How Deep will my GPR, Metal Detector, or Two-Box Reach?

OK, so we’d all like to know the answer to: “how deep into the earth can my electromagnetic sensor system penetrate?

The short answer takes two forms: 1) “not very far” and 2) “it depends“. The long answer is the topic of this article. The article is also focused on detection of targets such as buried metallic objects, though some of the modeling approaches and results will be relevant to other target types.

While some of the models and field equations here are applicable to near-zone sensors such as metal detectors and two-box cache detectors, the analysis in this article is really much more oriented at ground-penetrating radar (GPR), since it is the main technology that can afford an actuakl “view” of the subterranean environment in a compelling way. If not for limitations of existing architectures, GPR could achieve much greater penetration depth, even in mineralized soil.

The article is broken up into several topic areas, which build toward a final predictive model to answer our main question 2) with something better than a wishful guess. As you will see, finding moderate to small size targets at significant depths is an extreme challenge, and remains a mostly unsolved problem across a wide parameter space. There are hardware and signal processing methods that can enhance the basic signal model, but they are proprietary, and well out of scope for this basic introduction.

This article will use specific technical terms, often without additional definition, but the interested party should be able to follow along and learn the main concepts and end result without too much trouble. Just typing in the final model to get a detection range, like a 1980’s computer magazine program, is certainly do-able, but you may get misleading answers if you don’t first understand the underlying principles and valid parameter ranges.

The Topic Areas are:

1) Electromagnetic radiation from canonical (basic) structures (the transmitter)

2) Electromagnetic scattering (the reflection)

3) Free-space signal propagation (spreading)

4) Complex media signal propagation (lossy materials)

5) Dynamic range (biggest to smallest signal in our receiver)

6) Depth computation (our model)

7) Implications (what is and is not generally possible)

Not treated in the article are a number of relevant topics that address hardware design, stratified (layered) materials, de-embedding, and pattern recognition. These are important topics to consider once the basic model presented here indicates whether or not you will have “a snowball’s chance” of getting signal back from the target under the conditions that pertain to your situation.

1) Radiation.

There are two main structures generally considered when investigating electromagnetic radiation: the “stick” and the “loop”. We care about analyzing the stick and the loop because they allow us to estimate what happens with more complex shapes of actual objects. Sticks and loops produce spatial distributions of electric and magnetic fields, which is why antennas can be “pointed” to pick up a better signal sometimes. Sticks and loops also produce fields that decay (get smaller) at different rates with distance. Some fields are very strong up close, but decay quickly as we get farther from the source. Others start off weaker, but keep their intensity much better over distance. Old school metal detectors operating in the kHz represent the use of the former, while GPS and walkie-talkies represent the use of the latter “far-field” radiation modes.

We will see how these higher-frequency 10 MHz and up signals lose much more of their energy in the dirt than the low-frequency signals. We will also see that low frequency signals (long wavelengths) don’t bounce off of targets very well at all. It’s probably a good time to show the basic inverse relationship between frequency (f) and wavelength (lambda):

lambda=c/f

So the wavelength of an unbounded electromagentic wave is the speed of light divided by the frequency of the wave. Bigger wavelength, smaller (lower) frequency.

Continuing, a more formal name for the stick is the electric dipole, and a more formal name for the loop is the electric loop. Electric means that electrons are flowing, creating a current as charge moves on the conductive material. It can be demonstrated that the fields produced by an electric dipole are the dual of the fields produced by a magnetic dipole, where the magnetic dipole has the same fields as produced by an electric loop. The magnetic dipole is a fictitious concept, but using it as a thought experiment exposes some really interesting partial symmetry in the way that electricity and magnetism complement each other.

Electric (E) and magnetic (H) fields are vector quantities, so they have an intensity and a direction; kind of like your car has a mass and a direction of travel. Working with simple geometries and in simple materials, the individual direction components can be treated separately. So E will have an x, y, and z function, and so will H. Better yet, we can work in spherical coordinates, where each of E and H has a radial component, r, and two angular components (Theta and Phi). Some components may be zero, depending on the field configuration, and when we look at the “far zone”, we assume that all of the higher-order terms have died off, which makes spherical coordinates nice because the radial components always go to 0 in the far zone.

