Ian Von Hegner

Terrestrial Life in Light of the Copernican Principle

Keywords: astrobiology, Darwinian principle, extraterrestrial life, cosmological principle

Although many solar systems have been discovered, only one example of life is known. Thus, terrestrial life represents merely one data point. Consequently, extrapolating from terrestrial life to life elsewhere in the galaxy and beyond is often seen as a limitation in the search for different forms of life. Essentially, attempting to extrapolate from terrestrial life to life elsewhere implies that terrestrial life is representative of all life, reflecting a geocentric viewpoint. However, in accordance with the Copernican principle, the opposite holds true. Asserting that terrestrial life must differ from other forms of life in the universe is, in fact, the geocentric viewpoint. For if life elsewhere is not like terrestrial life, then it is ipso facto different life; more precisely, if terrestrial life does not represent general life, then that life must represent special life, which the principle states it is not. This study employs the Copernican principle as a probability assessment, addressing critiques rooted in the implicit assumption of the existence of different extraterrestrial forms of life. If various fundamental forms of life indeed exist, then differences in the probabilities of their emergence can be expected, forming a probability scale. This holds significance because it not only allows for insights into the characteristics of the majority of life elsewhere but also facilitates the establishment of boundaries for categories of life as we do not know it. Thus, the Copernican-Darwinian principle provides a valuable tool for astrobiology and the search for life in the galaxy and beyond.

Introduction

In 1543 Copernicus changed the millennia-old geocentric model by in a geometrical sense removing the Earth and hence its life from the centre of the universe and replacing it with the sun. Hence, in the heliocentric model, the Earth lacks a privileged position in the universe and is now recognised as one of eight planets orbiting the Sun in an elliptical path, with the Sun occupying one focal point. An extension of this argument of a moving Earth has subsequently continued to remove the notion of a privileged position.

Therefore, the Sun, positioned in the Orion arm of the Milky Way, is not privileged but is classified as a G-type main-sequence star [START_REF] Morgan | Some features of Galactic structure in the neighborhood of the Sun[END_REF][START_REF] Van De Hulst | The spiral structure of the outer part of the Galactic System derived from the hydrogen emission at 21 cm wavelength[END_REF], residing far from the galactic centre. The Milky Way forms part of the Local Group, [START_REF] Hubble | Mrs. Hepsa Ely Silliman memorial lectures[END_REF], which in turn is a constituent of the Virgo Supercluster [START_REF] De Vaucouleurs | Evidence for a local super,galaxy[END_REF][START_REF] De Vaucouleurs | The Local Supercluster of Galaxies[END_REF]. This supercluster is integrated into the larger Laniakea Supercluster [START_REF] Tully | The Laniakea supercluster of galaxies[END_REF]]. Indeed, the universe itself may be only one of many within a multiverse.

This development, codified by the Copernican principle, has significantly influenced astronomical and cosmological discourse. It is posited here that this principle can also inform our approach to life beyond the Earth.

Astrobiology is the multidisciplinary field that investigates the deterministic conditions and contingent events with which life arises, distributes and evolves in the universe [START_REF] Hegner | The indeterminacy bottleneck: Implications for habitable worlds[END_REF]. Thus, while not technically focused on the search for extraterrestrial life, astrobiology contemplates the profound question of life elsewhere in the galaxy and beyond, including the status of terrestrial life in this context. While this solar system may harbour only one inhabited world, this does not prevent the potential for life to flourish elsewhere. With an estimated 100 to 400 billion stars in this galaxy alone [START_REF] Williams | How many stars are there in the Milky Way? Universe Today[END_REF], with virtually each now known to likely be accompanied by planets (and likely moons as well), and over 2 trillion galaxies in the observable universe alone [START_REF] Conselice | THE EVOLUTION OF GALAXY NUMBER DENSITY AT z < 8 AND ITS IMPLICATIONS[END_REF], the room for potential extraterrestrial life is indeed considerable.

However, despite the abundance of known solar systems, the existence of life beyond the Earth remains unconfirmed, keeping the discourse within a geocentric framework. Thus, so far only one data point is known for life, terrestrial life, meaning that at present all discussion of the nature of extraterrestrial life, apart from pure speculation, can only occur from a geocentric position, i.e. to extrapolate from terrestrial life to life elsewhere. While using only one data point to draw conclusions is a commonly criticized approach, the criticism, as will be discussed in this work, is not as strong as usually assumed when incorporating the Copernican principle. This approach allows for reasoned discussion on the nature of extraterrestrial life, even in the absence of empirical data.

Hence, the search for life elsewhere in the galaxy and beyond involves many unknown unknowns. By employing probability theory we however restrict ourselves to known unknowns. Thus, boundaries can be placed upon an unknown by focusing on a known, terrestrial life. Thus, despite the term 'geocentric' inherently referring to a single world, it can nevertheless provide insights into life on a universal scale, as elaborated upon in the subsequent discussion.

Discussion

In this paper, Section 3 elucidates how the Copernican principle can justify the extrapolation from terrestrial life to life elsewhere. Section 3.1 delves into the asymmetry inherent between the potential forms of life and their abundance. Section 3.2 provides and analyses calculations supporting the notion that terrestrial life likely belongs to the general form of life in the galaxy and beyond. Section 3.3 explores the significance of extrapolation being grounded in chemical and biological evolution. Moving forward, Section 4 delineates the content of the Earth's one data point for life, arguing that the bacterial/archaeal form represent the general form of life. Section 4.1 explores what can be understood by other forms of life and the evolutionary mechanisms governing life. Additionally, Section 4.2 sheds light on the relationship between intelligent life and microbial life. Finally, Section 5 encapsulates the findings of this study and discusses their implications for applying the Copernican principle to gain insights into the majority of life in the galaxy and beyond.

The Copernican principle

Terrestrial life so far represents only one data point. It represents, as the phrase goes, life 'as we know it.' It is, therefore, plausible there are examples of life elsewhere in the universe that are very different.

Thus, extrapolating from terrestrial life to potential life elsewhere in the galaxy and beyond is commonly perceived as imposing limitations on the search for life elsewhere. In other words, to extrapolate from terrestrial life to life elsewhere is to say that terrestrial life represent universal life, thus embodying a geocentric viewpoint.

However, according to the Copernican principle, the inverse holds true. Arguing that terrestrial life ought to differ from other potential life forms in the universe is in fact the geocentric viewpoint.

The Copernican principle states that 'the Earth is not in a central, specially favored position .. the Earth is in a typical position' [START_REF] Bondi | Cosmology[END_REF]. Therefore, observations made from the Earth are merely indicative of observations made from an average vantage point in the universe. While Copernicus himself did not formulate this principle bearing his name, this was done by [START_REF] Bondi | Cosmology[END_REF], the principle is a retrospective extension of the ongoing deconstruction of any privileged status accorded to the Earth-a generalization that has proven notably fruitful. The Copernican principle pertains to the cosmological principle, according to which 'the universe presents the same aspect from every point except for local irregularities' [START_REF] Milne | Gravitation and World Structure[END_REF][START_REF] Bondi | Cosmology[END_REF].

The Copernican principle thus has its application in an astronomical and cosmological connection. However, the fact remains that like everything, life too forms part of the universe, and the application of the Copernican principle to life is thus considered a natural extension in this work.

Hence, the Copernican principle asserts that the Earth occupies no special position, and therein lays the central issue. If terrestrial life does not represent general life, it ipso facto represents special life. The situation is inversed, so to speak.

Thus, that terrestrial life is not general contradicts the principle. If the Copernican principle is taken seriously, then the critique of extrapolating from terrestrial life to life elsewhere is grounded in the assumption that terrestrial life is different from life forms elsewhere-a presumption directly contradicted by the principle, which posits that the Earth do not have a privileged status in the cosmos.

As terrestrial life so far constitutes only one data point, concerns emerge regarding extrapolation. The term extrapolation refers to the use of a data set to predict values in relation to this data. It can be broadly defined as 'an inference made about a system's behavior in a new range of variables, from experience in an old (familiar) range' [START_REF] Matalas | extrapolation, and surprise[END_REF]. Thus, extrapolation refers to predicting values beyond a specific range of known data points. It is related to interpolation, which can be defined as 'a method for estimating the value of a function between two known variables' [START_REF] Rukundo | Extrapolation for image interpolation[END_REF], or predicting values that are inside a range of existing data points, but extrapolation is subject to greater uncertainty.

Given the singular nature of this data point, extrapolating from terrestrial life is considered to introduce considerable uncertainty, thus constraining the search for extraterrestrial life in the galaxy and beyond.

