Earlier, we discussed the IP effect under the excitation of over-stable constant current (or DC pulse), which is characterized by the change of electric field with time (charging and discharging process), so it is also called IP effect in the "time" domain. According to the change of electric field with frequency (frequency characteristics), IP effect can also be observed under the excitation of alternating current, which is called "frequency domain" IP effect (Fu Liangkui,1991; Luo,, 1988). In the equipment shown in figure 1- 1- 16,

The power supply is changed to converter power supply, and the frequency f of the supplied alternating current is changed one by one (but the amplitude remains unchanged). According to the change of alternating potential difference between measuring electrodes with frequency, the induced polarization effect in frequency domain can be observed.

Figure 1- 1- 17 Frequency characteristic curves measured by several ore samples.

Fig. 1- 1- 17 shows the frequency characteristic curves (also called spectrum curves) measured on several ore samples in the above manner. Among them, the variation curve (amplitude-frequency characteristic) of the amplitude of the total field potential difference with frequency f shown in Figure (a) has a good corresponding relationship with the time characteristic described in the previous section: as f goes from high to low, the corresponding unidirectional power supply duration t (that is, the half cycle increases from zero, the induced polarization effect gradually increases, and as a result, the amplitude of the total field potential difference increases; But when f→0, T=→∞, and the IP effect is the strongest, so it tends to be saturated. For the limit case, there is the following relationship between the total field potential difference in time domain and frequency domain:

Geoelectric field and electrical prospecting

The change curve (phase-frequency curve) of total field potential difference δ relative to power supply current phase shift φ shown in figure1-17 (b) is characterized in that φ is negative at all frequencies (the phase of potential difference lags behind the power supply current), which shows that the impedance caused by IP effect has capacitive reactance property. When the frequency is very low or high, φ tends to zero; At medium frequency, the phase φ takes the extreme value. This is because the induced polarization effect tends to zero at high frequency and the total field is equal to the primary field, so there is no phase shift. When the frequency is very low, it is equivalent to the polarization saturation of unidirectional power supply for a long time. At this time, although the secondary field is the largest, it is "synchronous" with the current, so the phase shift of the total field is also zero.

It can be seen from Figure1-17 that although the basic behaviors of amplitude-frequency and phase-frequency curves of various rocks and ores are the same, different rocks and ores have different frequency characteristics. Rocks and minerals that charge and discharge rapidly in time domain have high frequency characteristics in frequency domain-the total field amplitude decays rapidly at a relatively high frequency, and the phase extreme value is obtained; On the contrary, rocks and ores with slow charge and discharge in time domain have low frequency characteristics in frequency domain-the rapid attenuation of total field amplitude and the extreme value of phase appear at lower frequency.

The experimental observation in frequency domain also shows that there is a linear relationship under the condition of current density that can usually be achieved in the field work of electrical exploration. Therefore, the total field potential difference normalizes the current and the device, and the AC resistivity independent of the current can be calculated:

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Where k is the device coefficient.

In the case of IP effect, δ varies with frequency, and generally there is a phase shift between δ, so it is frequency.

The complex function of velocity f (or angular frequency ω=2πf). Therefore, AC resistivity is often called complex resistivity and recorded as ρ (I Ω).

Obviously, the spectrum of complex resistivity has the same characteristics as the spectrum of δ mentioned above (the current amplitude is unchanged). (B) the relationship between amplitude-frequency characteristics and phase-frequency characteristics

In the theory of complex variable function, if the complex variable function ρ(s) is analytically finite on the right half plane of the complex plane S and has no zero point, it is called the minimum phase shift function. The real part Reρ(ω) and imaginary part Imρ(ω) of the minimum phase shift function ρ(iω) satisfy the hirt transformation:

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After some transformations, the relationship between the amplitude A(ω) and the phase φ (ω) of the minimum phase shift function ρ(iω) can be written as follows.

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The experimental data show that the complex resistivity ρ (I Ω) of rocks and ores approximately satisfies the minimum phase shift condition, so the spectrum of the real and imaginary parts of the complex resistivity is interrelated with the amplitude-frequency characteristics and phase-frequency characteristics, and can be transformed with each other. It can be seen from the formulas (1- 1-32) and (1-kloc-0/-33) that the imaginary part im ρ (ω0) or the phase φ(ω0) at a certain frequency ω 0 is equal to the real part Reρ(ω) or the logarithmic amplitude lnA(ω This weight function is a steep curve near ω=ω0, which shows that the derivative value at a given frequency ω0 is still the main influence. In fact, after some transformations of (1- 1-32) and (1-kloc-0/-33), the following approximate relations can be derived.

