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# The spatial inverse problem in Earth sciences

A retired clergyman approaching his 80^{th} birthday who recently gained a BSc in mathematics wrote in response to an interview in *Creation* magazine **39**(4):24–26 with geophysicist Dr Peter Vajda.

Rev. IC asked:

The article says you think laypeople (like me) would benefit from understanding “the spatial (3D) inverse problem in Earth sciences”. Would it be possible for you to give me some guidance on how to start looking into this problem? I really need to start at a basic level. I found your own story very interesting and thank God you found the right way. Many thanks.

Dr Peter Vajda responded:

Thank you for your email. Let me try to explain the basics very briefly.

Firstly, an inverse problem is one that begins with a set of observations and seeks to calculate the arrangement of factors that produced them. Examples include calculating the internal organs of a patient from the X-rays detected, or calculating the internal structure of the earth from its gravity field. It is called an inverse problem because we begin with the results and then calculate the cause. This is the opposite of the forward (or direct) problem, where we begin with the causes and calculate the results.

The problem is that many inverse problems in geophysics are non-unique. That is, there are a multitude of different possible causes that we can calculate from the observations, and we cannot tell which one matches the reality.

Many of the earth’s properties and its structures are inaccessible to direct observation, simply because an observer or an instrument cannot move throughout the earth freely to sample or observe.

An exception is drilling boreholes, but they sample the earth only at discrete points, and only to a few kilometers. When we want to know the 3D structure of the earth—including its material properties (such as rock type, chemical composition, mechanical properties, elasticity, viscosity, density, conductivity, temperature, etc.)—we have to use indirect methods to study these.

These properties, or parameters, and their distribution throughout the earth manifest themselves by physical fields observable (measurable) on the earth’s surface. For example, the distribution of the density of the rocks through the earth expresses itself by the gravity field outside the earth (including on its surface), and this can be measured by gravimeters.

With the forward problem we would begin with the three dimensional (3D) density distribution inside the earth. Then, we could relatively easily compute the resultant gravitational field on the earth’s surface. This is the forward, or direct gravimetric problem. It is solved simply by evaluating mathematically a Newtonian volume integral, which means that the gravitational attractions of each little volume compartment of the inner volumetric domain of the earth are added up.

The direct problem gives a unique solution. This means that for any given density distribution within the earth there is but one (just one) unique external gravitational field (including the gravitational field on the surface of the earth). However, when we turn the issue around, we get the inverse problem.

The inverse problem is: Given the known (observed) gravitational field on the earth’s surface, what is the density distribution in the earth’s interior? The point is that the inverse gravimetric problem is non-unique. In other words, given a measured gravitational field on the earth’s surface, there are many possible density distributions that could exist in the earth’s interior which would generate that particular gravitational field. Methods exist to solve the inverse problem. However, when we find a solution, we can only say that the solution we have found is just one of many possible solutions. But we cannot be certain what the real density distribution inside the earth is.

There are ways to help reduce the ambiguity of such inversion solutions, including the use of constraints. Constraints are assumptions or data from other disciplines such as geology, tectonics, seismics, magnetotellurics, etc. These help decide between all the admissible solutions, which are more realistic and which are less realistic. Also we can combine solutions from several methods such as gravimetric, seismic, electric, magnetotelluric, etc. These are called joint or integrated inversions, and help reduce the ambiguities, because each physical parameter and its respective field have their own ambiguities that differ among each other. This is how our current understanding of the earth’s interior has been developed, and we can trust this knowledge to a fair degree of certainty.

These inversion methods are routinely applied in practice with geophysical measurements. They are used to explore for raw materials such as minerals and hydrocarbons, mitigate natural hazards in geotechnical and geo-engineering applications, prospect for archaeological information, etc. However, in principle, these inverse problems remain non-unique. This is obvious also from the fact that we observe 2D information (a physical field measured on the surface of the earth) but we want to recover 3D information (distribution of a physical parameter inside the earth). It is just not possible to achieve a unique solution, and ambiguities always remain.

To express this in simple terms: The 3D information inside the earth (its structure and properties) is not uniquely accessible from the 2D information observed on the earth’s surface. This is a *spatial* inverse problem. It is relatively simple in that the measurements are made at only one specific time—the present. Even so, it is not possible to obtain a unique solution for the 3D spatial distribution from the 2D spatial information.

The situation becomes enormously more complex when we seek to determine the history of the earth or universe. That is a spatial-*temporal* inverse problem. The aim is to recover the knowledge of the earth’s deep past. This also is an inverse problem.

Given the knowledge about the earth (or universe) in its present state, recover its history. This is aiming at way, way too much.

By adding one more dimension, the temporal one (time) to the inverse problem the magnitude of the ambiguity (non-uniqueness) rises by one order (i.e. ten times more complex). The knowledge about the present state of the earth as a 3D instantaneous information is an analogue to the 2D surficial spatial information, while the 4D information about the earth (its history in 3D) is an analogue to the 3D spatial information.

Even if we *could* find some inverse solution to this 4D problem of the deep past, we can never know that the solution reflects reality. Even when we try to apply constraints to arrive at a solution for this temporal-spatial inverse problem, we find that the constraints are mostly metaphysical, based on unverifiable assumptions grounded on naturalistic, atheistic faith.

And when we consider the question of origins, this is much more of a problem than the problem of determining the deep past.

This is the reason we cannot put our trust in paleoscientific explanations about the deep past of the universe and the earth; or even about their origins. God was not only the true witness of their origin and the history, but he was also the One who did it—the Creator. We can put our trust in His word because it is the truth about our origins and our history.

Kind regards,

Peter

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