Читать книгу Electromagnetic Methods in Geophysics - Fabio Giannino - Страница 26

The AEM methods can be considered, as the airborne equivalent of the TDEM (or the FDEM) method, carried out on land. It was developed first for mineral exploration over vast areas in Canada and Australia. For a general overview of the various AEM systems, it is useful to read Siemon et al. (2009).

The methodology is currently not yet widespread, however there is a growing interest for those applications where the deployment of geophysical techniques over very large areas is required (for example for large‐scale engineering projects or groundwater mapping). At the same time, this technique should guarantee an economical advantage with respect to the application of land‐based techniques, and a comparable resolution in the result.

Further application of the AEM method in hydrogeology, as well as in other field of application where a higher degree of resolution is required, led to the fine‐tuning of systems capable of more detailed definition of the geophysical model, above all in the shallower layers. This need, was also due to the limited contrast in terms of electrical resistivity (or conductivity) that can be found in field of applications different from the mineral exploration (the application where the AEM was developed first), where the ore bodies show electrical resistivity several orders of magnitude lower than the hosting rock.

A good compromise to achieve the required resolution, has been reached by mounting the EM systems over helicopters. This occurrence, allowing for lower flying altitude and slower velocity than aircrafts, contributes to a better quality of the data and a higher resolution. Furthermore, the possibility to perform more complex processing and inversion techniques implemented over dedicated software, allowed to refine the data analysis and interpretation with respect to the more simplistic data analysis originally carried out for mineral exploration purposes, which is known as the so‐called “bump detection.”

In the EM helicopter borne systems (defined as HTEM), both the transmitting as well as the receiving loop, are carried below the helicopter in the same frame, at a height of about 30 to 40 meters above the ground level. In some systems, the receiver frame is independent from the transmitter frame.

AEM soundings are collected at approximately 1.5 seconds interval, which corresponds (taking into consideration the velocity of the helicopter) to a TDEM sounding (on land) approximately every 25 meters along the acquisition line. Actually, the sampling frequency may be defined during the post‐processing phase where a resampling can be performed with a finer stacking up to less than 5 meters.

The spacing between acquisition lines varies, from 50–100 meters to a few hundreds of meters, depending on the degree or resolution required and on the purpose of the survey.

AEM systems, may also host a GNSS, tilt meters, and altimeters used to control the position of the system, the inclination of the loops and the height from the ground, in continuous mode, allowing for the following correction during the data processing phase.

To summarize the phase of an AEM survey, starting from the data acquisition on the field to the data analyzed and visualized in office, the following can be concluded:

1 Once the most appropriate data acquisition system is selected based on the wave form emitted, and the survey is designed according to the purposes of the project, the data can be acquired on site.

2 EM data is visualized as Voltage variation in respect to time (along increasing time window). A first quality control is performed over each single 1D sounding, in order to verify that in no part of the survey area the data must be re‐collected because of a poor signal‐to‐noise ratio.

3 Then, each single 1D sounding is analyzed and filtered to enhance the quality of the signal, if needed. For each 1D signal a 1D inversion routine, is applied.

4 Finally, each single 1D model, is entered as input of a 2D or pseudo‐3D inversion routine to obtain 3D maps of the distribution of the resistivity.

Hereinafter, the principle and the basic equations related to the field quantities measured by AEM systems and the inversion will be presented.

As for the land FDEM and the TDEM method already presented, a transient magnetic field is generated by varying the flow of current around a transmitter loop. The EM field is thus generated, in order to diffuse downwards in the form of a plane wave. When the current abruptly interrupts the transmitter loop, this generates a variation in the magnetic field which induces an electromotive force (emf) in the medium (the half‐space below the ground encountered) according to Faraday’s Law of induction.

The magnitude emf is proportional to the rate of change of the primary magnetic field in the conductor. Hence, a current is induced in the conductor (the subsoil) like concentric horizontal eddy currents resembling smoke rings which run below the transmitter and diffuse through the medium (Nabighian, 1979).

As the time from the beginning of the primary generation passes, the induced current is weakened by the ground resistance and the current density increasingly weakens too. Therefore, currents decay over time (and depth for that matter). This phenomenon is, of course, a function of ground conductivity, and it occurs more slowly in highly conductive ground compared to poorly conductive ground (where currents diffuse and decay more quickly).

