Mon Taj Geophysics

montaj Geophysics v6.1 Provides a range of 1D and and 1D FFT FFT filt ers and statist ical to ols for wo rking with l arge-volume arge-volume geophysical geophysical data

TUTORIAL and USER GUIDE

www.geosoft.com

The software described in this manual is furnished under license and may only be used or copied in accordance with the terms of the license. Manual release date: 2/1/2005. Written by, Nancy Whitehead and Chris Musselman. Please send comments or questions to [email protected] Copyright Geosoft Inc. 2005. 2005. All All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form, or by any means, electronic, mechanical, photo-copying, reading, or otherwise, without prior consent from Geosoft Inc. Program Copyrig Copyrig ht Geosoft Inc. 2005. All 2005. All rights reserved reserved.. Geosoft and Oasis montaj are registered trademarks of Geosoft Inc. GEOSOFT, GEOSOFT, Oasis are trademarks of Geosoft Inc. Windows ®, and Windows NT™ are either registered trademarks or trademarks of Microsoft Corporation.

Geosoft Incorporated 8th Floor 85 Richmond St. W. Toronto, Ontario M5H 2C9 Canada Tel: (416) 369-0111 Fax: (416) 369-9599 Web Site: www.geosoft.com E-mail: [email protected]

The software described in this manual is furnished under license and may only be used or copied in accordance with the terms of the license. Manual release date: 2/1/2005. Written by, Nancy Whitehead and Chris Musselman. Please send comments or questions to [email protected] Copyright Geosoft Inc. 2005. 2005. All All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form, or by any means, electronic, mechanical, photo-copying, reading, or otherwise, without prior consent from Geosoft Inc. Program Copyrig Copyrig ht Geosoft Inc. 2005. All 2005. All rights reserved reserved.. Geosoft and Oasis montaj are registered trademarks of Geosoft Inc. GEOSOFT, GEOSOFT, Oasis are trademarks of Geosoft Inc. Windows ®, and Windows NT™ are either registered trademarks or trademarks of Microsoft Corporation.

Geosoft Incorporated 8th Floor 85 Richmond St. W. Toronto, Ontario M5H 2C9 Canada Tel: (416) 369-0111 Fax: (416) 369-9599 Web Site: www.geosoft.com E-mail: [email protected]

Contents Geosof Geosof t License Agreement

1

Finding More Help Inform ation

3

Contacting Technical Support

Chapter 1: System Capabilities and Concepts

3

4

Understandin g 1-Dimension 1-Dimension al Fast Fourier Transform Filtering

4

Summary of Fourier Filtering Capabilit Capabilit ies

5

Introduction to Filtering Concepts Concepts

6

Understanding the Mathematical Framework for Transforms

6

Relating Potential Fields to Transform Spectra

7

Understandin g the 1D-FFT 1D-FFT Filtering Process

Chapter 2: Quick Start Tutorial

10

12

Create a Proj ect

12

Loc ate the 1D - FFT menu s

13

Open a Database

14

Ap pl y Fi lt er t o Data D ata Chan nel

15

Chapter 3: Preparing Data and Applying Filters

18

Preparing Preparing Data Data for Filtering

18

Ap pl yi ng Stan dar d Fil F il ter s

19

Butterworth Filter (FFTBUTTW GX)

19

Gaussian Regional/Residual Filter (FFTGAUSS GX)

20

Cosine Roll-off Filter (FFTCOSN GX)

20

Lowpass Filter (FFTLOWP GX)

21

Highpass Filter (FFTHIGHP GX)

22

Bandpass Filter (FFTBANDP GX)

22

Vertical Derivative (FFTVDRV GX)

23

Vertical Integration (FFTVINT)

23

Horizontal Derivative (FFTHZDRV GX)

23

Horizontal Integration (FFTHZINT GX)

23

Downward Continuation (FFTCONT GX)

24

Upward Continuation (FFTCONT GX)

24

Magnetic Pole Reduction (FFTRPOLE GX)

25

Apparent Susceptibility (FFTSUSC GX)

26

Apparent Density (FFTDENS GX)

27

Multiple Filters (FFT1MULT GX)

27

Analytic Signal (FFTAS GX)

28

Hilbert Transform (HILBERT GX)

28

Power Spectrum (FFTPSPEC GX)

29

Chapter 4: Applying Custom Filters

30

References

32

Glossary

33

1

Geosoft License Agreement GEOSOFT agrees to supply the Licensed Program(s) as specified in my purchase order. Geosoft shall grant me a nontransferable, non-exclusive license to use the Licensed Program(s), subject to the Terms and Conditions herein contained. Should there be a separate signed agreement between you and Geosoft, or between your company and Geosoft, pertaining to the licensed use of this software, that agreement shall take precedence over the terms of this agreement. 1.

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LICENSE:

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TERM:

The Term of this Agreement shall commence on the Effective Date and shall continue until termination, as described in Section 6. 4.

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2 6.

TERMINATION:

This agreement shall terminate upon the termination date, if any, specified in your purchase agreement with Geosoft. This agreement may be terminated only upon thirty-days prior written notice to GEOSOFT. GEOSOFT may terminate this Agreement upon prior written notice effective immediately if I fail to comply with any of the terms and conditions of this Agreement. This Agreement shall terminate automatically upon the institution, or consenting to the institution of proceedings in insolvency or bankruptcy, or upon a trustee in bankruptcy or receiver being appointed for me/us for all or a substantial portion of my/our assets. 7.

EVENTS UPON TERMINATION:

I shall forthwith discontinue use of the Licensed Program(s), on the day Termination shall occur and agree not to resume such use in the future without written authorization from GEOSOFT. I shall uninstall and remove all software from my computer. Within thirty days after Termination, I shall destroy all physical and digital copies of the Licensed Program(s). This obligation relates, without limitation, to all copies in any form, including translations, compilations, derivatives and updated materials, whether partial or complete, and whether or not modified or merged into other materials as authorized herein. 8.

