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In sum, then, the leaf had to follow a curve having the same base, height, and length as the cycloid to be traced. That it might be the cycloid itself was quickly confirmed in a rough sketch of a demonstration that see Figure 3 :. Specifically, Wren showed that any arc FB of a cycloid FBA see diagram immediately following in the text is twice the chord FL subtending the arc FL through which the generating circle has rolled. Pascal's Histoire de la roulette made particular note of Wren's rectification see n.

Oxford, ; cf.

A draft of the revision HOC. In October , he again spoke of delay HOC. Thereafter, he said nothing about the treatise until , by which time the work had been expanded to include the center of oscillation and the compound pendulum. But the results of the first sea trials of Huygens' marine clock prompted him to withhold the text for commercial reasons see below. In the original Horologium Huygens had pointed to the development of a portable version of his clock as the key to transforming it from an astronomical instrument to a navigational tool. In particular, since it was accurate to within seconds over a day, the pendulum clock could measure small variations in the length of the solar day as marked by the sun's successive passages through the meridian.

When compared with theoretical calculations, the measurements offered astronomers empirical means of resolving longstanding questions about the parameters governing the inequality of the solar day and hence of settling on values of the equation of time to be used with the clock in recording solar events. An accurate table of the equation of time made it possible to use the sun and its shadow, instead of the stars, to regulate the clock.

That made regulation easier and more accessible to those using the clock. The phenomenon of the inequality of the solar day had been known since antiquity. Ptolemy rendered an account of it in Chapter 9 of Book III of the Almagest , and sophisticated sun dials had incorporated that account in the form of a figure-8 mean time meridian.

But Ptolemy's explanation could not be tested closely; as he noted, the difference over one solar day or even over several was too small to be measured. Hence, although he set out the mathematics for determining cumulative inequality over long periods, he did not compose a table of daily values. Copernicus followed the same form of presentation. Huygens' clock changed that. Although Parisian clockmakers had already begun to claim for their product that "solis menaces arguit horas" The Flammarion Book of Astronomy , New York, , p. The resulting corrections in solar time made measurements of the moon's motion and of eclipses correspondingly more accurate.

Huygen's various instructions for the clocks' use, especially his Kort Onderwijs see below, n. The principle of determining longitude is straightforward. The earth turns uniformly from west to east through o in one mean solar day of 24 hours. For every hour's difference in local time between two points there corresponds a difference of 15 o in longitude. If, then, a traveler carries with him a clock regulated to mean solar time and set to the sun at his starting point, he can determine his longitudinal distance from that point by comparing the time of the sun where he is to the time given by the clock.

After only a few days' travel, however, he will have to correct local time for the accumulated inequality of the solar day. Huygens first began investigating the equation of time as soon as he had built his first clock. He found some modern astronomers like Ismael Boulliau at odds with their predecessors over the structure of the phenomenon, and he used his clock first to decide between them. As the dialy differences accumulate, a clock regulated to the mean solar day I. For 10 February as base the cumulative inequality remains positive throughout the year, and Huygens chose it for that reason.

Or, in Copernican terms, from the eccentricity of the earth's orbit and the inclination of its axis to the plane of the orbit. Huygens, like navigators down to the present, found it convenient to analyze the phenomenon in Ptolemaic terms. The dates of the turning points are Huygens', which differ from the modern values reported in Flammarion's Astronomy New York, , p.

Huygens spent two years calculating and confirming his table of cumulative inequality, which by subtraction provided the inequality over any number of days, and sen out the first public copies in February After the clock had run continuously for several days, one then noted the time of the sun's passage through the meridian.

To that time one added the difference of the cumulative equation for the day of the second reading less that for the day of the setting; if the former was less than the latter, one subtracted the difference. The result was supposed to be ; any discrepancy, divided by the number of days the clock had been running, gave the daily advance or retardation.

When applied to finding longitude, the table first aided in calibrating the clock and then in correcting its readings for the elapsed inequality before they were compared with local sun time at sea. Hence the table formed as much a part of the navigator's kit as did the clock and the quadrant. At about the same time that Huygens made public his table of equation of time, he also announced an addition to his clock that made its regulation by the table even easier and more accurate: the sliding weight.