However, the application of the Copernican principle in this context shows that two implicit assumptions shape this criticism, meaning the criticism with or without the principle has fundamental differences for the discussion.

Firstly, the presumption of extraterrestrial life's existence, which, while trivial, forms a foundational premise for further discussion. If life is not assumed to exist elsewhere, then further discussion becomes redundant.

As per the second implicit assumption, extraterrestrial life must be different from terrestrial life. Unlike the first assumption, this one is non-trivial because if life elsewhere is not presumed to be different, then the critique becomes redundant. Therefore, a critique of extrapolating from terrestrial life is meaningful only if the existence of different life is implicitly assumed; extrapolation from terrestrial life in this connection is ipso facto justifiable if life is the same everywhere in the universe.

Thus, these implicit assumptions bring the matter straight to the core of the Copernican principle even if the extrapolation with the one data point in itself is not mathematically justifiable. As explained, this posits a fundamental premise that the Earth, along with its life forms, lacks a privileged position in the galaxy and beyond. If terrestrial life does not hold a privileged or special status compared to life forms that are prevalent throughout the universe, then it is reasonable, based on this rationale, to employ an extrapolation of terrestrial life as a foundation for discussing life elsewhere in the galaxy and beyond.

The form and amount of life

Given that only one data point has been established so far, extrapolating from terrestrial life to potential life elsewhere can be viewed as a geocentric approach. Nevertheless, asserting that life elsewhere in the galaxy and beyond differs from terrestrial life is, as explicated, in fact the geocentric viewpoint.

Extrapolation involves inherent uncertainty, as it unlike interpolation entails predicting values that are outside of a range of known data points, making inferences based on an assumed correspondence that may not necessarily exist.

Despite the current absence of empirical evidence regarding the existence of life elsewhere in the galaxy and beyond, the Copernican principle can still offer insights into the discussion through probabilistic evaluations. While this doesn't affirm the existence of life elsewhere, it provides a framework for meaningful discourse.

Thus, the crux of the matter lies in recognising that if life elsewhere diverges from terrestrial life, then it is ipso facto a different form of life. Put succinctly, if terrestrial life is not general life, it is ipso facto special life.

However, categorising some forms of life as 'general' and others as 'special' appears to stem from an anthropocentric perspective. Such classification prompts the question of what criteria delineate general versus special life forms.

A Darwinian principle can be posited, asserting that no single species holds a privileged position in nature [von [START_REF] Von Hegner | A limbus mundi elucidation of habitability: the Goldilocks edge[END_REF]. According to this principle, Homo sapiens do not enjoy any privileged status, mammals do not enjoy any privileged status, animals do not enjoy any privileged status, and so forth. Thus, there is no such thing as lower and higher life in evolution, there is only reproductive success.

The cosmological principle allows irregularities locally, which can be interpreted as the possibility of variations of forms of life. This suggests that in a galaxy hosting inhabited worlds with diverse forms of life, an inherent asymmetry exists in the distribution of these life forms. Despite the absence of any form of life holding a special status, variations in the abundance of different life forms across worlds are to be expected. Each form of life remains without special status, yet their distribution among different worlds varies in quantity.

When referring to 'different life' here, we extend beyond the diversity observed on the Earth. It is noteworthy that all known life on the Earth shares a common ancestry, with origins traced closest to the bacterial domain [START_REF] Madigan | Brock Biology of Microorganisms[END_REF]. Therefore, when we mention terrestrial life or the form of life, we specifically mean the bacterial/archaeal form, elaborated upon in section 4. The diversity in life emerges from the evolutionary process of speciation, where ancestral groups diverge into numerous subgroups over time, each adapting to distinct ecological niches, thereby accumulating differences. Thus, when asserting that life differs between inhabited worlds, it implies fundamental distinctions, such as variations in genetics and biochemistry, or even the potential existence of life devoid of conventional genetics and biochemistry, thus sidestepping anthropocentric perspectives.

In a galaxy housing inhabited worlds with diverse life forms, this diversity encompasses different classifications of life. If various forms of life exist, the question emerges: how much of one class of life exists compared to others? Due to deterministic and stochastic factors, these diverse forms of life cannot be equally distributed; there must be an asymmetry in their abundance.

Consequently, if distinct fundamental forms of life exist, they would also manifest varied pathways for their emergence, resulting in differences in the probabilities of their occurrence. While specific figures for these probabilities remain unknown, the concept holds true in principle.

Hence, a probability scale emerges wherein certain life forms exhibit a higher improbability of emergence compared to others, creating an asymmetry in the probability of their occurrence. Consequently, some life forms emerge more readily, while the emergence of others is so cumbersome that it faces a higher improbability.

Although these diverse life forms likely emerge due to the same known physical and chemical laws, it is the case that even if there are undiscovered laws of nature affecting the probabilities of the emergence of some of these forms, even then will there be a difference between all these forms of life in terms of probability.

Consequently, variations in the probability of different life forms emerging also translate into statistical differences in their prevalence between inhabited worlds and the abundance of forms of life in the galaxy and beyond. As a result, some forms of life will be rarer and exist on fewer worlds than others, introducing statistical inequalities among them. Thus, there is not equality between the different forms of life, meaning that life through this asymmetry between forms of life can appear special by existing on fewer worlds and in smaller quantities in the galaxy than other forms of life.

Moreover, stochastic events further influence the prevalence of different life forms across inhabited worlds. Thus, even if different forms of life were to emerge with identical probabilities, purely statistical events would ensure variations in their quantities. Thus, contingent events would ensure some forms of life being more prevalent than others in the group of inhabited worlds.

In this context, 'general' refers to the most prevalent form of life in the galaxy and beyond, characterised by its widespread existence among inhabited worlds. Conversely, it is possible for life to appear 'special' since it denotes the least prevalent form when compared to another more prevalent form. However, as discussed, different probabilities will be assumed for different forms of life, and these will determine the amount of different life.

Of course, the origin of life on each world are independent events, i.e. what happens on a world A is independent of what happens on another world B at the other end of a galaxy. However, although life emerges independently of each other on different worlds, the form of life for each of the mentioned probabilities can still be the same. Thus, a certain form of life, e.g. the bacterial/archaeal form, may only emerge that way, even on different worlds.

Thus, life being placed in the same class if possessing the same form, yet having emerged independently of each other on separate worlds, is in principle no different than e.g. two G-type main-sequence stars at opposite ends of the galaxy being treated the same way in astrophysics. These stars have emerged the same way and follows the same phases, thus, they are placed within the same type where an understanding of the sun makes it possible to understand another type of G-type main-sequence star as well.

Thus, although life emerges independently of each other on different worlds, if it has the same form, it will here be considered as belonging to the same class of life.

More than one form of life may potentially emerge on the same world, however, competition for resources and space often results in one form outcompeting the others. This dynamic may lead to one form becoming dominant, while the others persist as minorities, inevitably resulting in differences in their abundance within a given world.

The probability of forms

As its basic postulate, the Copernican principle states that the Earth, and thus its life, does not hold a privileged position in the galaxy and beyond. However, is this justifiable?

The crux of the criticism against extrapolating from terrestrial life lies in the implicit assumption that terrestrial life holds a special or privileged status, indicative of rare forms of life. If all life resembles terrestrial life, then the Copernican principle stands in its purest form. Alternatively, if certain irregularities are permitted in terms of differences in the form and abundance of life, as allowed by the cosmological principle, then we adhere to a quasi-Copernican principle known as the mediocrity principle. Under this principle, we assume mediocrity rather than initially presuming any phenomenon to be special or privileged.

Consequently, if an object is randomly selected from multiple categories, it is more likely to belong to the most numerous category than any of the less numerous ones. While it is not a given that terrestrial life cannot fall into the less numerous categories, it is more likely for it to belong to the most numerous category.

Thus, it comes down to probability theory, which can be illustrated by the following relatively straight forward example. It can be assumed for the sake of argument that life can emerge in four fundamentally different forms according to a probability scale: N lifeforms = P 1 > P 2 > P 3 > P 4 .

One estimate suggests that there could be at least 300 million potentially habitable worlds within this galaxy [START_REF] Bryson | The Occurrence of Rocky Habitable-zone Planets around Solar-like Stars from Kepler Data[END_REF]. Therefore, for the purposes of discussion, let us assume that life has emerged on 100 million worlds within the galaxy.