WARRANTY:

GEOSOFT does not warrant that the functions contained in the Licensed Program will meet my requirements or will operate in the combinations which may be selected for use by me, or that the operation of the Licensed Program will be uninterrupted or error free or that all program defects will be corrected. Each Licensed Program shall be furnished to me in accordance with the terms of this Agreement. No warranties, either express or implied, are made to me regarding the Licensed Program. THE FOREGOING WARRANTIES ARE IN LIEU OF ALL OTHER WARRANTIES, EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OR MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. 9.

LIMITATION OF REMEDIES

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3

Finding More Help Information There are several other functions included in the basic Oasis montaj help system that may be useful to your work. The entire documentation for the system is available through the online help system. This electronic library of information enables us to constantly update the information and provide you with the most up-to-date information available. The best way to find information in this system is to use the Search tab to perform a full-text search of all help topics. If you still can’t find the information you’re look ing for, the Online Books help system contains complete Geosoft manuals and tutorials in Adobe PDF format.

Contacting Technic al Suppor t The list below provides contact information for Geosoft Technical Support around the world. North America

Europe and North Africa

Geosoft Inc., 85 Richmond St. W., 8th Floor Toronto, Ont., Canada M5H 2C9

Geosoft Europe Ltd. 20/21 Market Place, First Floor Wallingford, Oxfordshire United Kingdom OX10 OAD

Tel +1 (416) 369-0111 Fax +1 (416) 369-9599

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Email: [email protected]


South America

Australia and Southeast Asia

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Tel: +27 12 347 4519 Fax: +27 12 347 6936 Email: [email protected]

4


Chapter 1: System Capabilities and Concepts The montaj Geophysics software extension provides a range of filters and statistical tools for working with large-volume geophysical data. S patial 1D Filters enable field geophysicists to process data by applying a variety of space domain filters (linear and non-linear). The 1D FFT Filter enables you to apply a variety of Fourier domain filters to one-dimensional (line) potential field and othe r data. A variety of geostatistical tools provide the ability for summary and advanced statistics, including histogram, scatter and triplot analysis, and the ability to subset data based on code or map group classification. This User Guide is targeted to Earth Science professionals who are familiar with geophysical data processing methods and who want to apply Fourier domain linear filters to process and interpret one-dimensional (line) po tential field data from magnetic and gravity surveys. To use this package effectively, you should be familiar with Fourier domain filtering concepts and methods including Fast Fourier Transforms. You should also have an appreciation of how to apply filters for numerical analysis and interpretation of line data.

Understanding 1-Dimensional Fast Fourier Transform Filtering Fourier filtering techniques are an important tool to geophysicists who work with potential field (gravity and magnetic) data. Fourier filtering techniques provide an effective means of processing and interpreting data that is broadband in nature (i.e. each observation contains the effects from all magnetic or gravity sources that produce a response above the noise level at the po int of observation). By transforming data to the Fourier domain (through the use of a Fast Fourier Transform, or FFT), we can deal with the data as a function of wavenumber, or wavelength. In this form, there are a number of manipulations that can be applied to enhance information that is of interest, remove information that is not of interest, or transform the data (see Bhattacharya, 1966, and Spector and Grant, 1970). For example, in the Fourier domain we might apply a Butterworth low-pass filter to data to remove short wavelength features that are not of interest. We might also upward continue or downward continue data to a new level that is more sensitive to the information of interest, or we might transform magnetic data from an observed geomagnetic latitude to the magnetic North pole. All of these procedures can be carried out using Fourier techniques. The montaj Geophysics 1D- FFT Processing System is a suite of Geosoft eXecutables (GX applications) that you can use to apply Fourier domain filters to one

Chapter 1: System Capabilities and Concept s 5

dimensional (line) data stored in an Oasis montaj database. Designed primarily for potential field data, montaj Geophysics provides you with the tools required to:

• •

Perform profile-based processing for interpretation and modeling purposes

•

Rapidly apply filters to one or more lines of data

Interactively select filtering parameters, display power spectra and identify optimal filters for processing and interpretation

Summary of Fourier Filtering Capabilities Generally, you can think of Fourier filters as falling within one of the following groups: 1. Sharpening filters that enhance the shorter wavelength features in the data. These include high-pass filters, downward continuations and vertical and horizontal derivatives. Such filters are normally used to enhance information from shallow geologic features. 2. Smoothing filters that enhance longer wavelength features in the data, normally by removing or attenuating the shorter wavelengths features. These include low pass filters, upward continuations and integrations. Smoothing filters are normally used to remove short-wavelength noise in the data or to remove the effects of shallow geologic features. 3. Geophysical transformations that convert data from one physical form to another. These include reduction to the pole for magnetic data and the calculation of apparent magnetic susceptibility or apparent density. Smoothing and sharpening filters are often combined to meet the needs of a specific problem. For example, a vertical derivative (sharpening filter that enhances nearsurface geology) might be combined with an upward continuation (smoothing filter that reduces the effect of noise in the data). Filters can further be described as geophysical filters -- those whose outcome has a geophysical basis or mathematical -- those that have a mathematical definition only. Since geophysical filters are based on the physics of potential fields, they are ideally suited to gravity and magnetic data whereas mathematical filters can be applied to any kind of data. Examples of geophysical filters are vertical continuations, vertical derivative or integration, reduction to the pole and apparent magnetic susceptibility or density. Mathematical filters include horizontal derivatives and the high and low pass filters -- Butterworth, Gaussian, cosine and simple high and low pass cutoff filters. In addition to filters, power spectra are often required. The montaj Geophysics system enables you to create a power spectrum of the data and to place the real and imaginary components of the Fourier transform into channels of the database. You

6


may then apply your own mathematical algorithms to the data using the math functions of Oasis montaj.

Introducti on to Filtering Concepts For mathematical convenience, the montaj Geophysics system applies filters in the Fourier or wavenumber domain. If you are not already familiar with the application of filters to one-dimensional data using Fourier Domain techniques, you may want to read the following summary of filtering concepts.

