This is the text of a speech given by Sylvia Asquith on 27th September 2017 at the Foundation of Art and Creative Technology (FACT) during the New Observatory Exhibition. Sylvia’s speech was followed by the screening of a short film by Yu-Chen Wang entitled “I wish to communicate with you”.
Good evening ladies and gentlemen.
My name is Sylvia Asquith and I joined the Bidston Observatory staff in February 1947 as Sylvia Brooks. It was a long time ago but I well remember those early days.
I was employed as a junior member of staff comprising six women and two men – Dr Doodson and Dr Corkan. As well as learning to be a meteorological observer I was introduced to two ammazing tide predicting machines! These were kept running continuously from 9 a.m. to 6.30 p.m. (4.30 to 6.30 being overtime) at 2 shillings and 6 pence per hour which is 12½ pence per hour in current money, except that we do not even have ½pence any more. The minimum wage obviously did not exist in those days!
During the wartime years the male members of staff were enlisted in the armed forces while the women gallantly went on with the work which was so vital at that time. They also did fire-watching on the roof and could tackle incendiary bombs very efficiently.
The Roberts-Légé machine was moved to a purpose-built underground room in the grounds and kept running from there as protection in case the Observatory was bombed. The Kelvin machine was already kept in a cellar. There were incidents of bombs falling on the roof and on the hill generally and on one occasion a landmine landed near the building causing windows to be blown out but without causing significant damage.
After the war Dr Rossiter and two female staff were demobbed and returned to their duties at the Observatory.
After joining in 1947, I received instruction on running a tidal machine, stopping at the correct moment and reading off the time showing at the zero point, and noting down high and low waters in succession. Times done first and the high and low to correspond. Also, you need to check the data when taking over from someone else in case they had got it a day out. All the wheels and pulleys are connected by a fine gold wire and represent forces of the moon and sun on the tide. As the biggest influence on the tide is the moon, that is represented by the largest wheel, “M2”, which has the largest amplitude on it. The names denote George Darwin, a relation of Charles, who was first to devise the method of harmonic analysis. The “M” and the letters nearest to it alphabetically refer to the Moon and similarly “S” to the Sun. The “2” means twice, etc.
The machines were stripped down and cleaned regularly by the three Doctors. However, one day they were away on business in London and the belt snapped! Someone remembered that bootlaces sewn together and carefully measured would be a good standby, so a staff member rushed to the shops and came back with laces and with a sewing machine supplied by Mrs Doodson from the house, and a new belt was made. Fitting was a bit tricky but we managed and the work was able to continue. The returning staff were most impressed by the ingenuity of their colleagues.
Returning to the process… The predictions came off the machine and the sequences were differenced and the differences smoothed by a senior staff member. The smoothed predictions were then typed up for publication in the Admiralty Tide Tables and photograph copies made of the originals. Yes, we even had a photographic studio and print room on site. We always tried to work two years in advance to allow for the checking and printing of predictions.
For all tidal predictions a tidal analysis is required using twelve months of actual height values and following the completion of this any year can be put onto the machine going as far back or as far forward as desired. That means that, provided we know the exact date in history for an event, we could identify the tidal conditions existing at that time.
Bidston was also a Met Office recording station and I was put in charge of the observers. I’m proud to say that we received two awards for our Met returns to London, indicating the very high standards, consistency and quality of our recording and reporting.
The one o’clock gun was a feature of the Observatory dating back to Victorian times giving an absolute and accurate timing to enable chronometers for shipping and people and businesses across Merseyside to set their own timepieces by. This was resumed in 1946 and fired electronically every day from Bidston to the gun at Morpeth Dock. Eventually this finished on July 18th 1969 and I had the pleasure of being the last person to fire the gun.
I returned part time in 1967 after ten years working from home – not so much of a modern concept as you may have thought – and continued as a Scientific Officer until retirement in 1990. Yes, I saw many changes in my 43 years association with Bidston, from a staff of 8 to a staff of 80 housed in two buildings. Today computerisation means that predictions now take microseconds where at the start using these machines they took about three days per port, but we are talking pre-computer times and these machines represented the height of technology in their own era and as such deserve their place in history.
