5.3.3 The general solution for plane waves incident on a layered halfspace. The general solution to the Helmholz equation in rectangular coordinates

Transcription

1 5.3.3 The general solution for plane waves incident on a laered halfspace The general solution to the elmhol equation in rectangular coordinates The vector propagation constant Vector relationships between vectors,, and Properties of uniform plane waves, characteristic impedance Reflection and transmission of plane waves at a plane interface T mode reflection and transmission at a plane interface TM mode reflection and transmission at a plane interface Reflection and transmission of plane waves incident at real angles of incidence Surface impedance formulation of reflection and transmission The general impedance of an n-laered half-space The general solution to the elmhol equation in rectangular coordinates In section 5. we derived the elmholt equation in either magnetic or electric fields. In the simple description of the MT method in section 5.3. we looed at a simple solution for the elmholt equation in when the horiontal derivatives were ero and there was onl one horiontal component of the field. ere we will loo at the general solution to the vector elmhol equation where there are no such simplifing assumptions. The behavior of magnetic and electric fields at the surface of a laered half-

2 space of arbitrar conductivit and dielectric in the laers as a function of frequenc and the angle of incidence of the field on the surface. This general treatment will consider the orientation of the electric and magnetic fields with respect to the interface and the impedance and reflection coefficient for such fields. We will use a right handed rectangular coordinate sstem with positive down. For an rectangular component of the field, sa, the elmhol equation is: 0 where we have used an i t e time dependence. [from here on the dependence is implied and will not be denoted eplicitl]. A classic method of solution is separation of variables: assume that the solution can be written as the product of three independent solutions in each coordinate vi: X Y Z Substituting this solution in the elmhol equation ields three independent ordinar equations: d X X d d X Y d d X Z d

3 and the condition that 0 implies that there are onl two independent equations since X Z Particular solutions to these ordinar equations are: so the general solution is: i i i X Ae, Y Be, and Z Ce i i i,,, e e e This is a Fourier transform and in fact we get to this solution more elegantl b simpl starting all over and taing the Fourier transform of the elmhol equation. Using the following Fourier transform pair: i e d i e d and appling the first directl to the elmhol equation, in other words taing the spatial transform, ields an algebraic equation in transform space:,,,,,, 0 or 0. The i are not independent so the inverse transform bac to,, space becomes: 3

4 i i i,,, 4 e e e Before going on to appl the general solution to particular problems, we will eamine the particular solutions and find out the phsical significance of the i parameters. The Vector propagation constant First, to simplif the ensuing algebra, we will consider a two dimensional model of the arth-air interface and a field above the interface that has no variation in, so is ero. [Over a uniform half-space the coordinates can alwas be rotated such that ais coincides with the direction in which the field is invariant]. Consider the particular solution for a component of the electric field in the direction: i i i 0 0 e e e The braceted term in the eponent has the form of a vector dot product of a position vector r i and a vector propagation constant i where i, j and are unit vectors in the, and aes respectivel; this solution has the form i r 0 e The following setch illustrates these vectors in two dimensions: 4

5 Figure ere we can identif and as the rectangular components of a vector maing an angle with the vertical,, ais. In this illustration and can be interpreted as the rectangular components of, sin and cos impling values limited to. But be careful because in the general solution or can have values far greater than suggesting meaning for that is not intuitive. We return to these solutions later. A small digression at this point will help eplain wh, in later sections, we will stud things lie reflection coefficient and surface impedance in terms of. If we consider the full solution for a field, e iwti, and consider where a field of constant phase moves in space and time we have: d t a constant. And so V, the phase velocit is in the dt direction. From the relation that wavelength,, V f, we find that such a 5

6 wave has a spatial wavelength of f. So a wave propagating in the direction has an implicit horiontal wavelength. The general solution, which is an integral over all is made up of the superposition of waves of different horiontal wavelength. We will see how the reflection and refraction, and surface impedance, of such a wave depends strongl on. Vector relationships between, and We can now show the vector relationships between the fields and the vector propagation constant b taing the Fourier transform of the individual Mawell equations. For eample the full as: equation can be written out in i i i Taing the Fourier transform of each of these ields:,,,,,,,,,,,,,,,,,, i i i i i i i i i 6