Understanding the Mathematical Framework f or Transforms Mathematically, the Fourier transform of a space domain function f(x) is defined to be: ∞

f (ω ) =

∫ f ( x) ⋅ e

iω x

dx

−∞

The reciprocal relation is f ( x ) =

1

∞

∫

f (ω ) ⋅ e

2π −∞

− iω x

d ω

where ω is an angular wavenumber in radians per ground_unit (for x in ground_units). The wavenumber in cycles per ground_unit (r ) is simply ω /2π . A line of data in the space domain can be thought of as a sequence of magnetic or gravity values at points along a straight line, each point separated by a constant distance. Such a line can be transformed to and from the wavenumber domain by use of a discrete Fast Fourier Transform (FFT). The equivalent data set in the wavenumber domain is commonly called a Transform. A Transform of a line is consists of real and imaginary amplitude values as a function of wavenumber in cycles per original distance unit. In addition to these basic mathematical concepts, there are a number of other generalized properties in Fourier domain processing, including:

•

A given potential field function in the space domain has a single and unique wavenumber domain function, and vice-versa.

•

The addition of two functions (anomalies) in the space domain is equivalent to the addition of their Transforms.

•

The energy spectrum is a 1-D function of the energy relative to the wavenumber.


Relating Potential Fields t o Transform Spectra The Fourier transform of the potential field due to a prismatic body has a broad spectrum whose peak location is a function of the depth to the top and bottom surfaces and whose amplitude is determined by its density or magnetization. You can relate the peak wavenumber (ω ' ) to the geometry of the body according to the following expression:

′ = ω '

ht hb

ln( hb ht ) hb − ht

where

is the peak wavenumber in radians / ground_unit is the depth to the top is the depth to the bottom

For a bottomless prism, the spectrum peaks at the zero wavenumber according to the expression: f (ω ) = e

− hω

where ω is the angular wavenumber in radians / ground_unit and h is the depth to the top of the prism. For more information, refer to Bha tacharia, 1966. For a prism with top and bottom surfaces, the spectrum is: f (ω ) = e −

ht ω

− e − h ω b

where ht and hb are the depths to the top and bottom surfaces respectively. As the prism bottom moves closer to the observation point at surface, the peak moves to higher wavenumbers as illustrated in the following figure: 1 no bottom top = 4 bottom depth 36 20

12 8

0 0

wavenumber

1

8


Considering the spectrum of a fixed size prism, as the prism depth increases, the peak of the spectrum shifts to lower wavenumbers (the space domain anomaly becomes broader) and the amplitude of the spectrum decreases: 1 thickness = 4

top = 4

8

16

0 0

wavenumber

1

When looking at the spectrum above, it is important to note that the amplitude of a deep prism does not exceed the amplitude of the same prism at shallow depth at any wavenumber. The effect of increasing the depth is to shift the peak to lower wavenumbers. Because of this characteristic, there is no way to separate the effect of deep sources from shallow sources of the same type by using wavenumber filters. You can only distinguish sources if the deep sources have greater amplitude or if the shallow sources have less depth extent. When considering a line that is long enough to include many sources, you can use the log spectrum of this data to determine the depth to the tops of a statistical ensemble of sources using the relationship (see Spector and Grant, 1970): log E (k ) = 4π hk where h is the depth in ground_units and k is the wavenumber in cycles / ground_unit. You can determine the depth of an 'ensemble' of sources by measuring the slope of the energy (power) spectrum and dividing by 4π. A typical energy spectrum for magnetic data may exhibit three parts -- a deep source component, a shallow source component and a noise component.


The following figure illustrates the interpretation of an energy spectrum into these three components: ENERGY SPECTRUM

deep sources Nyquist shallow sources

noise

wavenumber (cycles/ground unit)

10


Understanding t he 1D-FFT Filtering Process When applying Fourier domain filters to a line of data the following steps are typically applied: 1. Trend Removal

A first order trend is calculated by least-square fitting a straight sloping line to the data. This trend is removed and the trend coefficients are saved as part of the zero wavenumber component of the transform. This means that changes to the zero wavenumber in the Transform will also effect the first order trend.

2. Line Extension

The line length is extended by a minimum of 10% up to the next power of two, which is required by the FFT process used in Oasis. The minimum 10% is to provide a minimum extension within which to predict data that makes the function periodic.

3. Prediction

The extended line and any holes in the line are filled in by predicting data using maximum entropy prediction (Burg, 1975). Maximum Entropy Prediction insures that the predicted data has the same spectral character as the real data. The prediction further insures that the function has the same value and slope at each end. The line is then considered smoothly periodic because copies of the line can be place end to end without a discontinuity.

4. FFT

The data is that transformed from the space domain to the wavenumber domain using an FFT. The wavenumber increment of the resulting transform will be 1/(line_length). For example, if the extended line length is 4096 points, and the data fiducial increment is 10 ground_units, the wavenumber increment in the Fourier domain will be 1/(4096 * 10) or 0.00002441 cycles/ground_unit. The maximum wavenumber will be at the Nyquist frequency, which is 1/(2*data increment). In the previous example, this would be 1/20, or 0.05 cycles/ground_unit. The FFTOUT GX will place the real and imaginary components of the transform into channels of the database. The FFTPSPEC will calculate the power spectrum and place the result in a power spectrum channel. The other filters will carry on to the next step.

5. Filter Application

The specified filter is applied to the data in the Fourier domain.

6. Inverse FFT

The filtered wavenumber data is transformed back to the space domain using an inverse FFT.

7. Trend Replacement

The zero wavenumber amplitude is used to determine the trend to replace in the data. Many filters will set the zero wavenumber to 0.0, which results in no trend being replaced in the data.

8. Masking

The original data channel is used as a mask to return the filtered data to the same length and to place dummies in the final filtered data. The result is placed in the output channel specified by the GX


The Fourier domain filtering process (as outlined above) is built in to every ID-FFT filter. Note:

You can use the FFT1D|Advanced Usage menu item to go through the step by-step Fourier domain filtering process. This step-by-step method enables you to apply a user defined filter formula or math expression.