Among the various life forms, life form 1 prevails as the dominant one, inhabiting 80 million worlds. Following this, life form 2 inhabits 10 million worlds, while life form 3 inhabits 6 million worlds, and life form 4 inhabits 4 million worlds Thus, there is a total of 8 × 10 7 + 1 × 10 7 + 6 × 10 6 + 4 × 10 6 = 1 × 10 8 inhabited worlds in the galaxy, where one world out of 1 × 10 8 can be drawn C 1 ways. Hence,

n(S) C 1 = 1 × 10 8 (1)
The probability of randomly drawing life form 1 is as follows: Let A be the event of getting life form 1 on a world. There are life form 1 on 8 × 10 7 worlds in the galaxy, and 1 world out of 8 × 10 7 worlds can be drawn C 1 ways. Thus, n(A) C 1 = 8 × 10 7

(2)

Therefore, the probability of drawing life form 1 on a world is as follows:

P(A) (3)
For the other life forms, the same method can be followed:

The probability of randomly drawing life form 2:

P(B) (4)
The probability of randomly drawing life form 3:

P(C) (5)
The probability of randomly drawing life form 4:

P(D) (6)
There is also the following illustrative example. The probability of not randomly drawing life form 1: Let E denote the event of not getting life form 1 on a world. There are 1 × 10 7 + 6 × 10 6 + 4 × 10 6 = 2 × 10 7 non life form 1 on 8 × 10 7 worlds, and 1 non life form 1 world out of 2 × 10 7 non life form 1 worlds can be drawn C 1 ways. Hence n(E) C 1 = 2 × 10 7 (7)

Therefore, the probability of not randomly drawing life form 1 is:

P(E) (8)
For the other life forms, the same method can be followed:

The probability of not randomly drawing life form 2:

P(F) (9)
The probability not randomly drawing life form 3:

P(G) (10)
The probability of not randomly drawing life form 4:

P(H) (11)
Hence, considering terrestrial life as more likely to belong to the most numerous category rather than any of the less numerous categories appear statistically sound. From a statistical standpoint, the probability that terrestrial life among 100 million inhabited worlds falls within one of the 80 million worlds is notably higher. Consequently, the Copernican principle becomes a statistical argument-a measure of probability concerning both the form and abundance of life, comparing terrestrial life to life elsewhere.

Although the numerical values provided were arbitrary, as we lack empirical knowledge of life beyond the Earth, the principle still applies, regardless of differences in actual conditions or the actual abundance of forms of life. This also applies if the different forms of origin of life are not probable events, as these still have different probabilities compared to each other. This still does not make any of the forms of life lower or higher than others, it is merely a statistical difference between different inhabited worlds and the difference in their abundance in the galaxy and beyond given by the determinism of physical and chemical processes as well as the contingency of biological events.

Thus, the Darwinian principle integrates into the Copernican principle, which underscores the absence of privileged species in any real vantage point, emphasising instead statistical differences in the occurrence of forms of life.

However, it is not a given that terrestrial life cannot belong to one of the less numerous categories. Among 100 million inhabited worlds, terrestrial life may potentially be among the 4 million worlds, rather than the 80 million worlds. Thus, on the probability scale, there exists life whose emergence stems from less likely events, placing it in the minority of the universe's forms of life in terms of probability and numbers across numerous worlds. However, does terrestrial life belong to it?

The bacterial/archaeal form appears to be so rudimentary and elementary that it can be argued to be the one that emerges through the most likely events, which will be discussed in greater detail in section 4.

Yet, terrestrial life could potentially belong to the rare life. It may be special, there is no empirical data to support either option yet. Consequently, criticism regarding extrapolation from terrestrial life in this regard is meaningful.

Nevertheless, if compelled to make a wager, a prudent approach would be to bet on terrestrial life belonging to the most numerous category, i.e. terrestrial life is not privileged. It is therefore more justifiable to assume that terrestrial life is not special. Along this line of reasoning, criticisms regarding extrapolation from terrestrial life lose their potency. Ultimately, the decision rests on whether to embrace probability theory as a guiding principle in astrobiological research.

Chemical and Biological evolution

Extrapolation arguments may be strong or weak depending on how far one venture from known data points, potentially leading to less precise predictions.

Since the majority of life elsewhere has the form of terrestrial life, extrapolation will ipso facto be strong from its grounding in knowledge of chemical evolution, i.e. the natural assembly of complex molecules into living entities; and biological evolution, i.e. the natural ability for these living entities to use materials in the environment to generate variable copies of themselves.

Biological evolution provides profound insights into life's mechanisms, its propagation, adaptation to changing environments, and the dynamics of species emergence and extinction. Meanwhile, chemical evolution provides an increasing knowledge on the origin of life on this planet, and through this offers insights into the majority of life within the galaxy and beyond.

The significance of terrestrial life as a representation of general life cannot be overstated. It suggests that much can be inferred about extraterrestrial life based on the understanding of life on the Earth, even though life has not yet been located elsewhere.

However, it is important to realize that since terrestrial life belongs to general life, this implies that there can be special life, i.e. different life, of which, on the basis of terrestrial life, nothing may be said. Thus, since a minority has forms of life fundamentally different from the form of terrestrial life, then extrapolation will be weak, as it may not be possible to ground it in knowledge of terrestrial life. Until research into chemical evolution may unveil insights into these potential other forms of life, which will also make it possible to formulate numbers on the probability scale for life's mutual probabilities, criticism of extrapolation from a solely terrestrial perspective is justifiable, as this geocentric approach may hinder the recognition of different life forms, even if they are present in front of us.

Yet, since this life belongs to the minority in the galaxy and beyond, knowledge of the majority of life can still be obtained since terrestrial life belongs to the life which emerges most easily and exists on the largest number of worlds. Thus, it does not impair the possibility to work with the majority of life elsewhere, and the advantage will therefore far outweigh the disadvantage of this approach. The Copernican principle can thus be meaningfully applied even though life may exist whose form escapes the current understanding of life. Thus, it is still a sound approach.

It is also conceivable that is terrestrial life that holds a special status, and an extrapolation of this life may thus potentially prevent us in seeing the different, yet more general life elsewhere. However, as scrutinised through the lens of the Copernican principle, it is more likely that terrestrial life is not privileged, meaning it is still a more reasonable bet that terrestrial life holds the general status.

The extrapolation will be weak in terms of specifying what types of species there can be on other worlds besides the bacterial/archaeal form, which is due to the way evolution operates, i.e. specifying the specific ecosystems on other worlds.

Thus, evolution is about local adaptation to immediate environments. These environments are changing effectively on random vectors through geological time. This means that even if there is determinism of local adaptation for any given moment, since it is determinism of local adaptation to fluctuating environments that are effectively stochastic through geological time, it will be a stochastic pattern for the history of life [START_REF] Gould | The Structure of Evolutionary Theory[END_REF].

Thus, on an inhabited world the history of life will be unpredictably contingent since that history follows a stochastically changing vector of environmental circumstances through geological time. But the extrapolation will be strong with regard to the general overview, where the bacterial/archaeal form will be prevalent.

Contrary to the view that one example of life represents the geocentric limitation, this one example do not represent a limitation in our knowledge of life in the universe. Thus, even in the current absence of data on life elsewhere, it is the case that since terrestrial life likely belongs to the majority, we possess an extensive knowledge regarding the nature of most life potentially existing in the galaxy and beyond. As our understanding of chemical and biological evolution continues to increase and refine, it is according to the Copernican principle justifiable to utilize that knowledge of terrestrial life as search criteria for life elsewhere in the galaxy and beyond.

The data point content

It has been reiterated that terrestrial life only represents one example, a single data point from which we extrapolate. However, a closer examination of life on the Earth reveals a vast diversity of organisms, many of which exhibit morphological differences so significant that one might not readily discern their relatedness.

Despite this apparent diversity, all known life on the Earth shares a common genetic and biochemical structure, pointing to a common origin [START_REF] Madigan | Brock Biology of Microorganisms[END_REF]. This commonality underscores there is indeed only one common data point for life on the Earth, rooted within the bacterial domain from which all terrestrial life has diverged into an ever-branching bush.

When discussing the Earth as one data point, its form of life, we are effectively referring to microbial life, such as bacteria and archaea. Indeed, the Earth is fundamentally a microbial world, with multicellular life emerging relatively late in its history. Thus, if all plant and animal life were somehow eradicated, life on the Earth would remain essentially unchanged, but the eradication of microbial life would quickly lead to a lifeless world. Thus, this is the age of microbial life, which occupies virtually every available niche on the Earth, which it will continue to do until the phases of the sun towards the red giant stage put an end to this, the oldest terrestrial life.

However, while this holds true for the Earth, it doesn't necessarily mean it holds true on other inhabited worlds. The crucial question becomes whether the emergence of the bacterial/archaeal form represents the least improbable path.