At this point, the secondary magnetic field generated by the induced currents and the rate of change of the secondary magnetic field, induces a voltage that is measured as a function of time that is sensed by an active receiver sensor (a receiving coil).

The uppermost layers have got a strong influence on the measured signal that is given in terms of nV/m2 at the receiver coil. As time passes and the current penetrates deeper into the ground, then the measured signal provides information on the conductivity of the lower layers, of course. And this is the way a receiver coil records information on the electrical conductivity as a function of depth. This type of information is, in fact, called the TEM sounding.

The period of time in which the transmitter is turned off is called ramp‐off. The on–off time is “designed” to emit signal at selected frequencies. The bandwidth of the system does not correlate directly with the system’s base frequency (which is the time window applied for collecting data), and it is largely determined by the frequency range of the primary EM field. The latter is not easy to establish, and to overcome this difficulty most TEM systems implement a linear shut‐off ramp for the current (Anderson, 1989).

The duration of the off‐ramp is inversely proportional to the high frequency bandwidth. A long linear ramp has the effect of making early time responses resemble a step response. A short linear ramp instead retains the characteristics of an impulse response (Palacky and West, 1991).

When deploying Airborne Transient EM systems these must be capable of handling the great variety of the earth’s responses with different ground conditions and they must cover very wide dynamic range to be effective. A good compromise between a rapid ramp off of the transmitter current and gradual shut off of the transmitter current represent a solution for resulting in early off‐time gates, which is appropriate for mapping near surface resistivity, on one side, and deeper ground penetration on the other. This is of course challenging to achieve at the same time and for one single configuration.

The timing at which the receiver coil TEM measurements are taken are quite narrow, and they are referred to as “time gates.” The early time gates are narrower than those at a late time because they occur when the transient voltage is changing rapidly. Also, at early time gates the SNR is higher and does not need a larger gate to be sampled as happens for late times. The early time gates thus provide information about the near surface, while late time gates provide information about the deeper subsoil.

The gates are spaced with a logarithmically increasing time period in order to minimize distortion of the transient voltage and improve the signal/noise (S/N) ratio at late times, a method called “log‐gating” (Siemon et al., 2009).

The mathematical expressions defining the components of the EM field are defined by the electromagnetic phenomena and are governed by Maxwell’s equations.

As already mentioned in Chapters 2.2 and 2.2 the relationships between electric field E, current J, and electric displacement D are described in two of these, while a second pair describe the relationships between magnetic field H, magnetic induction B, and magnetic polarization M. In quantitative terms, these four constitutive relations are:

(2.4.1)

(2.4.2)

(2.4.3)

(2.4.4)

where σ is electric conductivity, ε dielectric permittivity, μ magnetic permeability, and χ magnetic susceptibility.

The four parameters comprehensively describe the electromagnetic properties of a material. The first relation is effectively the well‐established Ohm’s law in a microscopic context. According to Maxwell’s laws, an alternating current induces secondary currents in a conductive earth. These secondary currents in turn generate secondary magnetic fields, measurable using EM receivers.

The coupling between the E and H fields is described by Ampere’s and Faraday’s law.

The way in which an electric current can generate an induced magnetic field is described by Ampere’s law. If the electric field E is unstable and varies over time then there will be an additional current in the medium known as the displacement current, proportional to the variation of the electric field E. This proportional factor is known as the dielectric permittivity ε. Consequently, an additional contribute, dD/dt, acts to induce the magnetic field H. Since the displacement current acts in exactly the same way as the conductive current J, the total current will be J+ dD/dt. The Maxwell–Faraday equation is a generalization of Faraday's law, stating that any magnetic field that varies through time will be accompanied by a spatially‐varying, non‐conservative electric field, and vice‐versa. The emf induced in a coil is equal to the negative of the rate of change of the magnetic flux.

All AEM data collected on site, are addressed as input for inverse modelling (the inversion data processing) allowing to define a model of the subsoil resulting. In the frequency domain systems, the secondary magnetic field for a stratified subsurface caused by an oscillating (frequency f) magnetic dipole source in the air is calculated using the formulae below (Ward and Hohmann 1988). As mentioned, these are based on Maxwell’s equations and the issue here is to solve the homogeneous induction equation in the earth for the electromagnetic field vector F:

(2.4.5)

(2.4.6)

ω = 2πf is the angular frequency, ε is the dielectric constant, ρ the electrical resistivity, and λ is the wavenumber.