12 Chapter 2: Quick Start Tutori al

Chapter 2: Quick Start Tutorial This tutorial is designed to explain how to apply 1D-FFT filters in Oasis montaj. The procedures are designed to show you how you can apply 1D-FFT utilities and filters to channel data in a database.

• • • •

Create a project (page 12) Locate the 1-D FFT menu (page 13) Open a database (page 14) Apply filter to data channel (page 15)

These files are provided on the CD-ROM in the Oasis montaj\data\1dfft directory. Before you begin the tutorial, you will need to copy the database file (fft_mag.gdb) to your working directory.

Create a Project In order to access the menus in Oasis montaj, you must have an open Project. An Oasis montaj "Project" encompasses every item in your working project; from the data files in your project (databases, maps, and grids), to the tools used (including auxiliary tools such as histograms, scatter plots etc.), to the project setup including the menus you have displayed and whether you are working on a map or profile and the state in which you left it the last time you used it. The project also controls your working directory. Projects are saved as (*.gpf) files. If you open an existing project from a directory, the system assumes that all your project files are located in the same directory. To streamline your work, as well as keep it organized, you may wish to make sure that your project file is in the same directory as the other files you want to use. We recommend that each project you work on have its own project (*.gpf) file. If you use a number of applications or addon tools in Oasis montaj that have different menus, you can use the project to display only the menus you require. The Project Explorer tool enables you to browse as well as open any project item. The Project Explorer has two tab windows, the Data window that includes all data files included in the project and the Tools window that organizes and maintains the project tools. To access the Tools window click the Tools bar on the bottom of the Project Explorer. To return to the Data window, click the Data bar on the top the Project Explorer.

Chapter 2: Quick Start Tutori al 13

Important Note: Workspace files (*.gws) used in Oasis montaj prior to version 6.0 can be easily converted to Project files (*.gpf) simply by opening them in Oasis montaj 6.0. On the Open Project dialog (File|Project|Open) select File of Type as "Workspaces (*.gws)" and when asked if you want to convert the old workspace into a new Oasis montaj project file, select "Yes". The workspace file will be converted to a project file and all associated workspace information will be transferred to the new project file. In addition, the workspace file will remain untouched so that it can be opened in previous versions. T O C REATE A P ROJECT : 1.

Start Oasis montaj.

2.

On the File menu click Project and then click New. The New Project dialog is displayed.

3.

Specify a name and directory for the project. For example, name the project 1D FFT and specify the working directory as D:\Tutorial. Oasis montaj will automatically look for your data in the directory containing this project. Make sure you copy the sample database file (fft_mag.gdb) for this tutorial to your project directory (D:\Tutorial).

4.

Click the [Save] button. The system saves the project and indicates it is open by opening the Project Explorer window, enabling the buttons on the Main toolbar and adding menus to the menu bar. These are visual clues indicating that you are ready to start working with the system.

Locate the 1D - FFT menus Once you have installed or updated your Geosoft eLicense to include the montaj Geophysics system the 1-D FFT filter menus will become accessible to you from the Oasis montaj X-Utility menu. The 1-D FFT filters enable you to apply a variety of Fourier domain filters to onedimensional (line) potential field and other data.

14 Chapter 2: Quick Start Tutori al FFT1D MENU: Butterworth Filter FFTBUTTW.GX (page 19) Gaussian Filter FFTGAUSS.GX (page 20) Cosine Roll-off Filter FFTCOSN.GX (page 20) Lowpass Filter FFTLOWP.GX (page 21) Highpass Filter FFTHIGHP.GX (page 22) Bandpass Filter FFTBANDP.GX (page 22) Vertical Derivative FFTVDRV.GX (page 23) Vertical Integration FFTVINT.GX (page 23) Horizontal Derivative FFTHZDRV.GX (page 23) Horizontal Integration FFTHZINT GX (page 23) Downward/Upward Continuation FFTCONT.GX (page 24) Magnetic Pole Reduction FFTRPOLE.GX (page 25) Magnetic Susceptibility FFTSUSC.GX (page 26) Density for Gravity Data FFTDENS.GX (page 27) Multiple Filters FILT1MULT.GX (page 27) Analytic Signal FFTAS.GX (page 28) Hilbert Transform HILBERT.GX (page 28) Power Spectrum FFTPSPEC.GX (page 29)

A DVANCED USAGE MENU: Preprocessing previewer FFTPREP.GX Re-fid channel to distance base DISTFID.GX Transform from space to wavenumber domain. FFTIN.GX Transform from wavenumber to space domain. FFTOUT.GX Re-fid channel to fiducial base REFID.GX

Open a Database You must have a database open in your working directory to apply the 1D-FFT filters to a data channel. T O O PEN A D AT AB AS E : 1.

2.

On the Data menu, click Open Database. The system will display the Open Database dialog box. Select the fft_mag.gdb file and click [Open].

Chapter 2: Quick Start Tutori al 15 3.

The fft_mag.gdb database will be displayed in the spreadsheet window.

Apply Filter to Data Ch annel You must have a database open in your working directory to apply any 1D-FFT filters to a data channel. In this example, we will apply a Butterworth filter to the Z1 channel. Note that, the entire Fourier domain filtering processes (see Understanding the 1D FFT Filtering Process page 10) is built in to every 1D-FFT filter. Note:

All 1D-FFT GX’s (utilities and filters) have their own dialog bo xes with unique parameters specific for each GX. To find out detailed information on each GX. Click the dialog [Help] button.

T O A PPLY F ILTER TO D AT A C HANNEL : 1.

On the X-Utility | 1-D FFT menu, select Butterworth. The system will display the FFT Butterworth Filter dialog box.

16 Chapter 2: Quick Start Tutori al

2.

Select the channel to filter (Z1) from the drop-down list. Specify the name of the output channel (Z1_butterfilt). You can specify values for central wavelength cutoff, and degree of filter and select either a low or high pass filter. For more information on these parameters click the [Help] button.