Theoretically, it is not necessary to observe the spectrum of each component at the same time, and it seems that the spectrum of any component is the same. However, the ability or resolution of each component spectrum to reflect IP characteristic parameters is not the same, and the difficulty of its observation technology is also different technically. Therefore, it is still a subject worth studying to select suitable components for observation according to geological tasks and actual conditions.

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(C) the relationship between frequency characteristics and time characteristics

Not only the frequency characteristics of each component (Reρ, Imρ, A, φ) can be converted to each other, but also there is a certain relationship between frequency characteristics and time characteristics, which can be converted to each other.

In order to better study the time characteristics, we simulate the practice in frequency domain, normalize the total field potential difference in time domain to the power supply current I and the charging process δ u (t) of the device, and calculate the resistivity:

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In the practice of electrical prospecting, the electrical conductivity and induced polarization effect of the earth can usually be approximately considered as linear and "time-invariant". In this case, the time-domain characteristic ρ(T) of step current excitation and the frequency-domain characteristic ρ(iω) of harmonic current excitation can be related by Laplace transform and inverse transform:

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In the formula, the complex number S is iω, and ρ(s)=ρ(iω) is the complex resistivity spectrum.

The formula (1- 1-36) can be used to realize the conversion between time characteristic ρ(T) and frequency characteristic. Therefore, frequency domain IP measurement and time domain IP measurement are essentially the same, and are equivalent in mathematical sense, and the difference is mainly in technology.

(4) Parameters characterizing frequency domain IP effect.

1. Complex resistivity spectrum

Because IP effect in alternating electric field is marked by the frequency characteristics of total field potential difference or complex resistivity, the complex resistivity spectrum in the whole (ultra-low frequency) frequency band where IP effect appears should be the most comprehensive parameter to describe IP effect in frequency domain. Complex resistivity method or spectrum induced polarization method proposed by K.L.Zonge et al. It is to study underground geological conditions by observing the real and imaginary parts of apparent complex resistivity or the spectrum of amplitude and phase in a fairly wide (ultra-low frequency) frequency band. The advantage of this method is that it can provide rich IP information, but to obtain a complete spectrum, it needs to be observed at many frequencies, so the production efficiency is very low and it is not suitable for general survey and prospecting.

2. Dispersion rate

By imitating the formula for calculating the polarizability in time domain (1- 1-23) or (1-kloc-0/-26), the "dispersion ratio" can be calculated according to the amplitude of the total field potential difference of two frequencies fD (low frequency) and fG (high frequency).

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Used to indicate the intensity of IP effect in frequency domain.

In the limit case, low frequency fD→0 and high frequency fG→∞, and the limit dispersion rate can be obtained according to (1-kloc-0/-30) and (1-kloc-0/-27):

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Considering the formula (1- 1-28), polarization usually accounts for only a small percentage, including

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That is, the (extreme) dispersion rate and (extreme) polarization rate are equal. For unrestricted frequency systems and time systems, the dispersion ratio P(fD, fG) and polarization η(T, t) are generally different; However, they are still positively correlated with (extreme) dispersion and polarization, that is, if some factor or condition increases or decreases the former, the latter will also increase or decrease accordingly. Therefore, extreme or non-extreme dispersion and polarizability have the same properties and can be expressed by (extreme) polarizability.

The observation of dispersion rate only needs to be measured at two frequencies, which is naturally much simpler and more efficient than full spectrum measurement. In the early development of IP method, J.R. Witt and others established a "frequency conversion IP method", which is based on observing the amplitude of total field potential difference of two appropriate frequencies in ultra-low frequency band to obtain video divergence Ps(fD, fG) to study underground geological conditions. This variation of frequency domain IP method, like time domain IP method, has always been the most commonly used method.

Step 3: Stage

As mentioned above, the induced polarization effect leads to the phase shift of the total field potential difference relative to the supply current, that is, the phase φ of the complex resistivity. Other things being equal, the stronger the IP effect, the greater the absolute value of φ. So phase φ can also be used as a parameter to describe the strength of IP effect. In fact, according to the formula (1- 1-34), the phase φ caused by IP effect is approximately proportional to the slope of amplitude-frequency curve or the change rate of electric field amplitude with frequency; According to the formula (1- 1-37), the dispersion rate P(fD, fG) is also directly proportional to the average slope of the amplitude-frequency characteristic curve between the frequencies fD and fG. Therefore, the phase φ of a certain frequency f is approximately proportional to the dispersion P(fD, fG) of two nearby frequencies. That is, φ, like P(fD, fG), is positively correlated with (extreme) polarizability η, which can be expressed by the latter.

In principle, phase measurement can be carried out at only one frequency, which is more convenient and advantageous than dispersion measurement. However, it is difficult to make high-precision field phase measurement instruments, so the phase induced polarization method, a frequency domain variant based on phase measurement, developed late and was not as widely used as time domain induced polarization method and frequency conversion induced polarization method.