READ(2,*) LPS,(WXPS(I),I=1,LPS),(WZPS(I),I=1,LPS) READ(2,*) LPG, WZGW,WXG(1),DGEOF READ(2,*) SRFMAX,SRF1,SRF2,NRDEL READ(2,*) ILNAP,ITPNAP,LTEXT,LSTEP DO 1 I=1,ILNAP READ(2,*) XNAP(I),ZNAP(I) 1 READ(2,*) NAPIS(I) READ(2,*) ILPS,(NRPS(I),I=1,ILPS) READ(2,*)LPAUSE READ(2,*) XMAX,ZMAX,XPOCZ,ZPOCZ,DELX,DELZ READ(2,*) RADIUS READ(2,*) XSIZE,ZSIZE

85 READ(2,*)xleft,zlower,xright,zupper READ(2,*) xl,zlow,xr,zup where: NAZWA3 the name of binary file with resulted ray trajectories from direct task (RAYEDH) TEXT - the name of model LW - column number of velocity field LX - row number of velocity field DX - interval of variable "x" DZ - interval of variable "z", ITP - indicator of calculation option: ITP=0 - drawing rays for assumed receivers ITP=1 - drawing rays for assumed values of "s" parameter along refractor ILMOD - the number of subareas of the model (max 10) LMOD(I) - number of points defining I-th area BX(K,I), BZ(K,I) - coordinates of points defining the subarea I-th DS - interval of ray tracing (in meters) above refractor DXMAX, DZMAX - precision of hitting the receiver (in meters) DSRF - interval of ray along refractor DF1 - minimal increment of source output angle during defining critical angle INTL - parameter of interpolation option: INTL=0 - cubic interpolation, INTL=1 - linear interpolation LPS - shot points number (WXPS(I),WZPS(I)) - coordinates of shot points LPG - receiver points number, WZGW - "z" coordinate of receivers (one value), WXG(I) - "x" coordinates of receivers DGEOF - receiver interval SRF1,SRF2 - the range of values (min and max) of parameter "s"(actual length of trajectory along the refractor for which we are drawing trajectories of rays) SRFMAX maximum value of s parameter ILNAP - number of descriptions ITPNAP - parameter with value 0 LSTEP - interval of plotting ray points (=1 - each point)

86 (XNAP(I),ZNAP(I)) - coordinates of I-th description NAPIS(I) - the text of I-th description ILPS - number of shots for graphical presentation NRPS(I) - identification number of I-th shot (I=1,ILPS) LPAUSE = 1 - drawing ray by ray with pause (= 0 without pause) XMAX, ZMAX - maximum coordinates of drawing (in meters); XPOCZ,ZPOCZ - initial coordinates (in meters) DELX, DELZ - increment of "x" and "z" (in meters) RADIUS - radius of circle used for graphical presentation of shots XSIZE,ZSIZE - parameters defining window dimensions (in the range ) for values 1.0 we have the largest available window xleft,zlower,xright,zupper - range of values (number of pixels) in the window xl,zlow,xr,zup window parameters: graphics on full window: xl=0.0, zlow=0.0, xr=1.0, zup=1.0 graphics on one quarter (top right)of window:xl=0.5, zlow=0.5, xr=1.0, up=1.0, for (0.0,0.0,1.0,1.0) we have full graphics window Example of input data of # RAYPLTH KLIN101r.PLT KLIN101r E V=1.2 km/s V=0.8 km/s V=2.5 km/s FI=0 deg

87 The example of ray trajectories against a background of assumed seismogeological model and positions of sources/receivers is presented in fig Fig The example of ray trajectories against a background of assumed seismogeological model and positions of sources/receivers

88 III. RESULTS OF THE SOLUTION OF DIRECT PROBLEM OF THE REFRACTION TOMOGRAPHY Most of the tomographic inversions are based on the ray approach to wave propagation. But the wave pattern on the records is the realization of wave process. So it is very important to ascertain that for considered models of medium and considered parameters of acquisition we can use ray approach to predict the proper behaving of the waves and to estimate the correct traveltimes. In the case of low velocity layer and small distances between source, receiver and heterogeneities we can observe - using ray approach - strong deviation of the rays, complicated ray trajectories, the shadow zones. All these features may not occur in real wave process of propagation. One of the most commonly used tool of investigation wave process of propagation is finite difference modeling. Using this tool we shall ascertain that for discussed models we can recognize the effects of near surface heterogeneities on first break traveltimes and that these effects may be properly estimated using ray approach. III.1. THE RAY APPROACH AND THE WAVE APPROACH TO SEISMIC WAVE PROPAGATION AND THEIR ROLE IN TOMOGRAPHIC INVERSION In the traveltime tomography we reconstruct the velocity fields using the differences beetwen real traveltimes picked from the records and modeled traveltimes estimated usually using ray approach. If the traveltimes are picked with errors or the modeled traveltimes are not proper we obtain serious errors of tomographic inversion. One of the most important question is what is the smallest feature that can be reconstructed by traveltime tomography. This smallest feature is dependent on traveltime uncertainties (frequency bandwidth of the signal) and medium velocity. Williamson (1991) stated that the size of the smallest feature which can be accurately reconstructed using traveltime tomography has a dimension that is about equal to the first Fresnel zone radius. He also suggested that detail finer than this minimum size will not necessarily disappear but may be significantly defocused. The first Fresnel zone radius is given by (L where L is the propagation distance and is the wavelength. Using 850 Hz as the high frequency limit and 4 km/s as the medium velocity, this Fresnel zone criterion suggests that the smallest resolvable feature at the center of the tomograms (L = 100 m) in the reservoir interval is approximately 20 m. Choosing a grid size less than 20 m may cause the inverse problem to be an underdetermined system.