3.

Click the [SetSampling] button. The Set FFT Sampling Parameters dialog box is displayed.

4.

Here you can specify the required distance increment and the interpolation method to use. For more information on these parameters click the [Help] button.

5.

Click [Back] to return the FFT Butterworth Filter menu. Click [OK] to apply the Butterworth filter and display the output butter_filt channel in the current database.

Chapter 2: Quick Start Tutori al 17

18 Chapter 3: Preparin g Data and Apply ing Filters

Chapter 3: Preparing Data and Applying Filters The following chapter explains how the montaj Geophysics system prepares the data for filtering and details the application and mathematics of each of the 1D-FFT filters.

Preparing Data for Filteri ng The distance unit used by all filters is the fiducial. This means that if you upward continue the data, for example, you must specify the continuation distance in fiducials, or if you calculate a vertical derivative of total magnetic field data, the units for the result are nT/fiducial. It is normally more convenient to work with your data in a real distance unit such as metres (or your ground_units). To do this, you must make sure the fiducial numbering of the channel you want to filter is based on distance. If your fiducial numbering is not based on a real distance unit, we recommend that you re-sample your data to a real distance base before applying filters. To resample the data to a distance base, select the AdvancedUusage|Re-fid to distance option in the 1-D FFT menu. The corresponding GX (DISTFID) looks for a channel named "_Distance" (containing the along-line distance of each point in the channel). If the "_Distance” channel does not exist, the system creates one based on the “X” and “Y” channels in the database. The GX then resamples the input data channel (using the distance channel as the new fiducial base) and creates a new data channel. The DISTFID GX also creates a channel with the same name as the original channel, but with extension “_fid” appended. This channel contains the original fiducial values for the new data channel. If you want to return the filtered channels back to the original fiducial base for other processing, you will need the original fiducial values. To return channels to their original values, select the AdvancedUusage|Re-fid back to fiducial option under the FFT menu and specify the filtered channel to convert back to a fiducial base. As the output channel, you may wish to specify the same input channel so that the system simply replaces the channel with the new data. As the fiducial base channel, you must specify the “_fid” channel (for example, "mag_fid") that you created from the original data. Finally, leave the start fiducial blank and specify the original fiducial increment.

Chapter 3: Preparing Data and Applyin g Filters 19

Applying Standard Filters This chapter describes the use of the individual filter GXs included in the montaj Geophysics system. Each of these filters starts with a line of data in the space domain and produces a new line of data in the space domain that is the result of applying the filter. If you would like to apply your own custom filters, refer to the Applying Custom Filters chapter. This chapter describes the application and mathematics of each of the Oasis montaj 1D-FFT filters. The mathematical expressions use the following terms: ω

Angular wavenumber in radians / ground_unit

k

Wavenumber in cycles / ground_unit (

N

Nyquist wavenumber [1/2d where d is the original data increment]

= 2π k )

Butterwo rth Filter (FFTBUTTW GX) The Butterworth filter is excellent for applying straight forward high-pass and low pass filters to data because you can easily control the degree of filter roll-off while leaving the central wavenumber fixed. If ringing is observed, you can reduce the degree until you are satisfied with the result. A common, but more complicated alternative is the Cosine filter (FFTCOSN). 1.0 n= 2

Lk

4

8

16

0.5

0.0

0

kc

N

Wavenumber (cycles/ground_unit)

L(k ) =

1

⎡ ⎛ k ⎞ n ⎤ ⎢1 + ⎜ k ⎟ ⎥ ⎣ ⎝ c ⎠ ⎦

Parameters: k 0

Central wavenumber of the filter.

n

Degree of the Butterworth filter function. By default, we assume a degree of 8.


0/1

Residual/Regional flag to specify if a residual high pass or a regional low pass is required. By default, the system applies a regional filter.

Gaussian Regio nal/Resid ual Filter (FFTGAUSS GX) The Gaussian filter is another smooth filter often used for low-pass or high-pass applications. 1.0

L(k) 0.5

0.0

L(k ) = 1 − e

0

k o 2k o 3k o Wavenumber (cycles/ground_unit)

N

2 ⎞ 2 −⎛ ⎜ k 2 k 0 ⎟ ⎝ ⎠

Parameters: k 0

Standard deviation of the Gaussian function in cycles/ground_unit (similar to a cutoff except that the function amplitude at this point is only 0.39).

0/1

Specify 0 or 1 for residual or regional, respectively.

Cosine Roll-off Filter (FFTCOSN GX) Because this filter has a smooth shape, and it does not alter the energy spectrum below the start of roll-off (or after the end of roll-off in high-pass mode), it is commonly used for simple high-pass or low-pass operations. To reduce ringing, you can increase the separation between k 1 and k 0. 1.0

L(k)

0.5

0.0

n=2

0

k0

1

0.5

k1


N


L(k ) = 1, for k < k 0

⎡ π ⎛ k − k 0 ⎞⎤ ⎜⎜ ⎟⎟⎥, for k 0 ≤ k ≤ k 1 2 k k ⎢⎣ ⎝ 1 − 0 ⎠⎥⎦ L(k ) = 0, for k > k 1 L(k ) = cos ⎢ n

Parameters: k 0

Low wavenumber starting point of the filter. (Cut-off wavenumber for high pass or start of roll off for low-pass.)

k 1

High wavenumber end point of the filter. (Start of roll off for high-pass or cutoff wavenumber for low-pass.)

n

Degree of the cosine function. The default is a degree of 2 for a cosine squared roll-off.

0/1

0 for residual (high-pass) filter; 1 for regional (low-pass) filter. The default is a low-pass filter.

Low pass Filter (FFTLOWP GX) As with the band-pass filter, you should use this filter selectively because it can suffer from Gibb's Phenomena (ringing). 1.0 pass

L(k)

reject

0.5

0.0

k0


L(k) = 1 , for k ≤ k 0 L(k) = 0 , for k > k 0

Parameter: k 0

Cutoff wavenumber in cycles/ground_unit. All wavenumbers above this value are removed.


