استفاده ازتابع تیپر برای تهیه نقشه رسانندگی الکتریکی ساختارهای زیرزمینی در برداشتهای مگنتوتلوریک

دسته ژئوفیزیک
گروه سازمان زمین شناسی و اکتشافات معدنی کشور
مکان برگزاری بیست و ششمین گردهمایی علوم زمین
نویسنده علی مرادزاده
تاريخ برگزاری ۰۱ آبان ۱۳۸۴

Abstract
    To construct a detailed electrical image of the subsurface structures using MT, three orthogonal components of the magnetic field in x, y, and z directions and also two components of the electric field in x, and y directions are simultaneously measured for different frequencies at several points along a profile. Then a complex MT impedance tensor, that relates the measured electric and magnetic fields, is used to calculate apparent resistivity and phase data at each frequency.  Beside information of the MT impedance tensor, and based on the relationship between the vertical and horizontal magnetic field, another frequency dependent complex tensor called tipper or magnetic transfer function can be calculated. It can be used to better delineate the subsurface conductive anomalies. In this paper it is shown how the magnitude and direction of the tipper function can be used to better map the locations of underground conductive anomalous zones.
Keywords: MT, electrical image, magnetic transfer function, tipper, impedance tensor
 
 
 
Introduction
In MT exploration three orthogonal components of the natural source magnetic field in x,y, and z directions and two components of the induced electric field in x, and y directions are simultaneously measured at a point on the earth’s surface.  For a two-dimensional (۲D) structure, the relationship between electric and magnetic field at a single MT site is then established as follow:
                                                                        (۱)
where Z(w) = is a complex matrix termed MT impedance tensor of
each frequency (Cantwell, ۱۹۶۰; Zonge and Hughes, ۱۹۹۱). For ۲D lateral inhomogeneities if one of the measurement axes (x, y) is along the strike, then
                                           Zxx = Zyy = ۰   and     Zxy ¹ Zyx        (۲)
otherwise, none of the impedance elements is zero.
Since the strike direction of an anomaly is seldom known at the time of a field survey, the field sensors are oriented along arbitrary axes (x, y).  To determine the strike of the conductivity structure an attempt is made to eliminate diagonal elements of the impedance tensor.  However, if the conductivity tensor is not symmetric or some noise is present, these elements do not vanish.  In such cases an attempt is made to minimize the diagonal elements.  To achieve this the measured impedance tensor is rotated (Swift, ۱۹۶۷; Kaufman and Keller, ۱۹۸۱) through a positive angle q as follows:
                                            Z¢(q) = R Z RT                               (۳)
where R= is rotation matrix, RT is it’s transpose, and q is the rotation angle in the clockwise and is chosen such that Z¢(q) takes approximately zero-diagonal form, i.e.  
                                            Z¢(q۰) =                        (۴)
The off-diagonal elements of Z¢(q۰) are known as the principal components of the impedance tensor. With this rotated impedance tensor we have the following equations for the apparent resistivity and phase data in x and y directions:
 
            r¢xy = ۰.۲ T êZ¢xy ê۲    ,     j¢xy = tan (Im Z¢xy / Re Z¢xy )
            r¢yx = ۰.۲ T êZ¢ yx ê۲    ,    j¢ yx = tan (Im Z¢yx / Re Z¢yx )     (۵)
 
To obtain an image of the subsurface electrical structures, MT measurements are normally done at several stations along a profile and after a series of mathematical calculation the apparent resistivity and phase data are extracted from impedance tensor.  Due to the effects of the near-surface inhomogeneities on the apparent resistivity data (Moradzadeh, ۱۹۹۷,۱۹۹۸a), and based on the relationship between the vertical and horizontal magnetic field, another frequency dependent complex tensor called magnetic transfer function or tipper is calculated. It is described below how the magnitude and direction of the tipper function can be extracted from MT data.
 
Magnetic transfer function or Tipper
In ground with lateral inhomogeneity like a ۲D conductor- resistor boundary, the electrical currents concentrate on the conductive side and according to the Ampere’s law an anomalous (vertical) magnetic field is created.  In such cases the relationship between vertical and horizontal magnetic field components at each frequency (Parkinson, ۱۹۶۲; Sims and Bostick, ۱۹۶۹) is given by
                                       Hz = Tzx Hx  + Tzy Hy                (۶)                                         
where Tzx and Tzy are complex, frequency dependent parameters called the vertical magnetic field transfer function or tipper because of the tilt of the magnetic field from the horizontal plane (Vozoff, ۱۹۷۲).  In a ۲D structure with strike in the x direction Hz = ۰ for the Transverse Magnetic (TM) mode, so equation (۶) for Transverse Electric (TE) mode simplifies to
                                              Hz = Tzy Hy                       (۷)
In the general three-dimensional (۳D) case the magnitude and direction of tipper are both rotationally invariant and defined (Jupp and Vozoff, ۱۹۷۶) as:
                                  çT ç= (çTzx ç۲  +  ç Tzy ç۲ )۱/۲             (۸)
in which Tzx = a + jb   and Tzy = c + jd.  The tipper direction can be written
                     