The Darwinian principle states that no single species holds a privileged position in nature. Thus, Homo sapiens do not hold a privileged position, mammals do not hold a privileged position, indeed, multicellular life does not hold a privileged position in nature. However, when considering terrestrial life, this principle encounters its limits at the level of bacteria and archaea. These organisms cannot for this one world meaningfully be contrasted against anything, because what can they truly be contrasted against? It cannot be stated whether they are privileged or general, they simply are! However, when the Darwinian principle integrates into the Copernican principle, a meaningful discourse emerges within a galactic context. Here, the focus shifts back towards the mediocrity principle, where no initial assumption of specialness is made.

Thus, that the bacterial/archaeal form belongs to the most numerous category may stem from that life cannot be simpler than the most rudimentary bacterial or archaeal form imaginable; this structural form seems to emerge from the simplest conceivable complexity of life. As such, it is plausible that the majority, if not all, of life elsewhere must necessarily emerge or be instantiated in a form akin to the simplest functional bacteria or archaea possible.

Hence, one could posit that the bacterial/archaeal form is identical to the universal standard for emerging life. However, the state of affairs is not quite as straightforward.

Thus, if we identify the universal standard for life with the bacterial/archaeal forms nature, then it appears we are identifying it with certain properties of the bacterial/archaeal form. If so, a dilemma appears: Is the bacterial/archaeal form the universal standard because it has those properties, or are those properties the universal standard because the bacterial/archaeal form has them?

Or alternatively: Did chemical evolution discover the bacterial/archaeal form because it is the universal standard, or is the bacterial/archaeal form the universal standard because chemical evolution discovered it?

These issues currently pose significant challenges, and it is therefore not a given that the bacterial/archaeal form cannot belong to the less numerous categories. However, when the Darwinian principle is generalized into the Copernican principle, it becomes more likely that this form indeed fall within the most numerous category, i.e. the general form. This suggests that certain forms of life may emerge with a higher probability compared to others, as discussed with the probability scale.

Life as we know it is a planetary phenomenon. However, speculation persists regarding the existence of space life, exemplified by fictional entities like the black cloud organism proposed by [START_REF] Hoyle | The Black Cloud[END_REF], albeit this may lean more towards a Lamarckian organism rather than a Darwinian organism, some scientific support exists for their plausibility [START_REF] Tsytovich | From plasma crystals and helicalstructures towards inorganic living matter[END_REF]]. Yet, even if such or other kinds of space life were to exist, differences in probability and quantity of different life forms would still persist. Thus, would the emergence of such space life prove more cumbersome than the emergence of the bacterial/archaeal form?

The bacterial/archaeal form appears to be so rudimentary and elementary that it likely emerges with greater ease compared to other life forms. Indeed, it is not beyond the realm of possibility that life arose in multiple waves in this form on the Earth. Thus, it has been proposed that bacteria and archaea represent two distinct origins of life on this planet [von Hegner, 2020]. Their cellular similarities stem partly from inherent constraints dictating their formation only in this way, allowing for some degree of variation, and partly from gradual gene exchange through lateral gene transfer.

Hence, it is plausible that these organisms belong to the most numerous category of life and lack any privileged status in the galaxy and beyond.

Another rationale for the potential inclusion of the bacterial/archaeal form within the most numerous category stems from its inherent simplicity. Thus, it is not merely the ease of its emergence that is decisive, but also the rapidity and range of its operation. This form of life exhibits distinct traits concerning a world, characterised as 'information explosive [START_REF] Hegner | The indeterminacy bottleneck: Implications for habitable worlds[END_REF]. Upon emergence, it swiftly proliferates and disseminates, eventually occupying every available niche within a world.

Under certain conditions, such life forms can even 'leak' from a world, as on the Earth where it may be propelled into space by external impacts [von Hegner, 2020], or potentially on the Saturnian moon Enceladus through internal plumes [START_REF] Porco | Could it be snowing microbes on Enceladus? Assessing conditions in its plume and implications for future missions[END_REF], if such life exists there. Hence, this form of life plays a dual role, planetary in its capacity to rapidly occupy and dominate a world, and interplanetary by potentially transferring to and colonising neighbouring worlds.

Therefore, its velocity and range serve as pivotal factors that could render it the most prevalent form of life in the galaxy and beyond.

The other forms of life

The assertion has been put forth that terrestrial life is more likely in occupying the most numerous category, thereby positioning it as the general form of life. However, this assertion prompts a critical inquiry: is it even possible to address categories of life as we do not know it, such as e.g. the example of 3 other forms of life? After all, it is an implicitly given condition that it is life as we do not know it.

However, since all these forms of life must necessarily be part of this universe, regardless of whether they emerge by laws of nature either known or not yet known, that objection entails a paradox. This can be analyzed in the following.

In space A, there is a set consisting of 'life as we do not know it', and in space B, there is a set consisting of things that are not 'life as we do not know it'. Thus, 'life as we do not know it' goes in space A, and 'life as we know it' goes in space B.

The question now emerges, where does space A belong? It is evidently not 'life as we do not know it'. This means it must go in space B.

However, this gives rise to a problem, because space B is defined to only contain things that are not 'life as we do not know it'. This means that the contents of space A cannot go in space B.

Since space A, or the set, is not, in itself, 'life as we do not know it', but, by its definition, contains 'life as we do not know it', we cannot effortlessly categorise where the set belongs.

This paradox is, strictly speaking, an illustration of the notion that there is no such entity as life as we do not know it, or, put differently, that the condition itself is inconsistent. In essence, it highlights the challenge in comprehending the concept of life as we do not know it, not solely because we lack empirical encounters with this 'dark biomatter', but also because it must necessarily be part of this universe and we thus are compelled to acknowledge that we know it. This serves a significant point, as life as we do not know it is not the same as saying there is life we cannot explain.

Thus, 'n is a bachelor' means 'n is an unmarried adult male'. As a consequence, 'n is a bachelor but n is not an unmarried adult male' is false on its face; and 'if n is a bachelor, n is an unmarried adult male' and 'if n is an unmarried adult male, n is a bachelor' are tautologies.

By contrast, 'life as we do not know it but it is life that we can explain' is not false on its face; and 'life as we do not know it is life that we cannot explain' and 'life that we cannot explain is life as we do not know it' are not mere tautologies. This is sufficient to dispose of the claim that 'life as we do not know it, and 'life that we cannot explain' has the same meaning.

The necessity of life being part of this universe, subject to its laws of nature, leads us to a related yet distinct inquiry. Life as we know it is an evolutionary phenomenon, with the Earth teeming with Darwinian organisms. However, it might be opinionated that this is merely due to terrestrial life being guided by evolution as we know it.

This begs the question: Is this evolution universal? Could the other forms of life differ not only in their origins but also in their operational dynamics, potentially influencing their prevalence throughout the galaxy and beyond?

To delve deeper into this inquiry, it is essential to clarify our understanding of evolution. Evolution by natural selection is fundamentally a theory about interactors; it is not about replicators. It is about individual organisms that interact with their local environment, where natural selection means differential reproductive success conferred by the fitness of possessing certain beneficial properties. Thus, Darwinian interactors can be replicators such as genes in some circumstances, while individual organisms are the interactors in most of circumstances.

Thus, while life as we know it is a DNA or RNA based phenomenon, organisms elsewhere in the cosmos might conceivably possess hereditary material in forms divergent from the standard 4-base pair DNA and RNA molecules, such as DNA and RNA molecules consisting of 8-base pair [START_REF] Hoshika | Hachimoji DNA and RNA: a genetic system with eight building blocks[END_REF], or even in a form other than genes. However, such deviations do not alter the fact that organisms with alternative genetic materials remain subject to the workings of evolution.

Thus, in one interpretation, evolutionary mechanisms are supervenient on the hereditary material, because while a given hereditary material specifies an evolutionary strategy, a given evolutionary strategy does not specify any finite set of hereditary material.

Thus, the question emerges: If life on different worlds exhibits difference at the hereditary material level but evolution acts the same at the organismal level, can it truly be considered fundamentally different in the universe? The answer hinges on the prioritisation of either level. However, if disparate forms of life are governed by evolution the same way at the organismal level, their differences may be inconsequential in the broader context of cosmic existence.

Intelligent life

Expanding the discourse to encompass life beyond the Earth naturally leads to considerations regarding the potential existence of intelligent life elsewhere in the galaxy and beyond. While our focus thus far has centred on the bacterial/archael forms of life, with multicellular organisms being comparatively recent additions, one might argue that the emergence of intelligent terrestrial life could confer a privileged status upon the Earth.