Highpass Filter (FFTHIGHP GX) As with the band-pass filter, you should use this filter selectively because it can suffer from Gibb's Phenomena (ringing). 1.0 reject

L(k)

pass

0.5

0.0

k o Wavenumber (cycles/ground_unit)

L(k ) = 0 , for k < k 0 L(k ) = 1 , for k ≥ 0

Parameter: k 0

Cutoff wavenumber in cycles/ground_unit. Removes all wavenumbers below this value.

Bandpass Filter (FFTBANDP GX) You can use the Bandpass filter to pass or reject a range of wavenumbers from the data. However, applying such a simple cutoff filter to an energy spectrum almost invariably introduces a significant amount of ringing (otherwise known as Gibb's Phenomena). We recommend that you use a smoother filter such as the Butterworth filter (FFTBUTTW). 1.0

L(k)

0.5

0.0

reject

pass

reject

k0

k1


L(k ) = 0, for k < k 0 L(k ) = 1, for k 0 ≤ k ≤ k 1 L(k ) = 0, for k > k 1


Parameters: k 0

Low wavenumber cutoff in cycles/ground_unit.

k 1

High wavenumber cutoff in cycles/ground_unit.

0/1

If 1, pass the defined band. Otherwise, reject the defined band. The default is to pass the band.

Vertical Derivative (FFTVDRV GX) The vertical derivative is commonly applied to total magnetic field data to enhance the most shallow geologic sources in the data. As with other filters that enhance the high-wavenumber components of the spectrum, you must often also apply low-pass filters to remove high-wavenumber noise. L(ω ) = ω

n

Parameter: n

Order of differentiation.

Verti cal Integrati on (FFTVINT) This filter calculates the vertical integral of the input transform. This is the inverse of the vertical derivative. The zero wavenumber is set to 0. L(ω ) = ω -1

Horizontal Derivative (FFTHZDRV GX) You can use the horizontal derivative for identifying geologic boundaries in profile data. L(ω ) = (ω i )

n

Parameter: n

Order of differentiation.

Horizontal Integration (FFTHZINT GX) The Gradient Inversion (Horizontal Integration) filter calculates the horizontal integral of the input transform. This is the inverse of the horizontal derivative. The zero wavenumber is set to 0.


L(ω ) = (i ω ) -1

Downw ard Cont inu ation (FFTCONT GX) Downward continuation enhances the responses from sources at a shallow depth (by effectively bringing the plane of measurement closer to the sources). Note, however, that it is not possible to continue through a potential field source. If the data contains short wavelength noise, this noise can appear as very shallow sources in the continuation. Such noise should be removed before attempting to downward continue the data. A Butterworth low pass filter set to between 1 and 1.5 times the depth can be very effective to remove noise before continuation. You should make a plot of the energy spectrum to determine the wavenumber at which sources (noise) appears to be more shallow than the depth of continuation. The energy spectrum is also a good guide for determining the depth to which you can continue data downward. 20.0

h = 16

Lk

h=8 h=4 h=2

1.0 0

N


L(ω ) = e

hω

Parameter: h

Distance in ground_units, to continue down relative to the plane of observation.

Upward Continuation (FFTCONT GX) Upward continuation is considered a clean filter because it produces almost no side effects that may require the application of other filters or processes to correct. Because of this, it is often used to remove or minimize the effects of shallow sources and noise in grids. Also, you can interpret upward continued data numerically and with modeling programs. This is not the case for many other filter processes.

Chapter 3: Preparing Data and Applyin g Filters 25 1.0

h=2

Lw

h=4 h=8 h = 16 0.0

0

N


L(ω ) = e

− hω

Parameter: h

Distance in ground_units, to continue up relative to the plane of observation.

Magnetic Pole Reduction (FFTRPOLE GX) Reduction to the pole is used in low magnetic latitudes to change an anomaly to its equivalent as would be observed at the north magnetic pole. This transformation simplifies the interpretation and visualization of anomalies from low magnetic latitudes. The reduction to the pole is: L(θ ) =

1 (sin I a + i cos I ⋅ cos( D − θ )) 2

where I

Geomagnetic inclination

I a

Inclination for amplitude correction (never less than I)

D

Geomagnetic declination

Parameter I a

Inclination to use for the amplitude correction. Default is ± 20. ( I a = 20, if I >0; I a = (-20), if I <0 ). If |I a | is specified to be less then |I |, it is set to I .

Reduction to the pole has an amplitude component (the sin(I) term) and a phase component (the icos(I)cos(D-θ) term). When reducing to the pole from equatorial latitudes, North-South features can blow-up due to the strong amplitude correction (the sin(I) term) that is applied when D-θ is π/2 (i.e. a magnetic east-west wavenumber). By specifying a higher latitude for the amplitude correction alone, this


problem can be reduced or eliminated at the expense of unde r-correcting the amplitudes of North- South features. An amplitude inclination of 90 causes only the phase component to be applied to the data (no amplitude correction), and a value of zero causes phase and amplitude corrections to be applied over the entire range.

Apparent Suscepti bi li ty (FFTSUSC GX) The susceptibility filter calculates the apparent magnetic susceptibility of the magnetic sources using the following assumptions:

• • •

The IGRF has been removed from the data. There is no remanent magnetization. All magnetic response is caused by a collection of vertical prisms of infinite depth and strike extent.

A susceptibility filter is, in fact, a compound filter that performs a reduction to the pole, downward continuation to the source depth, correction for the geometric effect of a vertical prism, and division by the total magnetic field to yield susceptibility. L(k ,θ ) =

1 2π F ⋅ Η (ω ) ⋅ Γ(θ ) ⋅ Κ (k )

Η (ω ) = e −hω Γ(θ ) = (sin I a + i cos I ⋅ cos( D − θ )) 2 ⎛ sin (aω ) ⎞ Κ (ω ) = ⎜ ⎟ ⎝ aω ⎠ (ω)

Downward continuation to h

(θ)

Reduction to the pole

(ω)

Geometric factor of a vertical prism -- (2a ∞ ∞ in dimension)

Parameters: h

Depth in ground_units, relative to the observation level at which to calculate the susceptibility.