7 Since, and and are now nown as components of a vector this set of equations can be written in a much more compact form: and the same process ields the second curl equation i Finall the divergence equations 0 and 0 and Ñ = 0 simpl transform to: 0 So for this propagating plane wave we can see that and must alwas be perpendicular to one another and that both are perpendicular to the vector direction of the propagation constant. These simple rules show that there can never be component of the field in the direction of propagation of a uniform plane wave field. 0 Properties of uniform plane waves To simplif the algebra involved let s consider the nature of a wave propagating in the direction. Since solutions for each component are simpl additive we can illustrate properties b choosing an component. First lets consider the solution for i.e. a solution to 0 7

8 is 0e ii t and from Farada s law i i so the ratio of the electric to magnetic field for a plane wave propagating in the positive direction is the characteristic or intrinsic impedance, denoted here b and is equal to: Note a couple of things about the characteristic impedance. We could have started this process from the elmhol equation in 0 vi. The solution is for a wave travelling in the + direction is 0e and now from Farada s law i i., so The component chosen affects the sign of the intrinsic impedance. iit If we had chosen a wave travelling in the negative direction the signs of the derivative would have reversed and so and Finall some sharp eed reader ma have tried to get to these relationships b starting with the elmhol equation in, sa, The solution is 0e ii t 0 vi. and now from Farada s law 8

9 i i And so i i which doesn t loo much lie. But wait, with a little multipling top and bottom and factoring watch what happens: i i i i i i i ither definition of the intrinsic impedance is o. Now let s loo at the propagation characteristics for various values of,, and. In general is comple, that is i i. Re Squaring both sides and equating real and imaginar terms ields the following relations: Im Re Im and Re Im. After a tedious bit of algebra we solve for Re and Im : After a tedious bit of algebra we find the following equations for R and I : Re 9

10 and Im There are three limiting cases to be considered. ) If 0 (free space) then Im 0 and Re ) If then Re Im 3) If, then is small and, [based on when ] In this high frequenc limit Re, the same value as in a nonconductive medium and Im. The sin depth is therefore. This last result is ver important because it means that in a conductive medium at ver high frequencies the sin depth becomes independent of frequenc. The sin depth ma be ver small but it depends onl on the 0

11 conductivit, dielectric permittivit and magnetic permeabilit. The phase velocit on the other hand is independent of the conductivit. We will return to these results in section 5. on Radar. There are a few details left to discuss about propagation in free space. Again we will consider an field propagating in the positive direction the form for which is e 0 iit Now let s consider this solution for fields in free space where 0 and so is real and equal to 0 0. The complete solution is: i 0 0t 0e This is a true propagating wave. A point of constant phase is defined b t = constant 0 0 and d dt V ph 0 0 the phase velocit ( in + ). The constants and ield a phase velocit of which is the velocit of light in free space. This remarable derivation is due to Mawell who predicted it from first principles far ahead of its accurate eperimental determination. The characteristic impedance of free space is: Ohms. 0

12 For completeness the basic solution we derived in section 5.3. for an component propagating in a good conductor where here. is repeated 0 the solution for which is (including the time dependence): e 0 i ti The sign must be chosen so that the fields do not grow eponentiall with. In this case we choose e -i so that on epanding we get the solution: e e 0 And we defined the sin depth,, as i t f 4 0 f We now have all the basic definitions, formulations and propagation characteristics to address what happens when a propagating field in one medium is incident on another medium. Our major interest in in plane waves incident on a laered earth. The schematic diagram below shows an incident field approaching a horiontal interface between medium and medium with a propagation vector. The position vector lies in the interface and the unit vector is normal to the interface and is b our coordinate convention positive down. We will assume from eperimental results that there is energ reflected from the

13 interface and some energ ma be transmitted across the interface into the medium. Figure We have drawn this diagram in the conventional wa used in most tets to show the incident field as a ra approaching the interface at an angle. Remember though that is a vector composed of an component and a component where and are can have values far higher than the geometric interpretation cos or sin. As drawn the diagram represents a field incident at what is termed a real angle of incidence. In the derivations that follow we will consider the general case of arbitrar and and highlight the case of real angles of incidence as we go. We define the plane of incidence as the plane containing and the unit normal to the surface n. Before going on to derive the amplitude relations of the reflected and transmitted fields we can derive some important properties of the fields at an interface from their vector representation. We have prescribed and incident, 3