89 Krajewski et al. (1989) suggested that a low velocity anomaly can be recognizable in a tomogram if it causes a traveltime delay along an affected raypath that is larger than the traveltime uncertainty. If we take 0.5 ms for the picking error, 4 km/s for the medium velocity, and assume a 15% velocity decrease as a result of the temperature change, then the minimum resolvable body size would be 11 m. So we can use a 9 x 9 m grid for the tomographic inversion. Additional condition was proposed by Lee et al. (1995): in the final velocity images, the velocity at cells not covered by more than five rays should be reset to the background velocity (Lee et al. 1995). Therefore in the process of constructing models we can t introduce too small features of the velocity fields. In the process of inversion we can t use too small dimensions of the cells to avoid underdetermination of the set of tomographic equation. If we want establish relation between ray approach and wave approach for considered seismogeological models we must take into account seismic signal. Using ray approach we assume so called high frequency approximation of wave propagation. It means that instead of signal with limited frequency band we assume propagation of spike. In real data we have signals with limited frequency band and dominating pick frequency. The problem arises what kind of the signal we should to use and which phase of this signal we must pick on field records (simulated in our analysis using finite difference modeling) to obtain good relation between model (theoretical) traveltimes and observed traveltimes. To answer this question the analysis has been undertaken with application of signals of different phase characteristics. Using these signals as input to Finite Difference Modeling (FDM) procedure the records were calculated for the considered models and the first break arrivals were picked. The resulting traveltimes of head waves and turning waves were compared with theoretically calculated traveltimes using #RAYEDH and Turning Ray Tracing procedure. To estimate the parameters of the first break signals before modeling the land seismic records were introduced as input data to Interactive Spectral Analysis procedure of ProMAX system. Two type of land seismic records were used: dynamite records and vibroseis records. In fig. 3.1 typical dynamite record (after AGC application) is presented and in fig. 3.2 typical vibroseis record from the same area (after application AGC and frequency filter ) is presented.

90 Fig Typical dynamite record (after AGC application) Fig Typical vibroseis record (after application AGC and frequency filter ) The results of interactive spectral analysis of first breaks of the records from fig. 3.1 and 3.2 are presented in fig. 3.3 and 3.4.

91 Fig The results of interactive spectral analysis of first breaks of the records from fig. 3.1 (the blue window defines the area of spectral analysis, the amplitude spectrum is displayed on the right) Fig The results of interactive spectral analysis of first breaks of the records from fig. 3.2 (the blue window defines the area of spectral analysis, the amplitude spectrum is displayed on the right) On the base of the results of interactive spectral analysis of many dynamite and vibroseis records it was established the representative signal of first breaks as the minimumphase and zero-phase Ricker signal with dominating frequency 40 Hz. To model theoretical records the model of near surface layer with low velocity anomaly presented in fig. 3.5 was chosen. The parametrization window of FDM procedure with used parameters is presented in fig 3.6. The selected modeled records are presented in figs

92 Fig The model of near surface layer with low velocity anomaly applied in FDM Fig The parametrization window of FDM procedure with applied parameters

93 Fig The selected modeled record for the case of minimum phase signal Fig The enlargement of selected modeled record for the case of minimum phase signal with result of picking phase maksimum (snap peak), for offsets m the effect of low velocity anomaly is visible (longer times of first breaks of head wave)

94 Fig The enlargement of selected modeled record for the case of minimum phase signal with result of picking phase zero (snap zero + to ) In fig the comparison is presented of the picked traveltimes for the case of zerophase signal (snap peak) and minimum phase signal (pick zero) with predicted traveltimes using ray approach (#RAYEDH). Fig Comparison of the picked traveltimes for the case of zero-phase signal (snap peak) and minimum phase signal (pick zero) with predicted traveltimes using ray approach (#RAYEDH).

95 The analysis of the results confirmed that the traveltimes predicted using ray approach are close to the traveltimes obtained in the process of first break picking of records with minimum phase signal and zero phase selected for picking. It was confirmed too that the behaving of the zero-phase records are very similar to minimum-phase records in the zone of first breaks. III.2. IMAGING OF NEAR SURFACE VELOCITY HETEROGENEITIES OF THE MEDIUM IN WAVE PATTERN OF ACOUSTIC MODELING For the purpose of the analysis of first and later breaks of the head waves on reflection records the 2-D finite difference acoustic modeling has been applied (Kasina, Nasar 2005). The main goal of this modeling was to define the effect of heterogeneities dimensions, velocity distributions in the near-surface layer as well as the seismic signal parameters on the wave pattern of head waves breaks and on head waves traveltimes. The analysis has been undertaken to assess the possibility of applications these breaks for recovering near-surface velocity distributions by means of head wave tomography. VELOCITY MODELS AND MODELING PARAMETERS The velocity model of near-surface medium was composed of three horizontal layers with parameters described in fig In the second layer the low velocity body was inserted of effective width 20 m. The velocity of this body named in the analysis velocity anomaly was changing during consecutive computation from 600 m/s through 800 m/s to final value of 1000 m/s. The thickness of the body was changing from 50 m through 20 m to 10 m. The velocity model (fig. 3.11) has been defined for 4001 CDP points with CDP interval equal 1 m. The off-end spread has been used with 400 receiver stations (channels) and receiver interval 10 m. The shot has been located at the left edge of the velocity model on the surface. The zero-phase Ricker signal has been used as point source wavelet with dominant frequency 50 Hz, 70 and 100 Hz. The input computation grid had dimensions 1 m x 1 m, and has been automatically modified during computation to effective grid dimensions protecting against numerical errors.