It was the practical payoff of another theoretical breakthrough, the determination of the center of oscillation of a compound pendulum. Neither the simple nor the cycloidal pendulum clock in itself demanded that breakthrough. Each already had provision for moving the bob up and down and thus for adjusting the clock to agree with the stars, with the corrected sun, or with another clock.

But in musing on the wider implications of his invention Huygens had imagined that the length of a simple pendulum that beat isochronically with the 1-second pendulum of a precisely calibrated clock could serve as a universal standard of measure: one third of that length would be a pes horarius , a 'clock-foot'. Huygens' colleagues in the Royal Society proved especially receptive to the notion and in began experiments on both simple and cycloidal pendulums.

In particular, pendulums of different materials seemed to behave differently. It is not clear from the evidence when Huygens hit upon the notion of the compound pendulum and the associated concept of the center of oscillation. His working notes show that he began with the case of two dimensionless weights swinging together on an inflexible weightless rods, and his solution established the paradigm for analyzing all more complex systems. What follows here preserves the content, but not the precise order of Huygens' derivation. On this fruitful principle, named variously after Torricelli and Mersenne, and on its central role in Huygens' mechanics, see Alan Gabbey, "Huygens and Mechanics", Studies on Christiaan Huygens Lisse, , To consider the dynamical effects of these kinematical results, suppose that, on reaching centerline AD , the weights B and C strike directly weights G and F respectively equal to them and standing at rest.

By the laws of impact, B and C will stop, and G and F will move off at speeds of vel. B and vel. C , respectively. Suppose further that G and F are then reflected upward at those speeds. By Galileo's law those heights are as the squares of the speeds, and OP provides a basis of comparison. Therefore, or similarly,. As the diagonal lines in the diagram suggest, Huygens imagined the bodies to bounce off immobile inclined planes.

He had used the same analytic device in his treatise on impact. From the situation of bodies falling through certain heights, communicating their motion to other bodies, and thereby driving the latter to certain heights, it follows, Huygens asserts, that the composite center of gravity of the spheres G and F , after they have taken motion from E and D and have converted it upward as far as they can, that this center I say rises to the same height as that of the center of gravity of the spheres B and C.

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In later versions Huygens eliminated the medium of impact and simply imagined the bodies constituting the compound pendulum to be simultaneously released from constraint and to be individually directed upward at the speeds acquired at the point of release. But the basic principle remained the same. At the start of the swing the pendulum's center of gravity, which is fixed by the weights of its constituent parts and by their distribution over it, lies at a certain height. Whether or not the parts remain connected over the length of the swing, they come to rest only when their common center of gravity has again reached that initial height.

The difficulties of applying the principle lie in determining the initial and final positions of the center of gravity under the assumption of dissolution during the swing. In the case of a uniform rod of length a swinging about one end, Huygens expressed these two positions by a triangular and a parabolic area, respectively, equating an initial condition of and a final condition of to determine that. As a corollary to this last derivation, Huygens noted that "whatever weight is added to the rod at that place, it will make the swings no faster or slower.

From here he could move toward his goal of regulating pendular motion. Clearly, weight added to the bar below the center of oscillation would slow its swings, and added above the center would speed them up. Consider first a weight Q added to the very bottom see Figure 5. If a is again taken as both the length and the weight of the rod, and n is to AD as the weight Q is to the weight of the rod, then n simply represents the weight of Q.

To the determination of the initial center of gravity, Q adds the term nd ; to that of the final center of gravity, the term Hence, , whence a value for x directly follows. They now yield the equation:. Of greater interest here than the solution of x is that for c :. Huygens took explicit note of the explicit corollary to this last result: for a given x less than the length of the simple pendulum isochronic with the compound pendulum consisting of weight Q and rod AD alone, c has two positive values and hence determines two different positions for weight H. But he did not pursue the physical implications of that mathematically dictated finding.