I hope that I have given you a flavour of the history and use of these machines and the fantastic team that I worked with to operate them.
We should always remember that the outputs from these machines were used by individuals and organisations across the world who depended totally on their accuracy to help ensure safey at sea and around coastlines.
Thank you for listening and I hope that you enjoy the film.
This short film, by Andy Lane, Andy Heath and Craig Corbett, is part of the Tide and Time exhibition at the National Oceanography Centre, Liverpool. The exhibition showcases two tidal prediction machines – the Roberts-Légé and the Doodson-Légé. The film also explores the history of tidal science in Liverpool and its development as a port.
There is a lot of renewed interest in tide prediction machines and, after many years hidden away in storerooms, some of the machines made in the UK are on permanent display once again. Kelvin’s original 10-component machine is now part of the new Winton Gallery for Mathematics at the Science Museum in London alongside Ishiguro’s storm surge simulator. Two of the machines that were used at Bidston can now be seen at the National Oceanography Centre building in Brownlow Street in Liverpool.
As you may know from articles mentioned in the Resources section of this web site, the tide prediction machines were a way of simulating the tide in terms of its many harmonic components. Each component would be represented by an amplitude and phase lag, called the ‘harmonic constants’, and the machine, which can be considered as a sort of analogue computer, would be programmed to run by providing it with these constants. Of course, the constants would differ from port to port.
That raises the obvious question of where people like Arthur Doodson, and the other operators of the machines, got their constants from in the first place. This short article reviews the main characteristics of one of the machines (the so-called Doodson-Légé machine now on display at NOC) and then attempts to answer the question of how Doodson obtained the constants.
The Doodson-Légé Machine
The Doodson-Légé machine simulates the variations of the ocean tide by representing the tide as a combination of 42 constituents, each of which has a particular amplitude (h) and phase lag (g). The values of h and g are ‘programmed’ into the settings of the 42 wheels, and the nickel tape that wraps around and connects all the wheels serves to sum up all the constituents.
What are these constituents? Mathematically, the total tide htotal at time t can be expressed as a sum of many cosine series (one for each constituent). We can write this schematically as:
htotal(t) = ∑i=1,…,42hicos(ωit − gi)
where hi, gi and ωi are the amplitude, phase and angular speed of constituent i, and ωi = 2π / Ti with Ti its period. The periods of the 42 constituents correspond to the known main lunar and solar frequencies which contribute to the tide. Most of them have values around either 12 or 24 hours (semi-diurnal and diurnal tides), some have smaller values (shallow-water tides) and a few have values up to a year (the long period tides). The two largest constituents in many parts of the world, including Liverpool, are called:
M2, with a period of 12 h 24 min (the main semi-diurnal tide from the Moon with a period of half a lunar day) and
S2, with a period of 12 h (the main semi-diurnal tide from the Sun with a period of half a solar day).
At Liverpool, M2 and S2 have amplitudes of 3.13 and 1.01 m respectively. Because our day is 24 hours, the S2 tide will repeat itself twice a day exactly at the same time every day (shown in red below). M2 has a larger amplitude and repeats twice a lunar day (a little later each time) as shown in blue. They combine by ‘beating together’ to give a classic ‘semi-diurnal’ tide where M2 and S2 together result in a total tide that is larger and smaller over a fortnight, called spring and neap tides.
You can appreciate that simply by combining the separate contributions of these two constituents (M2 and S2), we already have a curve which starts to look something like the real tidal variation at Liverpool over a fortnight.
The fact that the tide can be parameterised this way, as a simple addition of harmonics (but many more than two), made it technically easy to invent machines such as the D-L machine that could sum them up. Some machines could handle 40 or more constituents. That was important in order to be able to handle the many smaller constituents that contribute to the tide (not just M2 and S2). Also, in other parts of the world, the total tide can have very different characteristics to that at Liverpool and so the machines needed to be able to handle their particular constituents. See Pugh and Woodworth (2014) for a discussion of why these different types of tide occur.