96 Fig The velocity model of near-surface layer used in synthetic records calculation THE WAVE PATTERN OF SYNTHETIC RECORDS The case of the near-surface medium without velocity anomaly The synthetic record calculated for the layered model from fig but without velocity anomaly using zero-phase Ricker source signal with dominant frequency 50 Hz is presented fig for the case of split spread. Additionally the results of approximate interactive apparent velocity analysis have been presented in this figure for the purpose of wave identification. Analysis of the wave pattern of these figures lets us to draw the following conclusions: - the waves which can be identified without any difficulties in first breaks comprise the direct wave and the head wave connected with the deepest refractor with velocity 2500m/s, - the breaks of head waves are well distinguished in first and later breaks, - the head wave connected with the deepest refractor with velocity 2500m/s is dominated in first breaks and can be identified in broad range of offsets, - it is possible to realize the picking of head waves not only in first breaks but in later breaks too, - the differences in apparent velocity are as usual very useful in head waves identification.

97 The case of the near-surface medium with velocity anomaly In the next step the calculation of the records have been accomplished for the case of near-surface medium with velocity anomaly in the second layer. The results of the calculations for the selected thickness of anomaly (50 m) and for the selected velocity of anomaly (600 m/s) are presented in fig. 3.13, 3.14, It can be seen that the traveltime delay related to low velocity anomaly is clearly marked in the first breaks of the head wave propagating from the deepest refractor and in the breaks of head wave from the second boundary. Fig The synthetic record calculated for the case of the split spread and the velocity model without anomaly with results of approximate interactive apparent velocity analysis

98 Fig The synthetic record calculated for the layered model with velocity anomaly (thickness 20 m, velocity 800 m/s) using zero-phase Ricker source signal with dominant frequency 50 Hz; the zone of breaks in the area of imaging traveltime delays related to velocity anomaly is located in the range of offsets m Fig Enlargement of first breaks zone of the synthetic record from fig. 4 in the area of imaging traveltime delays related to velocity anomaly (thickness 20 m)

99 Fig Enlargement of later breaks zone of the synthetic record from Figure 3.14 in the area of imaging traveltime delays related to velocity anomaly (thickness 20 m) The relation of the traveltimes after changing velocity of anomaly to 800 m/s and 1000 m/s can be defined from the differences of first breaks traveltimes presented in fig These differences achieve the values from 6 ms to 16 ms ms in the range of anomaly offsets and their graph is nearly the same for pick frequency 50 and 70 Hz. Fig Traveltimes differences between first breaks of head wave for the model without and with anomaly for different values of anomaly velocity: a) V = 600 m/s, b) V = 800 m/s, c) V = 1000 m/s; dominant frequency 50 Hz, anomaly thickness 20 m The next model parameter taken into consideration was the thickness of velocity anomaly. It was changed from 50 m through 20 m to 10 m for the velocity 600 m/s. The

100 resulting difference curves of first breaks traveltimes have been presented in fig These differences achieve the values from 5 ms to about 18 ms in the range of anomaly offsets. Fig Traveltimes differences between first breaks of head wave for the model without and with anomaly for different values of anomaly thickness: a) h = 50 m, b) h = 20 m, c) h = 10 m; pick frequency 50 Hz, anomaly velocity 800 m/s In the last step the effect of pick frequency of source signal on the first breaks traveltimes has been analysed. Three pick frequency have been used during modeling: 50, 70 and 100 Hz. The difference curves of first breaks traveltimes are presented in fig We can observe some dependence of difference curve values on the offset. The average value of difference curve achieves the greatest value about 7 ms for the case of pick frequencies 50 and 100 Hz. For the frequency pairs (50 Hz, 70 Hz) and (70 Hz, 100 Hz) the average value is about 3,5 ms. Fig Traveltimes differences between first breaks of head wave for the model with anomaly for different values of pick frequency of source signal: a) t = t(50 Hz) t(100 Hz), b) t = t(50 Hz) t(70 Hz), c) t = t(70 Hz) t(100 Hz), anomaly velocity 800 m/s,anomaly thickness 20 m

101 The results of undertaken model calculations let to draw the conclusions important from the point of view of application seismic tomography for imaging near-surface velocity heterogeneities: - the breaks of head waves are well distinguished for the discussed near-surface layered medium in first and later breaks, - the main factors affecting the first breaks traveltimes in the model with velocity anomaly are the anomaly thickness and anomaly velocity; the change of traveltimes achieves values from about 6 to 18 ms for the discussed models, - analysis of first breaks traveltimes obtained for different pick frequencies of source signal confirms that the effect of variation of pick frequencies from 50 to 90 Hz may achieve several milliseconds.