But if one knows the length of a simple one-second pendulum, one also knows the length of a simple pendulum that deviates from it by any fixed interval, say, 1' over 24 hours. Moreover, from the results obtained above one can determine the distance along a rod at which to place a fixed weight so that the compound pendulum has a period of 1 second i. A small weight added to that pendulum at the center of oscillation will not affect its period. If placed elsewhere, the weight will change the period, and one can calculate where to place it to cause a precise change.

One can, that is, mark off on either side of the center of oscillation the lengths c at which the small weight effects any desired deviation. That is the principle behind the poids curseur , or sliding weight, that Huygens first announced to Moray on 30 December The phrase "depuis quelque temps" cannot point very far back, since by all evidence the revision of the Horologium carried out in did not include any consideration of centers of oscillation.

Hence, although the notes and calculations that appear to date from the fall of indicate that Huygens had made a start on the problem, he did not gain full control of it until sometime in It was probably then that he set down the derivations outlined above and calculated a table, found in his notebooks, which lists the settings of a sliding weight for gains over 24 hours of up to 2 minutes by 5-second intervals.

Huygens' first set of results made some headway in accounting for the discrepances uncovered by English experiments, but it was not until that he could explain why the size of the bob made a difference. By then he had developed the mathematical techniques to apply his basic principle to various solids, most important among them the sphere. He could not locate the precise center of oscillation of a pendulum consisting of a fine wire and a round bob.

It lay a significant distance from the bob's geometrical center, and for a fixed length of wire that distance depended on the bob's radius. The scale ruled out in the first plate of the Horologium oscillatorium , p. Horologium oscillatorium , Part IV, Prop. XXII , pp. By , then, Huygens had all the elements of a theoretical treatise on the pendulum clock.

But by then he also had reason to hope that the treatise could offer stunning evidence of the theory's practical value. In and the Royal Society had sent a marine version of the clock to sea to test its reliability and its accuracy as a means of determining longitude. The first results had exceed expectations.

The first marine pendulum emerged from the collaboration of Huygens and the Scot Alexander Bruce during November and December Two of the clocks went with Bruce to London early in and, after minor setbacks prompting the intervention of the Royal Society, accompanied Captain Robert Holmes on voyages first to Lisbon and then to Guinea and out into the Atlantic.

The second voyage in added a dramatic touch. Having set sail due west from the island of St. Thomas off the Guinea coast, Holmes went some leagues before taking a northeast course back toward the African coast. After several days, supplies of fresh water began to run low, and Holmes's fellow masters urged that he head for the Barbados. They reckoned that the squadron was still some leagues from the Cape Verde Islands.

But Holmes's clock placed him only 30 leagues distant, a day's run. He continued his course and hit land the next afternoon. Huygens learned later that the ships had been becalmed for a time after heading northeast and had drifted with the current some 80 leagues eastward; traditional piloting could not detect that motion.

He did not want to releasethe details of the clock, and he expected soon to have systematic data to back up the drama of the clock's seaworthiness. Again in the clock met with initial success in measuring the longitudinal difference between Toulon, Crete, and several intermediate points. The experiment would benefit from the lessons of the earlier voyages and, in particular, would silence the critical murmurs coming across the Channel in Oldenburg's letters. According to Huygens' account, which no one at the time denied, the voyage was a disaster.

Shortly after sailing, one of the clocks stopped in a storm. After a delay of some hours, the second clock also stopped. Again ignoring instructions, Richer simply abandoned the tests altogether and allowed the clocks to deteriorate in their mountings and eventually to crash to the deck. The ship was not properly fitted, nor wqas anyone suited to carry out the texts. It was, therefore, in a mood more of resignation than of triumph that Huygens decided in to proceed with the publication of the Horologium oscillatorium , which appeared in the spring of In it he presented the clock with the addition of cycloidal cheeks, a revised escapement, and an adjustable pendulum.