An occasionally espoused viewpoint posits that microbial life may be prevalent throughout the galaxy, yet intelligent life remains rare. However, this viewpoint is internally contradictory [von Hegner, 2022].

Thus, in microbiology, there is a growing recognition that microbial life can and indeed does exhibit characteristics of intelligence. Macromolecular networks can facilitate intelligent characteristics in microbial organisms such as decision-making, association and anticipation, self-awareness, robust adaptation, and problem-solving capabilities [START_REF] Westerhoff | Macromolecular networks and intelligence in microorganisms[END_REF].

Thus, the presence of microbial and protist forms of life on a world elsewhere in the galaxy does not preclude the existence of intelligent life. Consequently, the traditional conceptual dichotomy between microbial life and intelligent life is a misnomer that should be discarded. Indeed, viewing them as distinct categories hampers the search for intelligent life beyond this planet [von Hegner, 2022].

In this context, the crucial consideration becomes the emergence of the form of life, and it is the focus on the prevalence of inhabited worlds and the abundance of life therein that is important.

Conclusion

Since 1543, a geocentric approach has with great success no longer been used within astronomy and cosmology. However, while the geocentric approach has been abandoned there, it is in some ways the very same approach in light of the Copernican principle that, seemingly paradoxically, can give it a renaissance within astrobiology.

The Copernican principle is neither a law of nature nor a theorem, but is at present a heuristic, a working assumption derived from Copernicus' argument of a moving Earth, and therein lies its limitation. However, in this work the principle has been utilized as a measure for probability, as there may be irregularities in the form of different classes of potential life on different inhabited worlds and different amounts of life, thus, it will be mathematical in nature.

So although it is not yet known whether there is life elsewhere in the galaxy and beyond, the strength of the Copernican principle lies in its probability assessment. It allows, in light of the implicit assumptions, to frame the situation through the various inhabited worlds, and thus put some boundaries on what can be stated about life universally.

However, some issues persist. For instance, the numerical values used in the probability calculations was arbitrary, yet the underlying approach remains sound as it highlights the differences between potential forms of life. Yet, understanding how these potential differences correlate remains a challenge.

Thus, if terrestrial life remains the sole example in the cosmos, it would indeed hold a privileged status. However, the discovery of life or remnants thereof on one of the other solar system bodies may not be decisive for whether it is privileged. If this life is similar to terrestrial life, it could merely suggest mechanisms such as lithopanspermia, wherein life is transferred between the different worlds [von Hegner, 2020]. Thus, the geocentric extrapolation will merely slip into a heliocentric extrapolation.

Alternatively, the conditions within this solar system may be conducive only to the emergence of a certain form of life. This could mean that a galaxy could be divided into many different regions, such that conditions within different star systems may result in regions conducive to specific forms of life while inhibiting others. This does not imply privilege for certain forms but rather underscores that different star systems provide different conditions for the emergence of life.

If life is discovered on one of the other solar system bodies that is distinct from terrestrial life, then it applies that the quantity of life on one world will surpass that of another, not necessarily due to differences in the probability of life's emergence, one being more cumbersome than the other, but rather as a result of contingent events shaping each world's evolutionary trajectory.

Thus, the mere identification of life elsewhere, marking the attainment of the next data point, will not suffice to conclusively determine whether the Earth holds a privileged status.

In scenarios where numerous inhabited worlds exist however, the relationship between general and special forms of life becomes a non-trivial question.

Thus, if the majority of extraterrestrial life is similar to terrestrial life in terms of emerging in the bacterial/archaeal form, the quasi-Copernican principle, then this supports the basic premise in this article, since the extrapolation of terrestrial life then will be meaningful.

In a similar vein, if all extraterrestrial life emerge in the bacterial/archaeal form, the Copernican principle in its purest form, then this supports the basic premise even more, namely that an extrapolation from chemical and biological evolution is justified.

While it remains possible that terrestrial life belongs to the minority of life, statistics suggest otherwise. Even in the absence of confirmed extraterrestrial life, statistical analysis indicates a higher probability of terrestrial life representing the majority.

Thus, despite the arbitrary nature of exact estimates in the calculations, the principle allows us to delineate broad boundaries regarding the nature of life in the universe.

It ultimately comes down to whether one wants to take probability theory seriously and apply it as a guiding principle in research.

Thus, the criticism against extrapolating from terrestrial life to life elsewhere is, within an astrobiology context, not as straight forward as otherwise thought, since it is precisely because the Copernican principle states that the Earth, and by extension, terrestrial life, is not privileged that makes it so useful. It thus offers a fruitful and robust tool for the search for life elsewhere in the galaxy and beyond.

The Earth has in recent centuries repeatedly been stripped of its privileged status in the universe, which has often been regarded by some as a diminution in self esteem. However, even though the Earth's life here too loses a presupposed privileged status, this can almost paradoxically be argued to be a source of increased self esteem. Thus, that terrestrial life likely belongs to general life has tremendous significance, as it is precisely this that means that much can be known about the majority of life elsewhere in the galaxy and beyond on the basis of knowledge of this life.

In the context of astrobiological research, terrestrial life is special in being near-universal, allowing it to make inferences about the majority of life elsewhere in the galaxy and beyond. Thus, while the term geocentric by its very meaning is limited to one world, it still inform nearly universally about life. This recognition does not confer terrestrial life the status of being the center of life in the universe, it is still not privileged. It merely underscores terrestrial life sharing its form with the majority of life in the universe.

Astrobiology is in many ways the offspring of the Copernican and Darwinian revolutions, the great thresholds in history which adjusted humankind's understanding of the cosmos and life. It is therefore appropriate that a Copernican-Darwinian principle can allow us to make informed statements regarding the distribution of life in the universe. Thus, just as the heliocentric model unlocked the study of other worlds, since the idea that other planets could be worlds like the Earth conceptually did not exist in the minds of natural philosophers before then, and just as evolutionary theory unlocked the possibility of studying life across past, future and remote places in the present day, the Copernican-Darwinian principle unlocks the possibility to work with potentially inhabited worlds elsewhere in the galaxy and beyond.


Marina Terzi
email: marina.terzi@uphf.fr

Lynda Chehami
email: lynda.chehami@uphf.fr

Emmanuel Moulin

Vladislav Aleshin

Nikolay Smagin

Pump-probe

INTRODUCTION

In general, many structural health monitoring (SHM) and non-destructive testing (NDT) approaches are based on the comparison of test data to baseline data. The "baseline data" term intends here a set of data associated with a pristine structure subjected to all possible environmental and operational variations, such as fluctuations in temperature, variations in surface moisture or external loading, structure aging, etc. A lack of data in the baseline or wrong match between the test data and the baseline can adversely affect measured signals and cause false alarms.

Thus, baseline SHM methodology requires a large amount of data and therefore is compute-intensive. For this reason, new NDT concepts that do not rely on previously obtained baseline data are of particular interest. A few baseline-free (or reference-free) methods have been recently proposed.

One of the attempts to reduce baseline data amounts consists in using a so-called instantaneous baseline [START_REF] Anton | Reference-Free Damage Detection Using Instantaneous Baseline Measurements[END_REF]. In this approach, Lamb wave propagation along several paths is interrogated in pitch-catch configuration with a distributed transducers array. Common features in the undamaged paths are considered as instantaneous baseline and used for damage prediction.

Time reversibility of Lamb waves [START_REF] Park | Time reversal active sensing for health monitoring of a composite plate[END_REF] allows to use Time Reversal (TR) method for damage detection. In [START_REF] Sohn | Combination of a Time Reversal Process and a Consecutiv Outlier Analysis for Baseline-free Damage Diagnosis[END_REF] damage index is used to evaluate the difference between input and TR reconstructed signals for the distributed sensor array. The threshold for damage index distinguishing "undamaged" and "damaged" paths was established comparing damage indices of the paths with each other. In [START_REF] Park | Laser ultrasonic imaging and damage detection for a rotating structure[END_REF] the analysis is based on the comparison between stationary and rotating blade states. Dividing into forward and backward propagating waves, applying a standing wave filter makes damage being visualized. It was concluded [START_REF] Park | Understanding a time reversal process in Lamb wave propagation[END_REF], that TR is low-sensitive to environmental and operational variations but, on the other hand, there are significant limitations such as necessity to employ narrowband signals to minimize dispersion and a large number of transducers required for precise localization. Another similar approach relies on breaking of the Reciprocity principle in the context of delamination detection [START_REF] Huang | Baseline-free damage detection in composite plates based on the reciprocity principle[END_REF].