102 III.3. THE SOLUTION OF DIRECT AND INVERSE TASK OF THE REFRACTION TOMOGRAPHY FOR SELECTED MODELS AND STATICS ESTIMATION In this chapter several models of near surface layer have been used to solve the direct task of head wave tomography: the model of gradient layer with velocity anomaly over halfspace with constant velocity, the model of gradient half-space without any refractor, the model of constant velocity layer with anomaly over half-space with constant velocity, the model of two constant velocity layers (with velocity anomaly in the first or in the second layer) over constant velocity half-space, the model consisting of three constant velocity layers (with velocity anomaly in the second layer) over constant velocity half-space. For each model with a refractor the traveltimes and ray trajectories of head waves were defined using Fortran programs (#RAYEDH, #RAYEDT, #RAYPLT). For most of the models the velocity grid consisted of the cells with dimensions: 25 m in x direction and 8 m in z direction (125 columns and 26 rows). For multi-layer models the cells had dimension 1 m in vertical dimension. The points of rays were defined with the interval 0.5 m. The distance beetwen receivers had the value 50 m typical for land seismics. The calculations were realized for two shots placed outside of the spread (one shot on the left and one on the right) or for many shots for the variant of seismic profiling with split-spreads (48 receivers, shot interval 100 m, 10 shots). For the model of gradient half-space the traveltimes were calculated using Turning Ray Tracing process in ProMAX system. Additionally the wave pattern of the records was estimated using Finite Difference Modeling process in ProMAX system. The traveltimes of head waves estimated for the models with the velocity anomaly were used as input data to tomographic inversion realized with the help of #INVERSDH. The starting models were created from the models used in #RAYEDH through the elimination of the velocity anomalies. Generally it means that the starting models have been estimated using additional information from shot holes or we have realized standard refraction interpretation of first breaks (e.g. using commonly available Hampson-Russel program named GLI2D or GLI3D). It must be stressed that the main goal of presented analysis is the estimation of tomographic effectiveness in reconstruction of the velocity anomalies included beetwen refractors and the application of reconstructed velocity fields to static corrections estimation. Additionally, the Turning Ray Tomography process in ProMAX system was used for the case of the model of gradient half-space.

103 In the process of inversion by means of #INVERSDH the following values of main parameters were used selected during many tests: - the velocity grid consisted of the cells with dimenions: 25 m in x direction and 8 m in z direction (125 columns and 26 rows), - the number of iterations in gradient conjugate method: 30, - operator of convolutional quelling q=1.0/0.5, - the value of constraint coefficient λ=10, - the dimensions of spatial smoothing operator of resulting velocity fields: 2 2, - the number of inversion iterations: 3. For the purpose of quantitative evaluation of the inversion effectiveness the following errors were calculated: RMSE = R, i= 1 ( Vi est Vi,mod ) R 2 1/ 2 RMSDT = N, i= 1 ( t i est t i,mod ) N 2 1/ 2 where: V i,est the value of velocity estimated in the i th node, V i,mod the known value of model velocity in the i th node, R the number of nodes in the velocity grid, t i,est the traveltime for i th ray calulated for the velocity field estimated in the inversion process, t i,mod the observed (known from seismic modeling) traveltime for i-th ray, N the number of considered seismic rays The first error RMSE was estimated using #RMSDV Fortran program, the second error RMSDT was estimated in the #INVERDH Fortran program. Additionally the bitmaps of resulting velocity fields will be presented (using SURFER program) as well as the bitmaps of differences between assumed model velocity fields and resulting from inversion. Before tomographic inversion the refractor in starting models is removed and the layer below is replaced by the medium with velocity equal to the velocity a little above the refractor. This was done to avoid the effect of layer below refractor on behaving rays a little

104 above the refractor. As a result all the velocity fields resulting from inversion do not include refractor and do not include the layer below the refractor. At the end for each velocity field resulting from tomographic inversion the static corrections will be estimated using the #STATICS Fortran program and diplayed using GRAPHER program. The statics corrections (statics) are calculated assuming datum at the position of the refractor. III.3.1. The two layer model with gradient medium over refractor The first analysed model consists of gradient medium over half-space with constant velocity (fig. 3.19). The depth of the refractor is 176 m, its velocity 3500 m/s. In the first layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12 CDP s (300 m), velocity 800 m/s. The velocity in the first layer is incresing from 960 m/s near surface to 1670 m/s near refractor. This starting model was modified through changing anomaly velocity from 800 m/s to 600 m/s and 400 m/s for the case of low velocity anomaly. Higher anomaly velocities (1500 m/s and 1800 m/s) were considered too. Additionaly the anomaly depth was varied from 48 m to 80 m. The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 8 m in the vertical direction. Fig Two layer velocity model with velocity anomaly and gradient medium over constant velocity half-space, refractor depth 176 m The parameters used in #RAYEDH for calulations of traveltimes and ray trajectories had the following values: positions of shots 1000 and 2000 m, receiver interval 50m, spread of 24 geophones. The example of ray trajectories for velocity anomaly 600 m/s are presented in fig

105 Fig The ray trajectories for the model with velocity anomaly 600 m/s, 2 shot points, refractor depth 176 m; horizontal axis distance in m, vertical axis depth in m Behaving of the traveltime graphs for the considered 3 variants of velocity anomaly (400, 600 and 800 m/s) is presented in fig Fig The traveltime graphs for the considered 3 variants of velocity anomaly (400, 600 and 800 m/s), refractor depth 176 m

106 In the next figures the results of tomographic inversions are presented for anomaly velocity 400 m/s, refractor depth 176 m and for different iteration number. The position of assumed anomaly is marked as white rectangle. For the starting velocity model the traveltime error RMSDT was equal ms. Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV= 122 m/s, RMSDT= ms) Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV= 122 m/s, RMSDT= ms)

107 Fig The result of inversion in 1-st iteration after smoothing in the case of anomaly velocity 400 m/s (RMSDV=124 m/s) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 400 m/s (RMSDV=124 m/s, RMSDT=1.19 ms)

108 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st iteration) in the case of anomaly velocity 400 m/s (RMSDV= 119 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-st) in the case of anomaly velocity 400 m/s (RMSDV=117 m/s, RMSDT=0.33 ms) The results of statics calculations for different iterations in the case of anomaly velocity 400 m/s are presented below.