He also described with illustrations the newest, as yet untested marine model with a triangular suspension of the pendulum and a Cardan mounting for the clock. He condensed his Kort Onderwijs to a few paragraphs and recounted the results of the , , and trials. He then took up in separate sections his analyses of the fall of heavy bodies and their motion along a cycloid, the evolution of curves, and the center of oscillation. The last concluded with the theory of the sliding weight, the design of a 1-second pendulum as a universal standard of measure, and the measure of the vertical distance through which a body falls freely in a given time.

To this central theme he added a brief account of the conical pendulum with escapement and cheek to keep it on its tautochronic paraboloidal surface. As an appendix he offered his theorems --but not their demonstration-- on the measure of centrifugal force. Huygens at first had trouble getting additional details about the voyage. Moray took until early March to run down another ship's officer, who confirmed the account and provided enough information for Moray to think that the original figures needed revision ibid.

Not until 27 March did Moray report the news about the currents ibid.

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Kort Onderwijs aengaende het gebruyck der horologien tot het vinden der lengthen van Oost en West, O. Huygens composed the instructions even before learning the results of Holmes's second voyage. That translation did not appear until in the Philosophical Transactions Vol 4, no. Huygens undertook a French translation, but postponed publication until he could add data from later voyages. Huygens' various attempts to secure the rights to his clocks would shed light on the early history of patents and warrant careful study. Francis Vernon heard Huygens' response viva voce , indeed vivissima voce ; cf.

Huygens to [Duhamel? VII, , for Richer's report. That source details Richer's measurement in Cayenne of the length of a one-second pendulum. In the years immediately following the appearance of the Horologium oscillatorium Huygens left the marine clock and the method of longitude in abeyance and turned instead to further exploration of the phenomenon of tautochronic oscillation. The new line of inquiry began with a theretofore unnoticed corollary of his analysis of motion on a cycloid: the effective motive force acting on a body at any point of the cycloid is proportional to the arc length from the vertex to that point.

Within a year Huygens had transformed that theoretical insight into a practical mechanism. The eureka is dated 20 January for the first sketch of a watch balance regulated by a coiled spring. Various preoccupations over the next few years, not the least of these being bitter priority disputes with Isaac Thuret, Robert Hooke, and Jean de Hautefeuille over the invention of the spring balance, distracted Huygens from extending it beyond its initial application to pocket watches. Work on such a seq-going version reached by a state sufficiently promising to attract formal encouragement from the Directors of the Dutch East India Company as a possible means of determing longitude at sea.

Although the evidence is quite thin and circumstantial, it appears that Huygens' experiments with springs themselves and with spring-regulated clocks had by borne out Oldenburg's warning of springs are very sensitive to changes in temperature and humidity. Considering each cord as a pendulum with one third of the ring's weight as bob, he invoked the displacement principle to show that the motion would be tautochronic if the pendulum followed a certain parabola, which he then demonstrated to be very close to the actual space-curve generated by the endpoints of the cords.

Noting that the period but not the approximation to tautochrony depended on the square root of the length of the cords and on the size but not the weight for that size of the ring, he foresaw calibrating the pendulum by small weights movable radially on the ring; one of his drawings even shows small cheeks about the suspension of one of the cords, apparently to guide the tricorn pendulum into the proper parabolic paths.

Mersenne knew from musical experience that a string of given length and tension is tautochronic: whether strongly or weakly plucked, it sounds the same note. Harmonicorum libri Paris, 2 nd ed. Of HOC. For Huygens' analysis by means of the tautochrony of cycloidal oscillation, see HOC. It is not clear when or where Huygens gained his knowledge of what is known as Hooke's Law. Hooke himself did not announce the law -ut tension sic vis- until in his De potentia restitutive cf. Writing to Leibniz in HOC. There follows immediately on that sketch, found in Ms.

E, an almost day-by-day account of Huygens' dealing with Thuret from 21 January to 25 February. Although that account, evidently composed in retrospect, may shae some circumstances in Huygens' favor, one datum is fixed by external reference, namely, the anagram sent to Oldenburg on 30 January; cf. The various contentions dominate Huygens' correspondence of until his final settlement with Thuret in September HOC. To pursue the claims and their justification would go beyond the scope of the present paper, although in the case of Thuret it seems clear that he deserved credit for designing an effective escapement; cf.