14 reflected and transmitted field, sa vector electric field, b i r i r i r, and e e e respectivel. actl the same form applies to the vector magnetic field. The boundar conditions are that the tangential components of the field must be continuous across the interface. Thus, on the interface the sum of the tangential incident and reflected fields must equal the tangential transmitted field. First consider the incident field to have onl a component, i.e. onl a tangential component. Then the boundar condition ields: i r i r i r e e e This condition must hold for an position, r, in the interface and this can onl be assured if ; r r r. Now suppose that r locates an arbitrar point on the interface, it is on the horiontal plane in the above setch. Now choose another vector t also ling in the interface so that r can be written as r t n. Substituting this form for r in the above equation and noting the vector identit t n t n we arrive at: n n n This is the general form of Snell s Law. An immediate consequence of this relationship is that all the propagation vectors must lie in the same plane, the plane of incidence. Also, in the plane of the interface, is ero so 4

15 so again if r r r r and this holds for an point in the interface, an, then we have the important result that. To put this general form of Snell s Law in its familiar form we return to the representation of the incident fields in the above setch where the propagation vectors of the incident reflected and transmitted fields lie at angles, and with respect to the normal to the interface n. Because n n n can now be written Since incidence. sin sin sin then so the angle of reflection is equal to the angle of Further in non conducting media so Snell s Law can be written in the familiar form: and the phase velocit,v, is sin sin V V Another important result for plane waves in free space incident at a real angle to a conducting ground can be found from sin sin or sin 5 sin If >> then sin 0 and 90 no matter what the incident real angle. The refracted wave propagates verticall even for incident fields propagating horiontall. This condition applies for a range of beond.

16 For the wave in medium to propagate verticall and since then in terms of horiontal wavelength in the incident medium. In a practical case where the ground has a resistivit of 00 Ohmm and at,000 we find that the horiontal wavelength must onl be greater than ~700 m for the wave to be refracted verticall. This implies that if the incident field is actuall produced b a finite nearb source then it is simpl necessar to ensure that the bul of the integral for the fields comes from horiontal wave lengths greater than 700 m. This is a fundamental statement about using controlled sources to simulate MT results in sstems lie CSAMT and Stratagem. Reflection and refraction of plane waves at a plane boundar We now have some basic properties of incident, reflected and refracted fields and we will now derive the amplitudes of the reflected and refracted fields, and in the process determine the surface impedance of a laered media. An incident plane wave field has the electric and magnetic field vectors orthogonal in the plane that is perpendicular to the direction of propagation. The field vectors can alwas be broen into two component one of which is perpendicular to the plane of incidence and consequentl parallel or tangential to the interface. Since the boundar conditions are on the tangential components it is convenient to set up the reflection refraction 6

17 problem in terms of electric or magnetic field components that are perpendicular to the plane of incidence. This field decomposition has two important definitions: () An incident wave with electric field normal to the plane of incidence is called the T mode. The incident wave has an electric field transverse to the plane of incidence. () An incident wave with magnetic field normal to the plane of incidence is called the TM mode. The incident wave has an magnetic field transverse to the plane of incidence. We will see that the amount of incident field that is reflected or transmitted depends on the angle of incidence and on the mode. T mode reflection and refraction at a plane boundar We will assume that the coordinate sstem can alwas be rotated such that the plane is the plane of incidence. The T mode then has onl a component of and the TM mode has onl a component of. [Note that an incident field with and in arbitrar directions can alwas be broen down into a field with an component and an component perpendicular to the plane of incidence, solved for each mode separatel, and combined for the total field in the reflected and transmitted fields.] We will set up the problem for the component of,, The incident, reflected and transmitted fields are i r i r i r, and e e e The tangential continuous boundar condition simpl ields: 7

18 From n B i i and that the tangential component of t we find that (dropping the subscript): n n n Using the vector identit A BC BC A AC B and noting that 0 n n n because is perpendicular to the plane of incidence and therefore perpendicular to n, this boundar condition reduces to: n n n Using both boundar condition equations and solving for and (and noting that n i n ), we find that: n n n n and n n n These are the Fresnel equations in vector form for the reflected and transmitted field amplitudes for the T Mode. The transmitted field is 8

19 called the refracted field in studies of light. The ratios of and reflection coefficient and transmission coefficient and t respectivel. are the Noting that n and that above Fresnel equations into: and we can epand the and similarl TM mode reflection and refraction at a plane boundar Following the same derivation used above for the T mode we can obtain the Fresnel equations for the TM mode: now is perpendicular the plane of incidence and is in the plane of incidence. and 9