109 Fig The results of statics calculations for different iterations in the case of anomaly velocity 400 m/s, refractor depth 176 m In the next figures the results of tomographic inversions are presented for anomaly velocity 600 m/s, refractor depth 176 m and for different iteration number. Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV= 80 m/s, RMSDT= 2.85 ms)

110 Fig The differences between the true model and the reconstructed velocity Field in 1-st iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV= 80 m/s, RMSDT= 2.85 ms) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s, RMSDT=0.33 ms)

111 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-nd iteration) in the case of anomaly velocity 600 m/s (RMSDV=79 m/s, RMSDT=0.03 ms)

112 Fig The results of statics calculations for different iterations in the case of anomaly velocity 600 m/s, refractor depth 176 m In the next figures the results of tomographic inversions are presented for anomaly velocity 800 m/s, refractor depth 176 m and for different iteration number. Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 0.88 ms)

113 Fig The differences between the true model and the reconstructed velocity field in the 1-st iteration (without smoothing) inr the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 0.88ms) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV=53 m/s, RMSDT=0.08 ms)

114 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 800 m/s (RMSDV=52 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-ndt) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 0.01 ms)

115 Fig The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 176 m In the next figures the results of tomographic inversions are presented for anomaly velocity 1500 m/s, refractor depth 176 m and for different iteration number. Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV= 34 m/s, RMSDT= 0.53 ms)

116 Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV= 34 m/s, RMSDT= 0.53 ms) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 1500 m/s (RMSDV=36 m/s, RMSDT=0.06 ms)

117 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 1500 m/s (RMSDV=36 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-st) for the case of anomaly velocity 1500 m/s (RMSDV= 36 m/s, RMSDT= 0.01 ms)

118 Fig The results of statics calculations for different iterations in the case of anomaly velocity 1500 m/s, refractor depth 176 m In the next figures the results of tomographic inversions are presented for anomaly velocity 1800 m/s, refractor depth 176 m and for different iteration number. Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 76 m/s, RMSDT= 1.13 ms)

119 Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 76 m/s, RMSDT= 1.13 ms) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 1800 m/s (RMSDV= 74 m/s, RMSDT= 0.13 ms)

120 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 1800 m/s (RMSDV= 75 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-st) in the case of anomaly velocity 1800 m/s (RMSDV= 74 m/s, RMSDT= 0.01 ms)

121 Anomaly velocity (m/s) Fig The results of statics calculations for different iterations in the case of anomaly velocity 1800 m/s, refractor depth 176 m Table 1. The values of RMSDV ( m/s) for different values of anomaly velocity and for different options of smoothing application Starting Values First Iteration Second iteration Without smoothing Second iteration With smoothing Third iteration without smoothing

122 Table 2. The values of RMDT (ms) for different values of anomaly velocity and for different number of iterations Anomaly velocity (m/s) Starting Values First iteration without smoothing Second iteration without smoothing Third iteration without smoothing III.3.2. The two layer model with gradient medium over refractor (depth 120 m) The next analysed model consists of gradient medium over half-space with constant velocity (fig. 3.53). The depth of the refractor is 120 m, its velocity 2500 m/s. In the first layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12 CDP s (300 m), velocity 800 m/s. The velocity in the first layer is incresing from 960 m/s near surface to 1440 m/s near refractor. The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 8 m in the vertical direction. Fig Two layer velocity model with velocity anomaly 800 m/s and gradient medium over constant velocity half-space, refractor depth 120 m

123 Fig The ray trajectories for the model with velocity anomaly 800 m/s, 2 shot points, refractor depth 120 m; horizontal axis distance in m, vertical axis depth in m Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.71 ms)

124 Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.71 ms) Fig The result of inversion in 2-nd iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.06 ms)

125 Fig The result of inversion in 2-nd iteration (with smoothing after 1-st) in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s) Fig The result of inversion in 3-nd iteration (without smoothing after 2-nd) for the case of anomaly velocity 800 m/s (RMSDV= 58 m/s, RMSDT= 0.01 ms)

126 Fig The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 120 m III.3.3. The model of gradient half-space In the third analysed model of gradient half-space the velocity increased with depth from 960 m/s near surface to 1670 m/s near the bottom of the model at depth 200 m. In the gradient medium the anomaly is placed with low velocity 800 m/s.. The anomaly is located at depth range 48 to 80 m with effective width 12 CDP s (from 80 to 92 CDP, 300 m), The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 8 m in the vertical direction. The discussed velocity model is presented in fig. 3.61, the example of ray trajectories are displayed in fig In figs and 3.64 the records resulting from Finite Difference Modeling are presented.