Moreover, it should be noted -as it does not seem to be in HOC- that Huygens had heard from Moray in about an idea of Hooke's to use a spring instead of a pendulum to regulate a balance wheel. Huygens all but dismissed the notion at the time; cf. His replies of 18 September, HOC. Both the French privilege above, no. A highly imaginative device posing significant difficulties for mathematical analysis, the tricorn pendulum nonetheless proved practicable within a short time.

On 17 December drawings and a rough model went to the clockmaker Johannes van Ceulen, who soon produced two clocks with the new regulator. After some adjustments in design during the spring, Huygens had a clock capable of convincing the East India Company in July to pay van Ceulen and to place a boat at Huygens' disposal for trials on the Zuyder Zee.

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At about the same time Huygens began to sketch the first versions of what emerged in early as his balancier marin parfait. Two of the systems had originally been designed in to counterbalance the centrifugal force of a conical pendulum and to pull it back into a paraboloidal envelope: the first consisted of two piles of chain resting on a common support; the second, of two weights partially immersed in mercury.

In each case, the torque acting on the wheel varied as its displacement from equilibrium and hence produced tautochronic oscillation. As noted above, the spring balance and its successors prompted Huygens as early as to propose new sea trials of his method of determining longitude by means of clocks. The time the Dutch were the first to respond, starting in The finely detail instructions prepared by Huygens for one of the attendants, Thomas Helder, on 23 April incorporated not only all the procedures of the Kort Onderwijs but, with the memory of Richer's behavior perhaps still fresh set out explicit rules for the mounting, regulatin, and maintenance of the clocks.

4. Clocks at Sea

The ship Alkmaar carrying the clocks returned to Texel on 15 August Helder did not return with it; shortly after leaving the Cape he had died at sea. Moreover, on the outgoing boyage he had had trouble with the clocks, which had responded badly to heavy seas. The other attendant, Johannes de Graaf, picked up the experiment and brought home enough measurements for Huygens to plot a course that could be compared with that of the fleet's pilots.

Huygens' busy preparation, reported to brother Constantijn, 6 April , ibid. In remontoir clocks the escapement is driven directly by a small spring or weight, which is rewound -hence the name- at very short intervals by the mainspring or main weight. Ideally the clocks have the advantage of maintaining a small, steady force on the escapement and hence on the pendulum or spring balance.

Huygens was experimenting with both types of remontoir as early as cf. Huygens' three-color map is reproduced at the end of HOC 9; cf. Having examined the logs kept by both assistants, Huygens knew he could not fault their work. Instead, he concluded that he had to take seriously a perturbing effect of which he had first heard in about but which he had then considered unproved.

On the expedition to Cayenne -- on which Huygens had refused to send his new clocks -- Richer had carried out instructions to determine the length of a 1-second pendulum. The deviation was of the same order as the equation of time and hence had to affect the method of longitude. Although the foot differed in length from region to region, all systems subdivided it into 12 inches pouce, duym and lines lignes, linie. Richer was not alone in reporting variations in the length of the 1-second pendulum.

Yet, astronomers and geographers of the stature of Jean Picard doubted these findings and insisted that the length was constant. To account for the seeming failure of his clocks on the Alkmaar , he reasoned that the earth's rotation produced a centrifugal force that diminished the weights of bodies by a factor dependent on their latitude.

A body at either pole suffered no diminution; at the equator it underwent a maximum decreased of roughly of its weight. In navigational terms, a clock carried along the same meridian from higher latitudes toward the equator would appear to indicate a longitudinal shift to the east. Huygens was suspicious of Richer's failure to record the details of his experiment; cf. Huygens offered no derivation here, since he had already do so in inhis treatise De la cause de la pesanteur , which he was not revising for publication Paris, Huygens' theoretical calculations of the centrifugal effect agreed on the one hand with his treatise on the cause of weight, which he had composed in but which only now found reason to publish, and on the other with most of the reported lengths of 1-second pendulums.