I a

Pole reduction amplitude inclination. Inclination to which to use the phase component only in the reduction to the pole. The default is ± 20. If |I a | is specified to be less then |I |, it is set to I .

I

Geomagnetic inclination

D

Geomagnetic declination


F

Total geomagnetic field strength

Apparent Densit y (FFTDENS GX) This filter calculates the apparent density of the ground that would give rise to the observed gravity profile. The density assumes that the gravity profile is due to a set of rectangular prisms with a top at the level of observation of the gravity profile, a bottom at depth t , and infinite strike length. You may wish to downward continue the profile to be close to the tops of the assumed geologic model of interest before calculating the apparent density. You must supply the thickness of the earth model and the background density. L(ω ) =

2π G (1 - e - ω ) t

Parameters: t d G

Thickness, in ground_units, of the earth model. Background density in g/cm3 Gravitational constant.

Multiple Filters (FFT1MULT GX) The Multiple Filters option enables you to apply multiple 1D Fast-Fourier Transform filters to a channel. The Multiple Filters option allows multiple filters to be applied to a channel with a single process, in which, the data firstly get Fourier Transformed (FT) to the wavenumber domain, then multiple filters are applied (simple multiplications), and the data finally are Inverse Fourier Transformed (IFT) back to the space domain. Note that in the 1DFFT specific single filter GX (fftvdrv GX, fftbuttw GX, etc…) the application of each filter requires its own FT-filter-IFT process. This tends to add numerical noise to the data as the number of filters is increased. Using this GX for multiple filters will avoid this problem. In this GX, along with the input and output channel names, the user can select up to six filters to apply. A separate dialog is displayed for each filter type specified, in which the user is prompted to enter values for the parameters relevant to that filter. For a non interactive process application, the script parameters for each filter type specified can be referred to the relevant 1DFFT GXs. Information for each filter type is available by clicking the [Help] button on the individual filter parameter dialogs.


Analyti c Signal (FFTAS GX) The Analytic Signal option calculates the analytic signal of a channel. The analytic signal can be useful for locating the edges of remanently magnetized bodies and in areas of low magnetic latitude (Macleod et al., 1994). The analytic signal (as) of a profile is defined as: as =

dz ⋅ dz + dx ⋅ dx

where dz

is the vertical derivative

dx

is the horizontal derivative

The vertical derivative is calculated using the FFT process described later in this chapter. The horizontal derivative is calculated by applying a space domain convolution filter. The analytic signal is then evaluated from these two sets of data.

Hilbert Transfor m (HILBERT GX) The Hilbert Transform option does the Hilbert Transform by the means of FFT based on the following known relation: F[ H[f(x)] ] = -i sgn(w) F[f(x)]

(equation 7 of the reference paper list below)

Where: F[f(x)] is the Fourier transform of f(x), H[f(x)] is the Hilbert transform of f(x), and Sgn(w) = w / |w| = +1 for w>0, = 0 for w=0, = -1 for w<0. Firstly, the GX does forward FFT transform of the input channel. Three output channels are created (this is only for the none-array input channel case). They will have the same name as the input channel but extension "_r" and "_i" for real and imaginary components of the transform, and "_w" for the wavenumber in radians/fiducial. (Note that the trend has been removed before FFT.) Note that the output values are the real and imaginary components of the positive frequencies of the transform. Since we are dealing with the real-valued space domain problem, the negative part of the spectrum is simply the conjugate of the corresponding positive part, i.e h(-f) = [h(f)]*, and is not included in the output. The fiducial number will be in cycles/fiducial. The wavenumber channel will be in radians/fiducial.


Then, the one-dimensional Hilbert transform operator –i sgn(w) is applied to the FFT transformed data. Finally, the GX does the inverse FFT transform to obtain the Hilbert transform results into the output channel. For the real data practice, it is suggested to remove trend line based on all data points (the default) before FFT process to prevent the discontinuity from the data two edges. The removed trend will be replaced back in the same manner as it removed after FFT. For the real data practice, it is also suggested to expand the data 10% before FFT process to prevent the discontinuity from the data two ends. The GX will first extend data by the user required % points, then further extend to the number of the power of 2 for the FFT process. For instance, if the original data contains 60 points, then the values will be padded with 10%, or 6 points at the end, giving 66 points. This will then be extended to the next power of 2, or 128 points, to do the FFT. The extended area will be interpolated by Maximum Entropy Prediction (MEP) method. MEP samples the original data points to determine its spectral content. It then predicts a data function that will have the same spectral signature as the original data. As a result, the predicted data will not significantly alter the energy spectrum that would result from the original data alone. However, for particular synthetic data test the set of “remove mean value” trend line removal option and 0% expansion may obtain accurate results. Reference Paper: “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations”, Misac N. Nabighian, Geophysics, Vol. 49, No. 6, pp. 780-786, 1984.

Power Spectrum (FFTPSPEC GX) The Power Spectrum option will calculate the log power spectrum of a channel. Note that, the power spectrum will be a log base 10 of the power. The fiducial increment will be cycles per original fiducial. The profile is extended to a power of 2 times the length and the wavenumber increment will be 1/(length * fid_increment).

30 Chapter 4: Applying Custom Filters

Chapter 4: Applying Custom Filters The montaj Geophysics system also enables you to work with the transform directly so that you can apply your own custom filters. The Advanced Usage menu provides users with step-by-step 1D-FFT data processing sub-menus. For information on the 1D-FFT filtering processes see the Understanding the 1D-FFT Filtering Process page 10. These menu items are designed for advanced users who want to apply custom filters or math expressions to their data.