127 Fig The model of gradient half-space with velocity anomaly 800 m/s. Fig The example of ray trajectories for the model of gradient half-space with velocity anomaly 800 m/s. horizontal axis distance in, vertical axis depth in m

128 Fig The example of ray trajectories for the model of gradient half-space with velocity anomaly 800 m/s. horizontal axis distance in, vertical axis depth in m Fig The example of modeled record obtained for the model of gradient halfspace with velocity anomaly 800 m/s by means of Finite Difference Modeling; the first breaks picks are shown using red colour

129 Fig The enlargement of record from fig with first breaks influenced by low velocity anomaly (increased traveltimes) For the analysed model the direct task of turning ray tomography was solved using ProMAX process named Turning Ray Tracing for model with anomaly. The calculated traveltimes were then treated as input data to tomographic inversion carried out with the help of ProMAX process named Turning Ray Tomography. The gradient model without anomaly has been applied as the starting velocity model. To define the effect of the possible errors of the starting model estimation the assumed velocity gradient was decreased or increased by 10% and 20%. The calculations were realized for the case of many shots (20shots were used) taking into account every one shot, every third, every fourth and every fifth. The split-spread was use with 49 receivers (49 channels), receiver interval 50 m, shot interval 100 m, shot on position of 25-th receiver. The dimensions of velocity grid: 25 m in horizontal direction and 8m in vertical direction. The results of tomographic inversion for discussed model are presented in figs The estimated statics is showned in fig

130 Fig The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m), RMSDV= 60 m/s Fig The differences between the true model and velocity field reconstructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s, RMSDV= 61 m/s

131 Fig The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m); gradient of starting velocity model decreased by 10%, RMSDV= 61 m/s Fig The differences between the true model and velocity field reconstructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s and gradient gradient of starting velocity model decreased by 10%

132 Fig The result of inversion obtained in ProMAX system by means of the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m); gradient of starting velocity model increased by 10%, RMSDV= 60 m/s Fig The differences between the true model and velocity field constructed by means of the Turning Ray Tomography process in the case of anomaly velocity 800 m/s and gradient gradient of starting velocity model increased by 10%, RMSDV= 60 m/s

133 Fig The results of statics calculations using velocity fields resulting in ProMAX system from the process Turning Ray Tomography with application of many shots (every third shot, shot interval 300 m) and different errors of of starting velocity model gradient III.3.4. The two layer model with constant velocities The next analysed model consists of constant velocity (1200 m/s) layer over half-space with constant velocity 2500 m/s (fig. 3.73). The depth of the refractor is 120 m. In the first layer the velocity anomaly is placed with the following parameters: depth 48 m, width 12 CDP s (300 m), velocity 800 m/s. The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 8 m in the vertical direction.

134 Fig The two layer model with constant velocities, refractor depth 120 m Fig The example of ray trajectories for the velocity anomaly 800 m/s in first layer, horizontal axis distance in, vertical axis depth in m

135 Fig The enlargement of record with picked first breaks influenced by low velocity anomaly (increased traveltimes) Fig The result of inversion in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s, RMSDT= 1.10 ms)

136 Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s, RMSDT= 1.10 ms) Fig The result of inversion in 2-nd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 55 m/s, RMSDT= 0.17 ms)

137 Fig The result of inversion in 2-nd iteration (with smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 55 m/) Fig The result of inversion in 3-rd iteration (without smoothing-after 2nd) in the of anomaly velocity 800 m/s (RMSDV= 55 m/s, RMSDT= 0.02 ms)

138 Fig The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s, refractor depth 120 m III.3.5. The three layer model with constant velocities The fifth analysed model consists of two layered over half-space with constant velocity (fig. 3.82). The depth of the refractor is 176 m, its velocity 2500 m/s. In the first layer the low velocity anomaly is placed with the following parameters: depth 48 m, width 12 CDP s (300 m), velocity 800 m/s. The velocity in the first layer is 1200 m/s, in the second m/s. The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 1 m in the vertical direction. The same model but without anomaly was used as starting velocity model in first iteration. Additionaly the values of first and second layer velocities have been change (in the range of ± 5% and ± 10%) to simulate the errors of the starting model. The position of low velocity anomaly (1100 m/s) in the second layer was also taken into consideration.

139 Fig The two layer model with constant velocities over half-space with constant velocity, velocity anomaly 800 m/s Fig The example of ray trajectories for the velocity anomaly 800 m/s in first layer, horizontal axis distance in, vertical axis depth in m

140 Fig The result of inversion in 1-st iteration (without smoothing) for the sase of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 1.39 ms) Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 1.39 ms)

141 Fig The result of inversion in 2-nd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 52 m/s, RMSDT=0.07) Fig The result of inversion in 2-nd iteration (with smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 52 m/s )

142 Fig The result of inversion in 3-rd iteration (without smoothing) for the case of anomaly velocity 800 m/s (RMSDV= 53 m/s, RMSDT= 0.01 ms) Fig The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s in first layer, refractor depth 176 m

143 Fig The two layer model with constant velocities over half-space with constant velocity, velocity anomaly 1100 m/s Fig The example of modeled record of the two layer model with constant Velocities over half-space with constant velocity obtained by means of Finite Difference Modeling, velocity anomaly 1100 m/s in first layer; the first breaks picks are shown using red colour

144 Fig The enlargement of record with picked first breaks influenced by low velocity anomaly (increased traveltimes) Fig The example of ray trajectories for the velocity anomaly 1100 m/s in second layer, horizontal axis distance in, vertical axis depth in m

145 Fig The result of inversion for first iteration without smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s, RMSDT= 0.41 ms) Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) for the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s, RMSDT= 0.41 ms)

146 Fig The result of inversion for second iteration without smoothing for the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s, RMSDT= 0.01 ms) Fig The result of inversion for second iteration with smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s )

147 Fig The result of inversion for third iteration without smoothing in the case of anomaly velocity 1100 m/s (RMSDV= 56 m/s, RMSDT= 0.00 ms) Fig The results of statics calculations for different iterations in the case of anomaly velocity 1100 m/s in second layer, refractor depth 176 m