More importantly they brought de Graaf's raw data into general agreement with the course plotted by the fleet's pilots. The agreement was not complete; Huygens knew it could not be so. The pilots had based their calculations on an incorrect longitude for the Cape, which had meanwhile been determined quite accurately by means of the satellites of Jupiter.

Reviewed and generally approved by Burchard de Volder, Huygens' results encouraged the East India Company to undertake another trial, again under the supervision of de Graaf, in The outcome disappointed everyone. Although Huygens salvaged the measurements for the longitudinal distance from St. Iago inthe Cape Verde Islands to the Cape of Good Hope, and although he could cite errors in de Graaf's handling of the clocks, he had to admit that his marine pendulums did not perform well at sea.

He continued to look confidently toward yet further trials. See, for example, Huygens' note in the Acta eruditorum October , ff. To the end, Huygens' work on time and longitude at sea continued the interplay between theory and practice that had characterized his earliest efforts. Each sea trial pitted the calculable world of theoretical mechanics against the arbitrary reality of wind and water. Each trial yielded new events and data to be incorporated as perturbations of the mathematical model.

Yet, even while that combination of theory and practice led to ever more refined mechanics and timepieces, it also encountered the limits of Huygens' theoretical world and of the realm in which he was willing to practice. Faced in with the fact of geographical variations in the length of a one-second pendulum, Huygens turned to his mechanical theory of weight and to the measure of the perturbing centrifugal force. He did not turn back to the spring balance, which is immune to that perturbation. The factors that perturbed the spring resided in its chemistry, in the matter within it and in the effects of temperature and humidity on that matter.

Between these factors and the science of motion stood the tedious, dirty, empirical search of men like Mariotte, Boyle, and Hooke, a search in which Huygens had never taken part. As John Harrison showed, metallurgy, not mechanics, held the answer to longitude, and that answer lay behind Huygens' reach. Figure 1: Huygens' original diagram; HOC. Figure 6. Archimedes' Clock From the very beginning of Christiaan Huygens' career as a mathematician and natural philosopher his father referred to him as "my Archimedes", and friends and admirers soon followed suit.

For a scientific biography of Archimedes that emphasizes his mechanical investigations, see Ivo Schneider, Archimedes Darmstadt, Huygens began seeking patents for his method of determining longitude by means of clocks even before the first sea trials had taken place.

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Chl a patterns in the euphotic zone hold information about the processes that govern the vertical structure of phytoplankton abundance, along with other dynamic aspects of the pelagic ecosystem Sarmiento and Bender Knowing and characterizing the shape of the Chl a profile in the water column at each oceanic region and during a given time of the year, is key to calculate total chlorophyll per unit area of sea surface in the euphotic zone Williamson et al. The Chl a values in the euphotic zone are also required as input to primary production models Platt et al.

Assuming a water column with a homogeneous Chl a vertical distribution implies that biomass values can either be underestimated if there are one or two subsurface maxima or overestimated if the maximum is on the surface. Phytoplankton primary productivity occurs in the euphotic zone, which extends to 4. Platt et al. Such maxima are described by four parameters Fig.

Mignot et al. Vertical distribution of chlorophyll a B Z represented by a Gaussian curve with four parameters Platt et al. Richardson et al.


Chl a profiles with two, and even a few with three, maxima have been reported in different areas of the ocean Cullen and Eppley ; Cullen , and the reports of deep chlorophyll maxima do not always present only one maximum or are very deep, hence we will be refer to as Chlorophyll Maxima in the Euphotic Zone CMEZ. To apply the Gaussian fitting to a vertical Chl a profile, it should have only one maximum Platt et al. Hence the aim of this article is to propose an alternative equation to parameterize vertical Chl a profiles, including those with two maxima in the euphotic zone, fitted through the application of genetic algorithms GA.

These algorithms are applicable to the solution of a great variety of optimization problems based on the evolutive behavior of biological systems where each iteration yields a large population of potential solutions which are classified according to the metric analogous to Darwinian for these. GA refer to a computational technique inspired by the theory of evolution. This technique is implemented by applying certain cyclic operators in an iterative set.