Preprocessing previewer FFTPREP.GX Re-fid channel to distance base DISTFID.GX Transform from space to wavenumber domain. FFTIN.GX Transform from wavenumber to space domain. FFTOUT.GX Re-fid channel to fiducial base REFID.GX

For detailed information on these or any of the 1D-FFT dialogs, click the [Help] button. The FFTIN GX (FFT1D|Advanced Usage|FFT space --> Fourier menu item) transforms a channel to the Fourier domain and creates a real and imaginary channel and a wavenumber channel. You can then apply filters with the real, imaginary and wavenumber channels to create new real and imaginary channels. The FFTOUT GX (FFT1D|Advanced Usage|FFT Fourier --> Space menu item) transforms the filtered real and imaginary channels back to the space domain and masks the result against the original channel. The channels created by FFTIN have the same name as the original input data channel with the suffixes “_r”, “_i” and “_w” added. These are the real, imaginary and wavenumber channels respectively. The wavenumber channel is in units of radians/fiducial (2π cycles/fiducial). The fiducial numbering of the Fourier domain channels is in cycles/fiducial. Note that the FFTIN process does not remove a trend from the data. If your data contains an undesired trend, you must remove it before running FFTIN. It is also your responsibility to replace a trend in the data after processing. The process of applying your own filter is best illustrated through an example. Say we have an existing channel named “Mag” which is already sampled on a distance base (the fiducials are ground_units). We would like to calculate the first vertical derivative using the expression:

Chapter 4: Applying Custom Filters 31

L(ω ) = ω

n

where n is the order of differentiation (in this case, n=1). To apply this expression we must multiply the Fourier transform by the wavenumber: 1. On the X-Utility | 1-D FFT menu, select Advanced Usage|FT space -->Fourier and select “Mag” as the channel to transform. This creates channels “Mag_r”, “Mag_i” and “Mag_w”. 2. Select the entire “Mag_r” channel in the database (click three times on the “Mag_r” channel header). 3. Press the “=“ (equal sign) to enter an expression, and enter: “Mag_r*Mag_w” The “Mag_r” channel is replaced by the filtered real data. 4. Select the entire database for the “Mag_i” channel. Press “=“ to enter an expression, and enter: “Mag_i*Mag_w”. The “Mag_i” channel is replaced by the filtered imaginary data. 5. Select FFT Fourier --> space... and choose the “Mag_r” and “Mag_i” as the input channels, “Mag” as the original reference channel, and “dz” as the output filtered channel. The “dz” channel will contain the vertical derivative in nT/ground_units.

32 Glossary

References Bhattacharya, B. K ., 1966, Continuous spectrum of the total magnetic field anomaly due to a rectangular prismatic body. Geophysics, Vol. 31, p.97-121. Burg, J. P. , 1975, Maximum Entropy Special Analysis. Unpublished doctoral dissertation. Stanford University. 168p. Claerbout, J.F., 1988. Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting, Blackwell Scientic Publications, Palo Alto, California, U.S.A. Gupta, V. K., and Grant, F. S. , 1985, Mineral exploration aspects of gravity and aeromagnetic survey in Sudbury-Cobalt area, Ontario. SEG; The Utility of Regional Gravity and Magnetic Anomaly Maps, W. J. Hinze (Editor) p.392-411. MacLeod, I. N., Vierra, S. and Chaves, A. C. , 1993, Analytic signal and reduction to-the-pole in the interpretation of total magnetic field at low magnetic latitudes. Proceedings of the Third International Congress of the Brazilian Geophysical Society, November, 1993. McClellan, J. H. and Nawab H. , Complex General-N Winograd Fourier Transform Algorithm (WFTA), Programs for Digital Signal Processing, IEEE Press, pp . 1.7-1 1.7-10, 1979. Spector, A. and Grant, F. S. , 1970, Statistical models for interpreting aeromagnetic data. Geophysics, Vol. 35, No. 2, p.293-302. Wiener, N. , 1949, Extrapolation, interpolation, and smoothing of stationary time series. Cambridge, M.I.T. Press. Winograd, S. On Computing the Discrete Fourier Transform, Mathematics of Computation, Vol. 32, N0.141, pp. 175-199, Jan. 1978. Claerbout, J.F., 1988. Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting, Blackwell Scientic Publications, Palo Alto, California, U.S.

Glossary 33

Glossary Ar ray ch ann els

Special channels in columns of the spreadsheet that contain multiple channels of data. Array channels are represented graphically by a curve in the spreadsheet. See also c hannels, sub-array channels.

channels

In the Oasis montaj spreadsheet, a channel is essentially a column that contains a specific type of data.

column

In the Oasis montaj spreadsheet, a vertical line of cells that contain data.

database

See Oasis database

desktop

Background area in the Oasis montaj project. You can open and display a virtually unlimited number of Spreadsheet, Profile and Map windows in this area.

fiducials

Points accepted as fixed bases of reference. A mark that indicates the order in which each reading or sample reading was taken.

Graphical User Interface

Interactive software environment where functions are performed by selecting graphic objects.

grid

Collection of points along rows and columns that define a two-dimensional rectangular area on some plan, usually a ground plan.

Grids or Grid file

Files containing location (X and Y) and data (Z) values. Values are typically interpolated to create a regular and smoothly sampled representation of the locations and data.

groups

A set of graphics elements that make up a graphic component of the map. For example, a line path plot, a contour plot or a profile plot would all be separate graphics groups within the Data View.

GX or Geosof t eXecutable

Programmed process (identified by the *.GX file extension) used to process data in Oasis montaj.

Images or Image file

Files containing location (X and Y) and color values. The values are not interpolated. Standard PC file types created using imaging or electronic photo-editing techniques.

line

Linear array of observation points.

Main window

Primary tool used to create and maintain databases, display data and process data. Oasis montaj is a Graphical User Interface (GUI) system that provides all functionality required to process and display virtually any type of Earth Science data.

Map (*.MAP)

Geosoft-developed file that integrates all graphics elements (lines, polygons and text) and layers (base maps, data, grids, plots and images) constructed in Oasis montaj.

Mon Taj Geophysics

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