148 Fig The results of inversion for starting model with first and second layer velocity decreased by 10%. Third iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 169 m/s,rmsdt= 0.06 ms) Fig Statics results for starting model with first and second layer velocity decreased by 10% in the case of low velocity anomaly 800 m/s

149 Fig Statics results for starting model with first and second layer velocity increased by 10% in the case of low velocity anomaly 800 m/s Fig Statics results for starting model with first and second layer velocity decreased by 5% in the case of low velocity anomaly 800 m/s

150 Fig Statics results for starting model with first and second layer velocity increased by 5% in the case of low velocity anomaly 800 m/s

151 III.3.6. The four layer model with constant velocities The last analysed model consists of three layers with constant velocities over halfspace with constant velocity (fig ). The depth of the refractor is 199 m, its velocity 2500 m/s. The velocity in the first layer is 800 m/s. Second layer with velocity 1200 m/s, third layer with velocity 1500 m/s. The low velocity anomaly 800 m/s was placed in second layer. Additionaly the anomaly depth was varied from 74 m to 123 m. The dimensions of the velocity cell were 1 CDP (25 m) in the horizontal direction and 1 m in the vertical direction. The results of calculations for this model are presented in figs Fig The model of fourth layers with constant velocities.

152 Fig The example of ray trajectories for the velocity anomaly 800 m/s in second layer, horizontal axis distance in, vertical axis depth in m Fig The example of modeled record of the fourth layer model with constant velocities over half-space with constant velocity obtained by means of Finite Difference Modeling, velocity anomaly 800 m/s in second layer; the first breaks picks are shown using red colour

153 Fig The result of the inversion in first iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 60 m/s, RMSDT= 1.45 ms) Fig The differences between the true model and the reconstructed velocity field in 1-st iteration (without smoothing) in the case of anomaly velocity 800 m/s (RMSDV= 60 m/s, RMSDT= 1.45 ms)

154 Fig The result of the inversion in second iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.06 ms) Fig The result of the inversion in second iteration with smoothing in the case of anomaly velocity 800 m/s (RMSDV= 58 m/s )

155 Fig The result of the inversion in third iteration without smoothing in the case of anomaly velocity 800 m/s (RMSDV= 59 m/s, RMSDT= 0.00 ms) Fig The results of statics calculations for different iterations in the case of anomaly velocity 800 m/s in second layer, refractor depth 199 m

156 III.4. THE ANALYSIS OF THE RESULTS OF DIRECT AND INVERSE PROBLEM SOLUTION FOR MODEL DATA The analysed results of direct and inverse problem solution comprised the following data: the graphs of traveltimes, the bitmaps of velocity fields, the bitmaps of differences between assumed model velocity fields and fields resulting from tomographic inversion, the RMSDV and RMSDT errors defined in introduction to chapter III.3, the theoretical records calculated using Finite Difference Modeling process, the graphs of ray trajectories and the graphs of static corrections. The most important conclusions from the analyses relating to tomographic inversion effectiveness and to static corrections estimation are presented below. The analyses of the ray trajectories graphs of turning wave in the case of gradient medium with velocity anomalies confirmed that the behaving of rays in discrete velocity fields defined with application of finite difference methods is very complicated. The strong rays deviations and oscillations may be observed (e.g. fig.3.62a) causing essential and artificial increasing of wave traveltimes not observed in the theoretical records (e.g. 3.62b) obtained using wave approach (finite difference modeling). It means that applications of these traveltimes in the tomographic inversion for estimation the differences between observed (resulting from picking) and modeled traveltimes may be the source if serious errors and may decrease in many cases the effectivenes of turning wave tomography (figs ). It seems that application of wave approach (finite difference modeling) to solution direct problem of tomography let us to avoid some of these errors. Otherwise we must very carefully analyse the calculated traveltimes and remove some of the erroneous traveltimes before inversion. Additionally in the case of turning wave tomography we can observe anomalous extending of the resulting velocity anomalies in the main, horizontal direction of turning wave propagation. In the case of head wave tomography behaving of head wave rays is generally much more regular (figs 3.20, 3.54, 3.83, 3.85, 3.93). Such a behaving of rays in the case of thin low velocity layer does not create optimal conditions for suitable illumination of the velocity anomalies. To obtain good results of the tomographic inversion we need the rays going through the anomaly as well as the rays which omitt the anomaly zone. This condition may be satisfied only in the case of small velocity anomalies (figs ). In other case the resolution of tomographic solutions in the vertical direction is very weak and we can t define the proper position of anomaly in depth. Analysing the velocity fields resulting from head wave tomography in the case of different models of low velocity layer (chapter III.3) we can notice that the vertical resolution