Let’s look at the electric dipole first. The field components for the electric dipole are:

etheta

er

hphi

The constants C1 through C7 have been used to highlight the first, most important concept: the 1/r, 1/r^2, and 1/r^3 terms. By general definition, the 1/r terms are the “far-zone” terms, and the others are the “radiative near field” (1/r^2) and “near field” (1/r^3) terms. Note the vector direction of each of the terms. For example, there is no far-zone term for the radial-pointed electric field. So pointing your stick antenna directly at something is generally counter-productive. This works out fine for ground to ground communications systems since you usually don’t care about signals coming from directly overhead.

Now, if we looked at the intensity of the constants (get a textbook), we would notice something else interesting. Many of the near-field constants are bigger than the far-zone constants as r–>0 (as we stay closer and closer to our transmit antenna). We’ll see later how this matters a lot for standard metal detectors, since this is the main thing working in their favor, and how this and the other model considerations clearly demonstrate that they are a terrible choice for achieving any real depth.

So, far-zone fields only point in some directions (they are vectors), and only exist in certain directions (they have a spatial distribution of intensity). Far-zone fields start weak, but lose much less with distance from the transmitter (or scatterer).

Near-zone fields only point in some directions, and only exist in certain directions. Near-zone fields start very strong, but lose intensity very quickly with distance.

For the electric loop (virtual magnetic dipole), we have a different set of fields, but the same concepts still all apply. If you do a little accounting, you will see that the fields of the stick and the field of the loop are spatially complimentary, and where one is zero, the other has maximum intensity. Using both together is about as close to an omnidirectional antenna as you could get, but there really is no such thing as a truly omnidirectional antenna; despite that, manufacturers love to brand even a simple stick in that way when they sell them to you. Don’t get me started…

2) Scattering

When an electromagnetic wave impinges on a target object (“hits it”), it produces electric currents on the object, and the fields produced by these induced currents are called scattered fields. This is (mostly) how electromagnetic systems are used to detect far away objects, especially when far-zone fields are the primary detection mechanism. The principle of Duality can be used to consider that the fields produced by an object when it is used an intentional antenna are the same that the object produces as a scatterer, with the additional consideration of the direction and polarization of the impinging wave. Polarization is just how the vector field components of the wave are oriented with respect to the target. The most important thing to remember about polarization is that scattering is generally maximized when the vector direction of the fields (not the direction of travel, the field pointing direction) matches the longest feature aspect (side) of the target.

Scattering can be very complex to model specifically, but there is a general relationship between wavelength and target size that we will use here as we develop concepts build toward our

    “how deep?”

model. This general relationship shows that target scattering efficiency falls into one of three general categories as the impinging wavelength goes from being very large to very small compared to the target. These three categories are “Mie scattering”, “resonant scattering”, and “optical scattering”.

For Mie scattering, the wavelength is very large compared to the largest target dimension. In this case, the specific shape of the target matters much less than it’s overall size. As the wavelength gets larger and larger (lower and lower frequency), the scattering efficiency of the target drops precipitously, losing 20 dB of intensity every time the target size is cut in half. This is bad news for low-frequency electromagentic sensor systems, since even though such waves tend to penetrate more deeply into the earth, they do not bounce off of small targets very well. How small is small? Well, at 1 MHz, the wavelength is about 300 m (in air), so to be an efficient scatterer at this frequency, a target must be on the order of about 30 m to be efficient. A 3 m target would be about 60 dB weaker as observed at the receiver.

For targets right around a wavelength, scattering varies significantly due to resonance, but is around the highest non-directed value. As target size approaches a wavelength, shape also begins to matter, which results in directive effects. Different shapes will scatter the wave energy in a highly spatially-dependent way, and this spatial-dependent scattering also starts to depend on wave polarization much more. This scattering regime is where all of the “action” is in the field of antenna design. This regime will not be addressed in detail since a system seeking depth of penetration will tend toward the lowest usable frequency, and since it is much more difficult to generalize when antennas and targets are large compared to the wavelength of operation. The target scattering efficiency will be referred to in this analysis on a relative basis using the 20 dB per halving of target size rule.

3) Free Space signal Propagation

The most basic and fundamental model for electromagnetic wave propagation is that of a wave in free space. The wave can be expressed as a spherical wave function, and in general is treated as a “plane wave”, that is: a wave with only vector components orthogonal to the direction of propagation. For a point source located at the origin, the propagation is outward, along the radial direction. The action of the wave propagation from a theoretical point source is to spread the energy of the wave over the surface area of a sphere. The surface area of the sphere increases with distance from the point source, and changes as 1/r^2. This concept forms the core of the so-called Friis transmission model (free space propagation model), which can be expressed in terms of Loss or relative Received Signal, the one being the inverse of the other.