The nomenclature used typically refers to ecological concepts Holland The set that is being operated on is called the population and each element of the population is called an individual. Each individual is a tuple and represents a possible solution to the problem. The position of each element in the tuple is called a gene. The value of a gene in a particular individual is an allele.

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The operators most commonly used in the AG are selection, crossing, and mutation. Since these operators modify the population, it changes evolves over time. The state in which a population existing at the end of an iteration is called generation Alander The selection involves evaluating the fitness of each individual. In this context, fitness is a qualitative value that describes the goodness of a solution. Fitness is the criterion used to select individuals to be parents of the next generation.

The cross is the mechanism by which new individuals are generated from individuals of the previous generation. The mutation is able to change alleles in a random fashion. In this work, each gene represents a parameter of the fitted model and fitness is the inverse of the error adjustment Holland Implementing this GA technique stands out for its simplicity Walsh et al.

This technique has been scarcely used in oceanography and there are only a few references available, such as those from Zhai et al. Chl a profiles had one or two maxima which can be located at surface, near the surface, subsurface, or close to the bottom of the euphotic zone, which lead us to evaluate the four models described below:. Study area and location of sampling stations for the January—February cruise.

The number of stations sampled varies with the time of year. Model A one maximum without gradient :. Model B one maximum with gradient :. Model C two maxima without gradient :. The number of iterations performed by a GA to find a solution to a problem is of utmost importance.

In this work, fitting profile time depends on the memory size of the computer that is being used. To determine the optimal number for the best fit, we evaluated profiles from January cruise by implementing all four models with different iterations: , , , , , , and The flowchart used in the GA for obtaining each fitting vertical chlorophyll profiles is shown in Fig.

The population number of individuals and the number of iterations were handled in the same proportion same number of genes and number of iterations. Profile evaluation was performed five times for every iteration number for each model. The results of the Chl a profile fitting were compared by means of the root mean square error RMSE which in turn provides the measurement of the differences in average between the predicted and the observed values Pielke Once we established the optimal number of iterations as , we sought to find the model that best fit the chlorophyll profile data from the four CALCOFI cruises in With the aim of choosing one model that fit the highest number of profiles, we calculated the differences between the RMSE values of models C and D Table 1.

It should be considered that if the percentage of the difference is positive, model D has the lowest value of RMSE. We analyzed in situ data from a total of profiles, from four cruises in CALCOFI 1, winter; 3, spring; 7, summer; 11, fall to determine whether one or two maximum of Chl a were present. This decision was based on the individual observation of each single profile, which was then classified in one or other category one or two maximum.

Once model D was defined as the most suitable, with optimal number of iterations as , we fitted the profiles of the remaining cruises. All profiles were evaluated five times with the model D, obtaining the same RMSE after the third evaluation.

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Number of profiles with one maximum bottom and with two maxima top for each cruise. Figure 6 a—d shows the in situ chlorophyll profiles and the results of the four models. Models A and B adjusted well for the first maximum but do not consider the second one. In Fig. Figure 6 c shows a profile with a maximum Chl a concentration at surface, and second maximum at 50 m. The four models had a good fit of surface values, but models A and B do not fit the second peak.

Figure 6 d shows a profile with decreasing Chl a values with depth, but with a maximum concentration around 60 m. It is interesting to note that Model A reproduced fairly well the subsurface maximum but the surface value was very low. Vertical Chl a profile fitted with four models a , b , c , and d. The black dots represent the in situ data. Also, include transect, station, date, and time of the profile. Figure 7 a—d shows the profiles fitted by models C and D. The first two profiles Fig. Both models have low RMSE values and the differences between them can be associated to slight difference in the profiles slope.

Vertical Chl a fitting with models C and D a, b, c, and d. The black dots represent in situ data. The Chl a vertical distribution in the euphotic zone has been reported in different accounts as a homogeneous distribution; a subsurface maximum gradually decreasing with depth; a subsurface maximum with low surface and at the bottom of the euphotic zone; a deep Chl a maximum and; on some occasions, a double maximum Cullen and Eppley ; Platt et al.