Another idea is applied in an instantaneous baseline group of methods based on mode conversion [START_REF] Kim | Instantaneous reference-free crack detection based on polarization characteristics of piezoelectric materials[END_REF]. It benefits from polarization characteristics of collocating PZT transducers attached to the opposite surfaces of a thin plate. Guided waves excited across different transducer pairs al-xxx low damage identifying.

In this paper, we describe a baseline-free imaging method capable of contact acoustic nonlinearity (CAN) localization in thin plates. This method is based on a pumpprobe experiment, in which information of damage is provided by differences in high frequency (HF) probe signals measured under varying low frequency (LF) pumping. Defect is localized by means of the back-propagation algorithm developed in [START_REF] Terzi | Baseline-Free Repetitive Pump-Probe Experiment for Structural Health Monitoring[END_REF]. This imaging method has been applied (see Chehami et al. [START_REF] Chehami | Experimental Study of Passive Defect Detection and Localization in Thin Plates from Noise Correlation[END_REF][START_REF] Chehami | Nonlinear secondary noise sources for passive defect detection using ultrasound sensors[END_REF]) for linear defects such as holes or rigid inclusions. Numerous linear SHM techniques rely on damage indices based on parameters such as change in wave velocity and amplitude, mode conversion, wave scattering, etc. However, the impedimental factor of these linear techniques is their susceptibility to operational and environmental variabilities. Moreover, linear techniques are primarily not sensitive to nonlinear defects (such as cracks, delaminations, debondings and other CANs), which means they cannot be used for early warning.

Here, the back-propagation algorithm has been adapted for a model defect mimicking the behavior of a real CAN. In the reported experiments, a steel ball pressed against an aluminum plate is used to mimic a CAN. Similarly to a crack or delamination, the ball excited by the LF pumping slightly modifies the propagation conditions for the HF probe. These weak changes are considerably enhanced by the application of the back-propagation algorithm due to multiple averaging over different loading states.

In the Section 2, the experimental procedure and signal processing are described in greater detail, as well as results of imaging. Section 3 presents concluding remarks.

REPETITIVE PROBING EXPERIMENT: SETUP AND SIGNAL PROCESSING

Experimental setup

In the presented experiment study acoustic waves and vibrations are excited in a thin aluminum rectangular plate (1 m × 0.5 m × 3 mm) horizontally suspended with elastic strings on a metallic support cage. Instead of real defect we use a 1 cm steel ball pressed against the top plate surface with elastic steel ruler fixed on the same support. (see Fig. 1.).

In the plate two acoustic signals are generated (see Fig. 2): a high-frequency probe (100 ms burst with one cycle of 5 V peak-to-peak sine at 20 kHz) and a lowfrequency harmonic pump in the frequency range from 0.5 to 4.5 Hz (which is much lower than f probe ). The probe signal is emitted by piezoelectric wafer glued to the plate surface. The pump wave is excited by a vibrating shaker firmly attached to the plate with the use of two polyurethane foam slices for preventing a direct mechanical contact that can be detected as a defect. In addition to the emitting piezoelectric wafer, there are N r = 7 receiving transducers of the same kind fixed at known positions. The signals are amplified and then recorded using National Instruments acquisition board (500 kS/s, 8 channels). The duration of the high frequency probe reverberated signal is 50 ms

1 fpump .
Exited by LF vibrations at f pump produced by the shaker, the ball experiences periodic oscillations which constantly modify propagation conditions for the HF Lamb wave. The pump parameters and precompression are selected in such a way that the ball never loses the contact and does not produce squealing.

Signal processing: Repetitive pump-probe algorithm

In this series of experiments, M sets of acoustic signals are recorded at different times corresponding to different loading states. The elapsed time between two acquisitions is about 0.45 ± 0.1 s. This particular time lapse should be less than the ball movement period. Besides, the period should not be not a multiple of the time lapse. In this situation, each recorded signal will be slightly affected by a changing ball-plate contact state which corresponds to the influence of LF pumping on a real defect.

Let us denote by s j m (j = 1...N r ) the m-th signal recorded at receiver r j . Signals registered at the same receiver at different times are subtracted from each other. Each differential signal ∆s j p = s j m1 -s j m2 produced by acquisitions number m 1 and m 2 (p ≡ (m 1 , m 2 )) is then backpropagated using a classical imaging algorithm (see [START_REF] Chehami | Detection and localization of a defect in a reverberant plate using acoustic field correlation[END_REF]). The idea is to compensate for the phase of the wave propagating from the emitter to a receiver via the candidate defect position (called "pixel"). In case of damage located at the selected pixel, an averaging of compensated signals over all possible emitter-receiver pairs results in a narrow peak at t = 0. However, if at the selected pixel no defect is present, the sum is destructive resulting in a noise-like waveform. By interrogating all pixels at the plate and by the multiple application of the back-propagation algorithm, a full image can be obtained.

More specifically, a back-propagation function for the given difference ∆s j p in frequency domain is defined by:

bpf p (x,y) (ω) = Nr j=1 ∆s j p (ω) e i k(ω) d Erj (x,y) , (1)
here d Erj (r) is the distance between the emitter E and the j-th receiver via the possible defect position r(x, y). The wavenumber k fulfills the dispersion relation of A0 Lamb mode (flexural wave). The candidate positions are organised in the rectangular grid with a spatial step ∆x = ∆y = 5 mm. After returning to the time domain through an inverse Fourier transform, the pixel intensity at position r(x, y) that is calculated in time domain as an integral of the backpropagation function squared over time window T 0 :

I p (r) = T0/2 -T0/2 bpf p (r) (t) 2 dt, (2)
where T 0 is typically the inverse of the bandwidth. The intensity (2) is determined for each acquisition pair. In order to take into account all acquisition pairs and accumulate all available information, one has to sum up all individual contributions and get

I(r) = P I p (r) P , ( 3
)
here P is number of loading states pairs. At the defect location, i.e. when r = r D , I(r) reaches a local maximum. Due to the incoherent summation (3) (summation of intensities) the I(r) presents clear defect imaging (Fig. 3). At the defect location, i.e. when r = r D , the difference between scattered fields under different loading states reveals a local maximum. Thus, an intensity of the differential signals will be much higher at the defect location, than elsewhere.

Results and discussion

The results of the measurements and signal processing are shown in Fig. 3 for different LF values. In this figure, the defect is marked with a circle, the square indicates the shaker position, the stars correspond to HF receivers.

As observed, a clear localization is obtained for the pump frequencies f pump in the range between 0.5 and 4 Hz (Fig. 3). Increasing pump frequency higher or equal to 4 Hz results in no localization as shown Fig. 3d. This threshold is related to a mechanical resonance of the support system accompanied by squealing and ball detachments.

The LF response of the structure is such that for pump frequencies f pump =0.5, 2. movement of the ball relative to the plate occurs. When the movement is purely vertical, the reverberations are affected only by the Hertz-type contact nonlinearity, it simulates better the real crack with varying stress. An important fact is that the localization quality stays approximately the same regardless of the ball movement type (compare 3a,b,c and 3d).

There is a speckle-like background noise in all images, especially in Fig. 3c. It is mostly related to averaging of intensities in (2) and (3) that we had to use due to unknown signs of the defect influence on the structure for differential signals from each particular acquisition pair. Indeed, averaging of intensities using so-called "incoherent sum" does not suppress noise on the deterministic part of the signal; filtering out noise in the ranges t < -T 0 /2 and t > T 0 /2 solves the problem only partly.

Therefore, to improve the images signal-to-noise ratio, we propose to take into account the signs of bpfs using coherent sum as described below.

Signal processing: Random sign compensation

In many obtained images peak-to-background ratio is not optimal due to high sidelobes. In order to improve it the random sign compensation is proposed.

Purposeful non-synchronization between the pump vibrations and probe recording entails randomization of the defect state probing. The sign of the defect strength (the influence of the defect on the structure) is arbitrary and unknown in the experiment, i.e. we don't know if bpf p (r) (t) experiences a positive or a negative peak.

Meanwhile, the sum in (3) is incoherent, which means that the phase information of backpropagation function (i.e. of scattered field) is discarded. Thus, squaring of backpropagation function in (2) causes the sign loss, which in turn adds to sum in (3) a destructive component. If signs of the defect strength ε p under different loading states were known (p refers to the pair of loading states), it would be possible to calculate the coherent sum of backpropagation functions with correct signs:

BP F (r, t) = P p=1 ε p bpf p (r, t), (4)
and then calculate the integral:

I(r) = T0/2 -T0/2 |BP F (r, t)| 2 dt (5)
instead of (2). However, as the signs of the "defect strength" (further: "sign sequence") are unknown, the suggestion here is to choose them randomly and obtain an image as it is described above. After that one should test quite a large number of different sign sequences. Then comparative study is based on the concept of the "image contrast" which is retrieved from:

C(ε p ) = max r I(r)| εp 1 N pixels r I(r)| εp , (6)
here N pixels stands for the number of considered candidate defect positions.