20 The amplitude reflection coefficients are defined for each mode as r T and rtm. It is quite clear that these reflection coefficients are different for each mode, and that the reflected (and for that matter the transmitted) fields can have a different phase than the incident field. This can be simpl illustrated with the limiting case of normall incident fields, i.e. when 0. For added simplicit let. Then r T and rtm So in this simple situation we see that the sign of the coefficient is different for each mode. This has particular significance when the incident field is in free space and and the field is incident on a conducting halfspace for which i. In this situation, which is tpical for most earth conductivities and frequencies less than a megahert, and so. The reflected field is reflected in phase and is essentiall the same amplitude as the incident field. This means that the total magnetic field on the ground surface is ver close to twice the amplitude of the incident field and is independent of the ground conductivit. On the other hand so the electric field is reduced almost to ero. Just how small the net surface electric field is can be found from the transmission coefficient: 0

21 For simplicit suppose and assume normal incidence, i.e. 0, then and if then i 4 i e The net surface field has a phase shift of 45 degrees and for tpical ground conductivit of 0.0 S/m and for an angular frequenc of,000 (f = 69 ) the net field is reduced b.88 *0-3. Neither result should be surprising since these are eactl the relationships that hold in the limit at the boundar of a perfect conductor. What is important to note about this result is that the small residual electric field on the boundar is ver sensitive to conductivit. In all geophsical measurements emploing incident plane wave fields it is the electric field that has information about the conductivit of the ground. The graphical window associated with this section has an option to calculate the T and TM reflection coefficients for arbitrar for an values of.

22 Reflection and refraction of plane waves at incident on the ground at real angles of incidence When the plane wave is incident at a real angle of incidence the Fresnel equations can all be rewritten in terms of sines and cosines of the angle of incidence. For the T mode the reflection and refraction coefficients become: cos sin r T cos sin t T and for the TM mode the are: and r TM cos cos sin cos sin cos sin t TM cos cos sin The graphical window associated with this section has an option to calculate the T and TM reflection coefficients for arbitrar for an values of or.

23 Surface impedance of a laered ground The practical problem with plane waves from distant sources is that we have no information about the incident electric and magnetic fields themselves-we onl have the total fields on the interface. We have seen that the observed magnetic field is twice the incident field and that it is the electric field, while much reduced, that is sensitive to the ground conductivit. It would seem that measuring the electric field would provide the information needed. But the incident fields are essentiall random in amplitude so the measured amplitudes var widel in time and cannot be related directl to ground conductivit. The solution is to reference the electric field to the magnetic field b measuring the ratio of orthogonal tangential and fields on the ground surface. We defined this ratio earlier as the surface impedance and we will briefl re-derive the reflection and transmission coefficients in terms of the characteristic and surface impedances, and then derive epressions for the surface impedance of a laered medium. We will adopt a revised notation to be compatible with the n-laered programs in the graphical windows of this section. First consider again a T mode solution for an arbitrar medium i: i i i e e i i i for a wave travelling in the positive and direction. For a wave travelling in the negative direction a solution is: i i i e e i i i The component of the accompaning magnetic field is obtained from i so 3

24 i i i i i i i The ratio, is the characteristic impedance, it = i. For a wave i it it travelling in the negative direction. i For the TM mode we have the solution in ; and from i i i e e i i i i i we derive the characteristic impedance for the TM mode to be; itm ii i. i We can now set up the interface problem in terms of impedance. Let the free space above the interface be denoted now b the inde 0 and the ground b inde. At =0, our solutions for and have to satisf the boundar condition that each is continuous at the interface. This leaves the two equations: T 0T T ZT We have introduced an important concept in recogniing that on the interface the total and fields define the surface impedance. For this simple model of two media the surface impedance is the characteristic impedance of the second medium. In the multilaer model to follow the boundar conditions are met with the surface impedance of a laered model 4

25 so ZT laered model. will be replaced with the general epression for the impedance of a Combining these two equations we arrive at a new epression for the reflection coefficient: 0 Z 0 T T 0 ZT T Similarl the TM reflection coefficient becomes: 0 0 TM Z TM 0 0TM ZTM These are the Fresnel equations rewritten in terms of the characteristic and surface impedances. The program behind the graphic window uses these epressions with either a general or a real angle of incidence and uses the general n-laer impedance formula of the net section of Z. Note, as we saw before in the general derivation of the reflection and transmission coefficients, these equations are functions of the spatial wave number. as are the characteristic impedances themselves and the surface impedance. To bring us bac to the practical matters of magnetotellurics, it is the surface impedance that we actuall compute from measurements of electric and magnetic fields on the ground surface. Then, as we saw in the simple introduction to MT, we assume for purposes of representing the data that the computed impedance at a given frequenc is for a simple half-space model. This allows us to calculate an apparent resistivit via A Z 5