157 of velocity fields is very weak but horizontal resolution is very good. Then although we cannot define the proper depth of anomaly we can estimate quite well its horizontal position. Additionally we can observe very interesting compensating effect in vertical direction (figures 3.23, 3.30, 3.36, 3.42, 3.56, 3.76, 3.85, 3.89, 3.96): decreasing of velocity in one zone (connected with low velocity anomaly) is compensated by increasing of velocity outsite this zone (below and above). Thanks to this compensation effect the static corrections defined as always for vertical propagation of rays are in all cases estimated with good (fig. 3.28) or very good accuracy (figures: 3.34, 3.40, 3.46, 3.52, 3.60, 3.80, 3.89, 3.91, 3.100) if several iterations of tomographic inversion is applied and proper spatial smoothing of velocity fields is done. Even if we introduce errors in the velocities of starting models the resulting the graphs of static corrections are moved by constant value in vertical direction retaining the same shape (figures: 3.95, 3.100, 3.105, 3.110). If we analyse the behaving of RMSDV and RMSDT errors of resulting velocity fields and obtained differences between calculated and assumed traveltimes we can observe that in consecutive iterations RMSDV errors do not change seriously while RMSDT errors are decreasing very fast. Decreasing of RMSDT errors correlates with better estimation of static corrections. The proper estimation of starting model which should be very close to the model without velocity anomaly - plays very important role in the inversion of first breaks by means of head wave tomography. But this is the well known problem of tomographic inversion. We cannot reconstruct the boundaries when the propagating rays are perpendicular to these boundaries. To define position of these boundaries we must use commonly used, very effective tool known as GLI inversion based on standard refraction interpretation of traveltimes. In the case of medium consisting of several gradient layers we can use also turning ray tomography which is very effective tool for defining horizontal layering.

158 III.5. THE TOMOGRAPHIC INVERSION OF FIRST BREAKS ON FIELD RECORDS The evaluation of the tomographic inversion effectiveness is very complicated task because the facters affecting this effectiveness are connected not only with many parameters of inversion but in great part with the quality of input data and with the type of seismogeological model of investigated medium. This is the main reason why all the serious analyses of this type which appeared in the geophysical papers were based only on model data (e.g. Dyer and Worthington 1988, Ivansson 1985, Krajewski et al.1989, Peterson et al.1985, Philips and Fehler 1991). In practice we are first of all interested in the evaluation of the usefulness of inversion results (e. g. velocity fields) in other essential processing procedures (statics estimation and migration). If we want to realize such a evaluation in broad range much volume of field data from areas of different seismogeological models (in our case models of near surface layer) must be included. Taking into account that presented dissertation has mainly the form of model study it was very difficult to select such a big volume of field data and include these data in the presented analysis. However some attempt was undertaken to show the main aspects connected with application of refraction tomography to inversion of first breaks on field records. The selected vibroseis field records from the area of investigations are presented in fig and These are raw field records after application only AGC and bandpass filter ( ). They were recorded with split-spread geometry using 120 channels, receiver interval 50 m, shot interval 100 m.. The parts of records with visible influence of statics are marked using red rectangles. Fig The vibroseis record from shot 184, parts of record with visible influence of statics is marked using red rectangle

159 Fig The vibroseis record from shot 201, part of record with visible influence of statics is marked using red rectangle Fig The vibroseis record from shot 207, part of record with visible influence of statics is marked using red rectangle

160 Fig The vibroseis record from shot 215, part of record with visible influence of statics is marked using red rectangle From analysed records two of them have been selected with numbers 184 and 215. The positions of channels of first record (no. 184) and 1-60 of second record (no. 215) were the same. The result of first break picking on these two records are displayed in fig Fig The vibroseis record from shot 184 with marked results of first break picking

161 Fig The vibroseis record from shot 215 with marked results of first break picking The picks from right half of the first record (184) and the left part of the second one (215) were selected as input data to tomographic inversion. These parts of records are presented in fig Fig The parts of two records selected as input data to tomographic inversion In the area of investigation besides vibroseis lines also dynamite lines were recorded. The graphs of uphole times, elevation and shot depths for dynamite line placed near analysed vibroseis line are displayed from the line database in fig The data from these graphs were used to construct the starting model for tomographic inversion.

162 Fig The graphs of uphole times, elevation and shot depths for dynamite line Fig The graph of elevation for dynamite line Fig The graph of uphole times for dynamite line From the analysis of the above graphs the parameters of starting model have been evaluated: - the depth of the bottom of LVL was equal to the constant depth of shot holes (18 m), - the velocity in the first layer was calculated from uphole times (800 m/s), - the velocity in the second layer was estimated from apparent velocity of first breaks (1800 m/s).

163 The graphs of traveltimes resulting from first break picking are presented in fig These times were the input data to tomographic inversion. The #INVERSDH program was used to realize inversion with the folowing parameters: - the velocity grid size: 397 x 20, - the cell dimensions: 25 m x 1 m, - number of iteration in conjugate gradient method: 300, - constraint coefficient λ = 10, - parameter of convolutional quelling: q = 1.0/0.5, - spatial smoothing operator: 50 m x 2 m, - number of inversion iterations: 1. Fig The graphs of traveltimes resulting from picking vibroseis records 184 and 215 The velocity fields resulting from inversion before and after smoothing are displayed in fig The estimated statics calculated for these two velocity fields using the bottom of starting model as datum (depth 18 m) are presented in fig

164 Fig The velocity field resulting from tomographic inversion Fig The velocity field resulting from tomographic inversion after spatial smoothing

165 Fig The estimated total statics calculated (before and after smoothing) using the bottom of starting model as datum (depth 18 m), horizontal axis positions of receivers (common for two shots), shot statics was the same The estimated total statics was applied to traces of two analysed records using Hand Statics procedure. The part of records before and after statics are displayed in fig Fig The part of record 184 before statics (on the left) and after statics (on the right)

166 Fig The part of record 184 before statics (on the left) and after statics (on the right) Analysing the results we can state that the velocity field resulting from tomography includes some low and high velocity anomaly. Generally the values of statics are small for so thin low velocity layer. On the records we can see some small improvement in the case of record 215 in the red rectangle. But such a small range of analysis cannot be the base of evaluation the effectiveness of tomographic inversion.