The image with highest contrast is considered as a final result (the corresponding sign sequence is an optimal one).

In the Fig. 4 the results without and with random sign compensation are shown for 1000 different sign sequences. It is worth noting that the application of the sign compensation technique increases the signal-to-noise ratio in 15 times; speckle-like artifacts are significantly reduced.

CONCLUSION

A reference-free defect localization technique based on a pump-probe SHM experiment is proposed. Preliminary measurements were conducted in a thin aluminum plate having an artificial defect (Hertz contact: steel ball pressed against the plate). The repetitive probing experiment is robust to locate the position of the contact nonlinearity related to the defect. The algorithm works for different defect positions, even for defects located outside of the zone covered by receivers or closely to the plate borders. For the certain pump amplitude below the frequency threshold of about 4Hz the defect can be successfully localized without a hypothetical reference. Signal to noise ratio of image can be increased in approximately 15 times by using the random sign compensation technique.

Future works will be focused on adaptation of the developed algorithm for passive SHM i.e. when weak ambient vibrations of a structure accompanying its operation are used instead of an active low frequency pump.

ACKNOWLEDGEMENT

This work was supported by the French National Research Agency (ANR). Grant number ANR-17-CE08-0013-01 (PANSCAN project).

xxxFigure 1 .

 1 Figure 1. Experimental setup.

Figure 2 .

 2 Figure 2. Typical pump-probe signals.

Figure 3 .

 3 Figure 3. Imaging localization results at different pump frequencies: (a) f pump = 0.5 Hz; (b) f pump = 1.5 Hz; (c) f pump = 2.5 Hz; (d) f pump = 4 Hz.

  xxx

Figure 4 .

 4 Figure 4. Imaging localization results at different defect locations at f pump =1.5 Hz without random sign compensation (left column) and with it (right column).


domain: spi info phys

text:
Marina Terzi

Lynda Chehami
email: lynda.chehami@uphf.fr

Emmanuel Moulin

Aleshin Vladislav
email: vladislav.aleshin@univ-lille.fr

Baseline-Free Repetitive Pump-Probe Experiment for Structural Health Monitoring

Keywords: Imaging, Non-linear acoustics, Pump-probe, Baseline-free NDT

In structural health monitoring, it is often necessary to make reference measurements on a baseline healthy sample. However, baseline parameters can be strongly affected by environmental and other conditions and significantly change over time. Here we propose a baseline-free damage localization method consisting of a repetitive pump-probe experiment in a metal plate and implementation of a back-propagation algorithm. In the reported experiment, a PZT patch generates a high-frequency (HF) probe wave at 20 kHz, while a continuous low-frequency (LF) pump of 1 Hz is produced by a shaker. Signals propagated in the plate are recorded using a set of PZT sensors glued at known positions. We use a steel ball pressed against the plate to mimic a nonlinear defect. The aim here is to produce a solid-solid contact that will be modulated by the LF pump, as would be the case, for instance, in fatigue cracks. To enhance this effect, we subtract signals recorded at different times (corresponding to different loading states) and apply the back-propagation algorithm that locates the origin of the modulation. The procedure is repeated several times to enhance the localization quality. In future, we plan to use the method with a real defect, thus exploiting more realistic contact acoustic nonlinearity.

INTRODUCTION

Structural Health Monitoring (SHM) refers to a permanent inspection of a system or structure in service and includes damage detection, structural integrity assessment, as well as damage evolution prognostics and lifetime estimations. Permanent monitoring should be capable of detecting fatigue damage that appears in materials under repetitive loading. Accumulated fatigue damage may lead to catastrophic consequences, especially when accompanied by environmental hazards or extreme events. A wide class of SHM systems uses ultrasound; modern ultrasonic SHM [START_REF] Johnson | Nonlinear Elastic Wave NDE II. Nonlinear Wave Modulation Spectroscopy and Nonlinear Time Reversed Acoustics[END_REF][START_REF] Mohd Aris | Structural Health Monitoring of a Pitch Catch Active Sensing of PZT Sensors on CFRP Panels: A Preliminary Approach[END_REF][START_REF] Zhang | Nonlinear mixing of ultrasonic coda waves with lower frequency-swept pump waves for a global detection of defects in multiple scattering media[END_REF] allows one not only to detect the presence of damage but also to locate its position. In that sense, SHM is similar to imaging techniques such as ultrasonography, laser vibrometry, shearography, etc.

Many of existing SHM techniques require additional measurements on a baseline intact sample for comparison with the actual sample under study. However, even a significant difference in measured properties does not necessarily indicate the presence of damage. The matter is that materials' characteristics can slowly evolve due to aging, environmental effects, etc. In addition, measurements in the original state of a structure are frequently not available or not possible. The problem can be solved by using nonlinear methods based on strong acoustic nonlinearity induced by damage. In this case, certain nonlinear criterions should be obtained instead of a simple difference in linear properties. Hence, nonlinear SHM is baseline-free.

Modern nonlinear ultrasound damage diagnostics techniques can be divided into 2 groups: "pass-fail" tests and imaging techniques.

"Pass-fail" tests serve for detection of the presence of the nonlinear defects (such as cracks, delaminations, deboning etc.). In recent publications [START_REF] Donskoy | Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing[END_REF][START_REF] Johnson | Nonlinear Elastic Wave NDE II. Nonlinear Wave Modulation Spectroscopy and Nonlinear Time Reversed Acoustics[END_REF][START_REF] Chigarev | Nonlinear frequency-mixing photoacoustic imaging of a crack[END_REF][START_REF] Novak | Nonlinear acoustic characterization of micro-damaged materials through higher harmonic resonance analysis[END_REF][START_REF] Zhang | Nonlinear mixing of ultrasonic coda waves with lower frequency-swept pump waves for a global detection of defects in multiple scattering media[END_REF][START_REF] Hilloulin | Small crack detection in cementitious materials using nonlinear coda wave modulation[END_REF][START_REF] Shui | Nonlinear ultrasonic evaluation of the fatigue damage of adhesive joints[END_REF] "pass-fail" tests have been conducted using diversity of methods from classic reference-free nonlinear resonance and harmonic generation [START_REF] Novak | Nonlinear acoustic characterization of micro-damaged materials through higher harmonic resonance analysis[END_REF] or nonlinear wave mod-ulation spectroscopy [START_REF] Johnson | Nonlinear Elastic Wave NDE II. Nonlinear Wave Modulation Spectroscopy and Nonlinear Time Reversed Acoustics[END_REF] to the late reference-based frequency mixing of coda waves with lower frequency-swept pump waves [START_REF] Zhang | Nonlinear mixing of ultrasonic coda waves with lower frequency-swept pump waves for a global detection of defects in multiple scattering media[END_REF][START_REF] Hilloulin | Small crack detection in cementitious materials using nonlinear coda wave modulation[END_REF] which is sensitive only for nonlinear defects.

As regards to the nonlinear imaging techniques providing damage localization opportunity, all of them are reference-free including nonlinear time reversed acoustic methods [START_REF] Johnson | Nonlinear Elastic Wave NDE II. Nonlinear Wave Modulation Spectroscopy and Nonlinear Time Reversed Acoustics[END_REF], nonlinear photoacoustic imaging [START_REF] Chigarev | Nonlinear frequency-mixing photoacoustic imaging of a crack[END_REF] and break of reciprocity principle [START_REF] Huang | Baseline-free damage detection in composite plates based on the reciprocity principle[END_REF].