                          fT =            (۹)
 
and for a  ۲D earth it simplifies to
                                     fT = tan(Tzy  / Tzx )                                      (۱۰)
As Hz is weak, typical tipper values are less than one.  Therefore, care should be taken during the field measurement to ensure a reasonable signal to noise ratio is attained.
Since the tipper direction points to the inhomogeneity, it can be used to determine its strike direction.  A convenient way to interpret the tipper is by means of induction arrows (Parkinson’s arrow). The magnitude and direction of these arrows can be computed from expressions (۸) and (۱۰).  Since the tipper is a complex quantity its magnitude and direction are defined for both real and imaginary parts.  Conventionally, the direction of real induction arrows is reversed (Parkinson, ۱۹۶۲; ۱۹۶۴) so that they point towards conductive features.  The tipper direction (fT) is measured positively clockwise from the local geomagnetic meridian and therefore varies in the range of -۱۸۰۰ to ۱۸۰۰.
In the following section it is shown how the tipper data can be used to refine subsurface conductive anomalies mapped by resistivity and phase data.
 
Usage of tipper function on real data
The pseudosections of TE mode data along a ۱۰۰ km E-W profile in South Australia (Moradzadeh, ۱۹۹۸b) are shown in Figure ۱.  The apparent resistivity pseudosection shows there is a conductive zone under WEI and YAT and another conductive zone beneath the eastern sites (i.e. PIC and BEN) at frequencies less than ۱Hz. The presence of these conductive zones are supported by high phase values in the phase pseudosection, however, phase data propose the eastern conductive zone is stretched under FRN site.  The existence of such anomalous zones and also their extension toward FRN can be checked further by tipper data.  Figure ۲ gives a pseudosection for the real part of the tipper function (i.e. rTzy) along the above profile.  Any sharp gradient in rTzy indicates lateral conductivity contrast and the location of the induced current concentration.  The rapid change of the rTzy under YAT (at high frequencies) and WEI at low frequencies marks the conductive zone between WEI and YAT.  The sharp gradient of rTzy beneath PIC (y=۲۰۰ km) which persists over a large frequency band (>۰.۰۱ Hz) and its low frequency (f <۰.۰۱Hz) variation between RUS and FRN (y=۱۷۰ km) all indicate that a conductive zone must be present in this location. 
The magnitude and direction of the real part of tipper function were calculated using equations ۸ and ۱۰ and after reversing of their directions they were presented in Figure ۳.  It can be seen that for frequencies less than ۱Hz the arrows of the site YAT and those of the MOU site are consistently towards where the western conductive anomaly is located. The great direction consistency of these arrows suggests that this anomaly is ۲D and elongated in the NS direction.  It also clear all arrows of beneath BEN are directed towards west whereas for frequencies more than ۱Hz all arrows under FRN are directed to the east where the second anomaly is located under BEN and PIC sites. The direction of the arrows under RUS for frequencies less than ۰.۱ Hz indicate that the eastern conductive zone is stretched under FRN site as phase data also indicate. 
 
Conclusion
In this paper it has been shown that the tipper or magnetic transfer function data are very important and must be considered in interpretation of MT data for imaging of the conductive anomalous zones. This complex function, relates vertical and horizontal components of the naturally occurring magnetic field measured at the earth’s surface. The tipper function is defined for each frequency and its magnitude and its direction together with the apparent resistivity and phase data could be used to acquire a detailed image of subsurface resistivity structures.   

چکیده
برای بدست آوردن تصویر دقیق الکتریکی ساختارهای زیر زمینی بکمک روش مگنتوتلوریک، سه مولفه عمود بر هم میدان مغناطیسی ،  در راستای x  ، y   وz  ،  و دو مولفه میدان الکتریکی القایی ،  در جهت x  وy   ، به طور همزمان و در چند محل روی یک پروفیل  برای بازه فرکانسی خاص اندازه گیری می شوند.  پس از اندازه گیریها ، تانسور مقاومت ظاهری (امپدانس) که یک عدد مختلط است برای هر فرکانس تعیین گردیده واز روی آن داده های مقاومت ویژه ظاهری و فاز محاسبه می گردند.  علاوه بر تانسور مقاومت ظاهری ، وبر اساس روابط بین مولفه های قائم و افقی میدان مغناطیسی اندازه گیری شده ،  تابع مختلط دیگری تحت عنوان تیپر یا تابع انتقال مغناطیسی می تواند برای هر فرکانس تهیه و از آن برای تعیین محل بهتر بی هنجاریهای رسانا استفاده کرد.  در مقاله حاضر ضمن شرح روش محاسبه تیپر در برداشتهای مگنتوتلوریک چگونگی استفاده از آن برای اکتشاف بهتر و دقیقتر بی هنجاریهای الکتریکی زیر زمینی همراه با یک مطالعه موردی  بیان خواهد شد.
کلمات کلیدی:  مگنتوتلوریک ، تصویر الکتریکی ، تیپر ، تانسور مقاومت ظاهری ،  تابع انتقال مغناطیسی
 

کلید واژه ها: میدانمغناطیسی مقاومتویژه مگنتوتلوریک ژئوفیزیک سایر موارد