Lfs = (4*pi*r/lambda)^2

A typical application of the core Friis physical model is to form an equation for absolute received signal:

Pr = PtGtGr(lambda/4*pi*r)^2

Where Pt is the power of the RF source, Gt is the transmitter antenna gain, and Gr is the receiver antenna gain. This formula will be specialized for our present application by eliminating the receive gain term, and replacing it with a scattering efficiency term based on the rule in Section 2).

Of course, we are interested in propagation in other than free space! We will address this in our simplified model in two ways: a) we will account for the dielectric loading of the ground material by modifying the effective wavelength of the wave, and b) we will introduce an attenuation term in the next section, Section 4).

The wave function of a wave in real ground materials, especially considering that they may have internal structures, is actually very complex. In our first-order model, we will at least modify the effective wavelength using the earth dielectric constant as so:

Lambda_eff = Lambda/sqrt(er)

This means that the wavelength of the wave in the ground is shorter than the wavelength in air, helping somewhat to increase the coupling to small targets.

4) Complex Media Signal Propagation

Complex media refers to materials with both a real and imaginary component to the effective dielectric constant, such as most all soil and rock materials. What does this mean in layman’s terms? It means that the media tends to “burn up” the wave energy, in addition to the wave spreading out as it does in free space. This effect is captured as an additional attenuation factor, alpha, the value of which depends not only on the media parameters, but also on the frequency of operation. This factor in our model has a lot to do with the preference for low operating frequencies. In essence, the less wavelengths that the wave travels to get from “A” to “B”, the less times it has to pick up and put down the stuff in the way (picking up and putting down burns power).

The frequency-dependent complex media attenuation factor depends on the media’s dielectric constant (epsilon), magnetic permeability (mu), and electrical conductivity (sigma):

alpha

This term, combined with the free-space propagation loss factor and target scattering efficiency, forms the core of our detection depth model.

5) Dynamic Range

One more thing that needs to be considered before discussing maximum depth is the practical matter of dynamic range. Dynamic range is the difference between the largest and smallest signals that a system can tolerate at the receiver before it overloads. Dynamic range is thus a matter of practicality, and depends on the design of the electronics and their use in the sensor system. Essentially, other than operating frequency, it is only dynamic range that is at the disposal of the designer to change in order to achieve more sensor penetration depth. Standard architectures have well-established limits to their dynamic range, and thus the current state of the art is limited in depth penetration to values which we’ll explore later in section 7.

The author of this article has experience with system design practices that have in the past achieved very high levels of dynamic range, including the use of certain special analog processing techniques. It is planned sometime to put those old techniques together with some new ideas, including advanced antenna design, to implement a portable prototype that is expected to have vastly improved performance relative to anything else currently available. Of course, that would take time, money and focus.

6) Depth Computation

We can now estimate the receive signal strength available to a sensor system versus frequency for a given depth and target size by combining the factors of sections 2), 3), and 4). This graph will show us the optimum operating frequency to use based on both wave propagation losses and target scattering efficiency.

We can re-compute the graph based on different target sizes and depths, or the interested reader may re-cast the model in terms of these parameters in order to create a graph with a different independent variable. So, if you know the size of your buried target (treasure!?), then you could compute the maximum signal that you would get back at the receiver versus depth, for example, and then compare that signal strength to your transmit power and dynamic range. This is why the transmit power does not matter, in a way, since our real limitation is most often dynamic range. Noise of course does play a role, as does interference, but there are often ways of dealing with these issues that are not as much of a hard practical limit as is that pesky dynamic range.

The combined model uses a compact function notation that includes “_dB”, where _dB relates the decibel value of the function, relative to some unit. For power (signal strength), this unit is milli-Watts.

depth_model

The model for received power (PRX) indicates that it is determined by the 2-way free-space loss (Friis spreading loss modified with shorter wavelength), the soil attenuation (alpha) in dB/m times the distance (r) in meters, and the relative radar cross section (RCS) of the target (that old 20 dB length/2 when it’s below 0.1 lambda in size). That’s it.