In this paper, we describe a similar imaging method based on a pump-probe experiment, in which of information of damage is provided by differences in measured HF probe signals. These differences are induced by a changing defect loading state which, in turn, is influenced by powerful LF pumping. The slight differences in HF reverberations are then used for localizing the defect. This is done by the application of the back-propagation algorithm developed in [START_REF] Chehami | Detection and localization of a defect in a reverberant plate using acoustic field correlation[END_REF] for thin plates. The algorithm is based on the fact that a signal propagating in a plate following the path emitter-defect-receiver experiences certain phase transformations depending on the path length. These transformations can be compensated for by introducing an inverse phase factor corresponding to back-propagation. As a result, after a subsequent averaging over various emitter-receiver pairs a nonzero magnitude related to the emitted probe wave will be retrieved. However, if at this hypothetic defect's position (called pixel) there is actually no defect, the sum is destructive. By interrogating all pixels at the plate and by the multiple application of the back-propagation algorithm, a full image can be produced. This imaging method has been applied (see Chehami et al. [START_REF] Chehami | Experimental study of passive defect localization in plates using ambient noise[END_REF][START_REF] Chehami | Nonlinear secondary noise sources for passive defect detection using ultrasound sensors[END_REF]) for linear defects such as holes or rigid inclusions and is adapted here for a model defect mimicking the behavior of a real nonlinear one. In our preliminary experiments, we use a steel ball pressed against an aluminum plate. Similarly to a crack or delamination, the ball excited by the LF pumping slightly modifies the propagation conditions for the HF probe. These weak changes are considerably enhanced by the application of the back-propagation algorithm due to multiple averaging over different loading states. In the next section, the experimental procedure and signal processing are described in greater detail. Results of imaging are presented in Section 3 followed by concluding remarks in Section 4.

REPETITIVE PROBING SETUP AND SIGNAL PROCESSING

In these series of experiments, the objective is to localize the origin of modulation due to a solid-solid contact (here it is a steel ball pressed against the surface of a plate) on a thin plate using ultrasound.

Experimental procedure

An aluminum rectangular plate (1 m × 0.5 m × 3 mm) is horizontally suspended with elastic cords on a metallic support. A 1 cm diameter steel ball is pressed against the top surface of the plate with elastic steel ruler fixed on the support cage (see Fig. 1).

The ball plays the role of a nonlinear defect (i.e., fatigue crack) to be localized. In order to find out the defect position, a pump-probe experiment has been conducted. One PZT patch (glued to the plate to provide acoustical coupling) generates a high-frequency (HF) probe wave which is a 100 ms burst with 1 cycle of 5 Vpp sine wave at f probe = 20 kHz (see Fig. 2-(a)).

At the same time, a vibrating shaker rigidly attached to the plate and to the support cage produces a lowfrequency (LF) pump wave which is a continuous 3 Vpp sine wave at f pump = 1 Hz (see Fig. 2-(b)). LF shaker parameters are so as plate continuously performs vertical oscillations, in a such a way that the position of the ball and the plate-ball contact loading state (stress, displacement) slightly changes all the time. The reverberated signals are acquired at 7 PZT patches similar to the PZT emitter, using National Instrument acquisition card (500 kS/s, 8 channels). The signals are amplified at reception. The total length of the HF probe reverberated signals is 50 ms which is much less than one period of the pump wave. The acquisitions are repeated for N = 20 times while HF and LF stimulus are excited. An average time between 2 acquisitions is 50 ms. These different acquisitions correspond to different contact loading states and defect position. An example of received exponential-decayed signal is shown in Fig. 3.

The idea of these different acquisitions is to collect as much data (information) as possible for different defect state without removing the defect. Elastic rubber cords attaching shaker to the support cage makes the shaker "invisible" in the resulting images (shaker-plate contact assumed to be negligible).

Signal processing: back-propagation algorithm adapted to repetitive probing experiment

Let us denote by s j m ( j = 1...N T ) the m th acquired signal at known receiver position r j , N T is number of receiving transducers. To enhance the effect of modulation, one should make subtractions between signals at different loading states, i.e., s j m+1s j m (m = 1...N acquisitions). We obtain then the subtractive matrix denoted by ∆s m . We note that for N acquisitions there are N s = (N 2 -N)/2 possible subtractions.

The second step is discretizing the given structure in a mesh grid with a given spatial step ∆x = ∆y. Then one should compensate wave phase shift due to wave propagation via point (x,y) (pixel). The principle is to take the element of index j of ∆s m matrix and back-propagate it according to the current pixel position (x,y). The back-propagation function (bpf) of ∆s m is estimated from Eq. ( 1):

bp f m (x,y) (ω) = N T ∑ j=1 ∆s j m (ω) e i k(ω) r E j (x,y) , (1)
here r E j (x, y) is the distance between the emitter E and the jth receiver via the pixel at position (x,y). The wavenumber k obeys to the dispersion relation of A0 Lamb mode (flexural wave). Matrix element ∆s j m (ω) is the Fourier transform of ∆s j m (t). Only the time interval that includes the direct propagation between the defect and the receivers is kept. In practice, due to the electronic setup, the response of the transducer toward itself cannot be measured. For this reason, the element∆s EE m (ω) is not considered to compute bpf. To obtain the scattered energy in each pixel for a certain subtraction one should go back to the time domain through an inverse Fourier transform, the pixel energy at position (x,y) is then obtained by integrating the backpropagation function over time T 0 (accordingly to the approximation of back-propagation in an infinite plate), i.e.,

E m (x, y) = T 0 /2 -T 0 /2 bp f m (x,y) (t) 2 dt, (2)
where T 0 is typically the inverse of the bandwidth. For pixels located on the defect, this process will be equivalent to a numerical back-propagation of the signals to zero time (instant of the emission) followed by a coherent summation of the back-propagated signals. This will lead to a constructive sum and a local maximum of the pixel energy is obtained as shown on Fig. 4-(top). For pixels located elsewhere, the obtained energy will be made up of a summation of non-coherent contributions corresponding to the reverberations at the plate boundaries as shown on Finally, we get the cumulated energy by averaging Eq. ( 2) over all N s subtractions (i.e., all the pairs of loading states):

E(x, y) = ∑ N s E mn (x, y) N s . (3)
Localization results are shown and discussed in next section.

RESULTS AND DISCUSSION

The obtained results are shown on On the obtained images we can see that it is possible to localize defect inside and outside the transducers area. In comparison with the results of a conventional back-propagation algorithm applied for baseline experiments (see L. Chehami et al., [START_REF] Chehami | Detection and localization of a defect in a reverberant plate using acoustic field correlation[END_REF]), in the current images signal-noise-ratio (SNR) is lower: there are many artifacts. This can be interpreted by the lack of information about the modulation effects: in contrast to the array of emitters used for baseline experiments, in pump-probing experiments only one transducer plays role of emitter. Consequently, there are some preferred directions, mutual location of emitter, receivers and defect matters. Finally, we should mention that some problems can occur if during the experiment acoustic emission (squealing) appears.

CONCLUSIONS

Here we proposed a baseline-free repetitive pump-probe experimental procedure to detect a contact-like nonlinear defect. The localization of the origin of modulation based on adapted back-propagation algorithm has been described. Efficiency of the algorithm has been illustrated with localization of a 1 cm steel ball pressed against vertically oscillating aluminum plate. Imprecision of this algorithm is connected with an asymmetry of transducers geometry (only one of them plays the role of emitter), in contrary to conventional beamforming algorithm, where all the transducers are at the same time both emitters and receivers. Thus, in this configuration there are preferred directions, and a success of defect localization depends on mutual location of defect and transducers. Consequently, there is also dependency on the choice of emitter. For the optimized emitter and parameters of both pump and probe waves, defect can be successfully localized without an hypothetical reference (without damage).

These preliminary results are very encouraging for SHM, especially for nonlinear acoustics. Future works will be focused on the improvement of the SNR images and adapting the current algorithm for passive SHM, without need of ultrasonic emitter.

Figure 1 .

 1 Figure 1. (left) Pump-probe experimental setup. (right) Nonlinear defect "solid-solid" contact.

Figure 2 .Figure 3 .

 23 Figure 2. Typical signals: a) HF probe provided with PZT patch at 20 kHz, and b) LF pump provided with shaker at 1 Hz.

Fig. 4 -Figure 4 .

 44 Figure 4. Example of back-propagation functions when pixel (x,y) is: (a) on the defect position, and (b) elsewhere.

Fig. 5 .

 5 The positions of transducers are indicated by crosses and that of the shaker and defect by, respectively, a circle and a square. The origin of coordinates is taken at the left lower corner of the plate. The spatial pixel resolution is ∆x = ∆y= 5 mm. One of the transducers array is used as emitter for each figure.

Figure 5 .

 5 Figure 5. Imaging localization results obtained for a nonlinear solid-solid contact located at: (a) [0.695,0.264] m, and (b) [0.4,0.145] m. (a-right) Zoom on the receivers area. The positions of the shaker and defect are, respectively, indicated by a circle and a square. E is the emitter.

ACKNOWLEDGEMENTS

This work has been supported by the French National Research Agency (ANR). Grant number ANR-17-CE08-0013-01 (PANSCAN project).