When using horizontal‐coplanar coil pair (Transmitter plus Receiver coil) with a coil separation r and at an altitude h above the surface, the resulting secondary magnetic field Z is given by (Yin and Hodges 2005):

(2.4.7)

(2.4.8)

The subscript 0 indicating that the various parameters are considered in free space.

The inversion of AEM data refers to a mathematical methodology that consists in estimating a conductivity (or resistivity) model from the EM data collected in the form of decaying magnetic field measured at the receiver coil (in nV/m²). Also, the EM response is sensitive to other properties of the materials, such as the magnetic susceptibility, the dielectric permittivity and the chargeability. And the noise, of course.

As for all the inverse problems, also the AEM inversion problem is an ill‐posed problem. In order to obtain a unique and stable solution, a procedure of regularization, by using a priori information, is necessary. In addition, the EM inversion is the non‐linear relation between the observed data and the geoelectric properties of the geological structures. Thus the solution is usually obtained by a numerically‐intensive iterative method (forward modelling).

All these issues, together with the limitation of the computer technology, help to explain why the inversion of the AEM data, in the last decades, was performed assumed 1D earth models. However, many efforts have been put through to addressing of these limitations by using increasing computer power and by the fact that the geological structures cannot always be approximated by a 1D model. This brings to more realistic inversions structures, turning the interest to higher dimensional models (2D and 2.5D) and to improvements of the existing 1D inversion algorithms.

The general approach (i.e. the starting model) to invert EM data is dual:

 The homogeneous half‐space, and

 The layered half‐space.

The result is given in terms of apparent resistivity and due to the skin‐effect (high‐frequency currents are flowing on top of a perfect conductor) the plane‐wave apparent skin depth:

(2.4.9)

increases with decreasing frequency f and increasing half‐space resistivity ρ_a. Therefore, the apparent resistivities derived from high‐frequency EM data describe the shallower parts of the conducting subsurface and the low‐frequency ones the deeper parts. There are several procedures for the layered half‐space inversion of HEM data available (Beard and Nyquist, 1998), which are often adapted from algorithms developed for ground EM data.

Finally, assuming that the lateral variability of the resistivity is often not very strong the model parameters can be tied together by constraints to increase the number of data variables per model and, thus, to enhance both, resolution and stability. Siemon et al. (2009), and Steuer et al. (2008) presented laterally or spatially constrained inversion (LCI/SCI) results for HEM data.

As illustrated above for the Frequency Domain methods, also the time‐domain theory is based on solving Maxwell’s equations given a set of conditions and assumptions.

The vertical magnetic field Hz in the center of a circular loop, which is a good approximation for a square or otherwise segmented loop of the same area, with radius a and current I, is:

(2.4.10)

with:

 h is the transmitter height

 z is the receiver height;

 J1 is the Bessel function of order one and

 λ and α0 are defined as before.

 RTE is the reflection coefficient and is a quantity expressing how the layered half‐space modifies the source field.

 Hz is expressed in the frequency domain because RTE is a function of frequency (analogue to R1).

When deploying an AEM system for an actual survey it is of a paramount importance the correct knowledge and setting of:

 The geometry of the used system

 Frame geometry

 Position (altitude, angle) S

 Shape of the transmitted current waveform

 Timing (to within 0.1 μs), and

 Bandwidths of the receiver system.

However, for visualization purposes there has been a tradition to present data as late‐time apparent resistivities using a central loop configuration at the surface of the earth. For this configuration an analytical solution exists for the model of a homogeneous half‐space. In this case R_TE becomes:

(2.4.11)

assuming quasi‐static conditions, the vertical magnetic field simplifies to:

(2.4.12)

Using the simple relation b = μ₀ h we can now solve for b_z by evaluating the integral and applying an inverse Laplace transform:

(2.4.13)

(2.4.14)

where,

 σ=1/ρ is the conductivity of the half‐space,

 t is the time window

 erf is the error function

For t that tend to 0 b_z = μ₀I/2°. The time derivative, or the impulse response, db_z/dt is found through differentiation to be:

(2.4.15)

When θ tends to 0, (at late times) the time‐derivative of the magnetic field can be approximated by:

(2.4.16)

The apparent resistivity ρ_a is derived from:

(2.4.17)

(2.4.18)