7) Implications

Now that we have the model and know something about the meaning of the parameters that drive it, we can have some fun with some example cases. In general, we will see that achieving large detection depth is very difficult. Our model here uses only the most fundamental physical relationships to establish our results, and our model represents an upper limit of possible performance. The real-world phenomenology will always tend to reduce detection depth compared to our model, except in very specialized edge cases that should not be depended upon in the development of a tool for general application.

Unless otherwise noted, we’ll use an earth relative dielectric constant of 3.5, a relative permeability of 1, and a conductivity of 0.001 Siemens/m (which is a bit optimistic).

Example 1: A 1 m long metal treasure chest maximum detection depth versus operating freqeuncy

GPR Depth Chart 1 m Target
GPR detection depth as a function of frequency for a 1m target in average soil.

The lower detection line is what an advanced GPR-type device could do (which does not yet exist…) and the upper is what could be expected from a current high-performance system. Taken alone, some would say that “lower frequency is better” for detection depth, but the scattering part of the model is important here as well.

Example 2: A 0.25 m metal object maximum detection depth versus operating freqeuncy

GPR Depth Chart 0.25 m Target
GPR detection depth as a function of frequency for a 0.25m target in average soil.

As we see here, the 100 kHz operating frequency is not even an option for detection of a 0.25 m target, even in easy soil conditions. Why? Two reasons: 1) scattering efficiency, and 2) our far-zone model. Recall from Section 1 that metal detectors rely on the near-zone 1/r^2 and 1/r^3 field components to achieve detection. These fields start out stronger, but die off much faster with distance. Even with a stronger start, the 0.25 m target size is likely just too small to see very deep at 100 kHz. Note that 100 MHz is now the “winner”, since it scatters better, even though 10 MHz was better in the 1 m target analysis. Interestingly, 1 GHz (or 1000 MHz) achives about the same performance as 1 MHz. Which would you choose for your sensor design in this case? Hint: antennas are much smaller and more efficient at 1 GHz.

Example 3: A 10 m long metal object maximum detection depth versus operating freqeuncy

GPR Depth Chart 10 m Target
GPR detection depth as a function of frequency for a 10m target in average soil.

This analysis result brings up an important point about models and their limits, which is similar to dynamic range considerations. The point is that the Friis free-space spreading model assumes we are some nominal fraction of a wavelength in distance for the 1/r spreading terms to dominate, and the spherical spreading model to hold. We see in this analysis that the spreading model is most likely to blame for the greater than 0 dB target return signal shown in the graph at lower frequencies – this is as fictitious as your receiver specification for sensitivity when it’s dynamic range is being overloaded by backscatter from the air-earth interface. Where the model matters most is where it crosses the detection sensitivity line, and so these results are still valid as an estimate of detection depth.

Example 4: Our 1m long metal treasure chest maximum detection depth versus operating freqeuncy, but in more conductive 0.01 mSiemens/m soil

GPR Depth Chart 1 m Target
GPR detection depth as a function of frequency for a 1m target in mineralized soil.

As expected, soil mineralization (conductivity) has reduced target detection depth, and in this example we also see an interesting balance of the attenuation and scattering portions of the model, in that 1 MHz and 10 MHz achieve about the same performance, as do 100 kHz and 100 MHz.

Example 5: Maximum detection depth versus frequency for the 20 ft. long corroded stack of gold bars still waiting ? meters underneath Victorio Peak…

GPR Depth Chart 20 ft Target
GPR detection depth as a function of frequency for a 20 ft. target in mineralized soil.

So there you have it: detection depth is a tricky business, nature does not want you to be able to sense things in the soil, and there are lots of tricks in getting even a conventional sensor architecture to work. Dynamic range enhacement is the key to victory, and there are some advanced methods available to the designer, but not many out there know how to bring it all together. And those that might, lack the funds or time to make it real.

In particular, the industry that has the most money to advance these technologies has the least motivation. The construction industry is quite happy with the status quo, where they call out the usual suspects (high-school educated utility beepers, conventional GPR jockeys, and metal detector guys), spray paint some lines, and throw their hands up when they hit something, which is OK, ‘cuz they done spent some money on the other stuff and shucks I guess that’s just how it is. Dig up the destroyed utility, hope nobody got hurt, repair it, bill the customer, and move on folks. Meanwhile, border security and advanced mineral prospecting and archaeology wish for new technology, but have nowhere near the funds to make it happen.

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