26 The practical impact of a finite horiontal wave number, 0 on the T and TM surface impedances and hence on the apparent resistivities can be appreciated in the following little analsis. We will assume that the earth is a good conductor so. The TM surface impedance of a half-space, medium, is: Z TM i i i but since i o and is the surface impedance for normal incidence we have the interesting result that as 0 increases beond the normal impedance, and the apparent resistivit calculated from it, goes up. For incident waves in the T mode ZT 0 0 So now as the horiontal wave number goes up the impedance and the apparent resistivit goes down. We have spent a lot of effort on the topic of arbitrar horiontal wave number because the solutions for the fields from finite sources are integrals over such wavenumbers and the behavior of the solutions can often be deduced from the behavior at a single wavenumber that is representative of the main contribution from the wavenumber spectra in the integral for the 6

27 full solution. This concept is dealt with full in section 5.4 where, for eample, we derive the solution for a line source in the direction. These fields are T and as we get closer to the source the solution requires shorter and shorter horiontal wavelengths to represent the large spatial variations in the field. In that case apparent resistivities calculated in this one will be lower than those epected from a normall incident plane wave. Other source would cause the apparent resistivities to go up. This issue is important in assessing the actual sources for magnetotelluric fields and whether apparent resistivities calculated from field data reall are source independent. The general impedance of an n-laered half-space We now derive the general form for the impedance of a laered halfspace. We will use the notation shown in Figure below. The solution for the TM mode in an laer is: ii ii ii i i i e e e ii ii ii i itm i i e e e From the general form of Snell s Law so in all ratios i of to these terms cancel and so will be dropped in the following derivation. 7

28 Figure At the surface of laer i, i and the above equations can be written in matri form vi: i ii ii i itme itme i i i A iii iii i e e i i i and at the bottom of the laer, i i ii ii i itme itme i i i B i i i iii iii e e i These two equations can be combined to ield an equation for the fields at i in terms of the fields at i both within laer i vi: i i i A B i i i 8

29 After a horrible algebraic eercise of multipling this out and noting that h we obtain: i i i i i i itm iihi iihi iihi iihi e e itm e e i iihi iihi iihi iihi e e i itm e e i itm Now multipl this out into two equations and divide one b the other noting that i i i i i i because and are continuous at the i interface: iihi iihi iihi iihi i i itm i itm i i ZiTM i iihi iihi iihi iihi i i itm i i e e e e i e e e e i i Finall if we divide top and bottom of the right hand side b iihi iihi i i e e we get the disarmingl simple form for the impedance at the top of laer i in terms of the impedance at the top of the laer, i+,below: Z itm itm Z Z tanh( i h ) tanh( i h ) itm itm i i itm itm i i To find the surface impedance of an n-laer medium start b calculating the surface impedance at the top of the laer that is on the basement, at depth n, and use the characteristic impedance of the basement ntm for Z ntm. Then b a recurring operation calculate the surface impedance at the top of each successive laer all the wa to the ground surface. 9

30 The T mode impedance is derived the same wa. We now start with an incident field: and the accompaning impedance: ii ii ii i i i e e e field is now written via the T characteristic ii ii ii i it i i e e e The same procedure used above leads to the same laer to laer impedance formula ecept with the T characteristic impedances and surface impedances. Z it it Z Z it it i i it it i i tanh( i h ) tanh( i h ) These epressions are used recursivel to find the surface impedance for an arbitrar horiontal wave number 0.[ for a finite angle of incidence)]. i i 0 ( i 0 sin The graphical window shows plots of the apparent resistivit derived from this impedance, and the phase of the impedance, for multilaered models. The resistivit, the permeabilit i and the susceptibilit i for each laer are entered as input as are the laer thicnesses and desired frequenc range. The default settings assume that the permeabilit and susceptibilit are free space values. Finall the impedances are calculated for either specified horiontal wavelength ( = / 0 ) or real angle of incidence 0. The default is normal incidence, 0 = 0. For an model the mode, T or TM, must be specified on opening the window. 30