Obtaining the Harmonic Constants to give to the D-L Machine
As mentioned, the D-L machine has 42 wheels (or constituents), which means that we need 84 numbers to ‘programme’ it (i.e. the amplitude and phase for each constituent for the port in question). Once it has been set up correctly, then it can be run to predict the tide at the port for many years in the future (or past).
But how did Doodson know what these 84 numbers were in the first place?
The 84 numbers come from analysis of previous observations of the tide at the port using a tide gauge. Normally one year of data was adequate, with observations of the water level every hour (i.e. about 9000 values in a year). The team of people who worked with Doodson (called his ‘computers’) usually worked with values of water level in units of 1/10 of a foot and expressed as integers. There is a lot of arithmetic involved in this work and integers are much easier to deal with than real numbers.
His method of analysis of the hourly values made use of ingenious arithmetical filters designed to emphasise the importance of each constituent in turn and so, after a lot of work, arrive a set of estimates of amplitude and phase for all 42 that could be used to programme the machine. The work was very labour intensive, involving endless integer arithmetic by someone who could add up properly. Analysis of a year of data could take a ‘computer’ a few days or a week. The procedures are described in Doodson (1928) and Doodson and Warburg (1941) although be warned that a reader has to devote some hours to understand them.
Now, the people in those days before digital computers could not readily handle 9000 hourly water level values in most of their work, and it turns out that for Doodson’s tidal analysis it is not necessary, as long as the data is good quality. Doodson invented a set of filters which would convert the hourly information into daily numbers, which are just as useful for the tidal analysis, and have 24 times less the bulk of the original data.
For the semi-diurnal constituents these filters are called X2 and Y2 and are a set of simple integer arithmetic weights applied to the hourly values for each day. (They can be thought of as representing the real and imaginary parts of the variations. For people used to studying satellite altimeter data from space the outputs of the filters are akin to the aliasing that occurs in tidal lines using ‘repeat track’ data.)
The method they used was to list the hourly values from hours 0 to 23 each day on a page, one line for each day, and then have a cardboard cut-out with holes for the hours which had to be multiplied by a filter weight. These cut-outs were called stencils. The weight value itself was written on the cardboard, black for a positive weight and red for a negative weight.
The X2 filter for a particular day used data for hours 0-23 on that day and also hours 24 to 28 (i.e. hours 0-4 on the next day), spanning 29 hours total. The integer weights were:
Note that the central value is 3 hours different from that of X2. So the two filters sampled orthogonal (or real and imaginary) components of the semi-diurnal variation.
Nowadays, we can easily apply these filters to our example Liverpool data using a computer and the daily record of X2 and Y2 time series for Liverpool then look something like:
You can see it has much the same information content as Figure 3 (i.e. variation over a fortnight and with two sets of amplitudes and phases) but with 24 times fewer numbers. The constant parts (the offsets) of the red and blue curves come from S2, because S2 is the same every day. And the cyclic parts come from M2, which varies over a fortnight. The red and blue offsets give the amplitude and phase of S2, and the amplitudes and phases of the cyclic parts give the amplitude and phase of M2.
So the first task of the ‘computer’ person was to calculate X2 and Y2 for each day (the work was done by hand of course, or sometimes with the novelty of an adding machine) and write the values for each day in a table with 12 columns (for 12 months of the year), with some columns having 29 rows, and some 30 rows. X2 (or Y2) values for days from the start to the end of the year (i.e. about 360 values) would be listed down column 1 first, then down column 2 etc. until the year was completed in column 12.
The second step is harder to describe but in fact is the most important. The X2 (or Y2) values in each of the columns of the 12 months in the first table were multiplied by different sets of integer weights to maximise the importance of the many different constituents with slightly different frequencies (called ‘daily multipliers’, see Table XV of Doodson, 1928). In addition, sums of the X2 (or Y2) values in each month, listed in each column of the first table, were multiplied by further several sets of integer weights for each month (called ‘monthly multipliers’, see Table XVI of Doodson, 1928). In our idealised example, after a lot of arithmetic, that readily leads to 4 numbers i.e. the 2 each we need for M2 and S2.
The different constituents with periods around 12 hours (other than M2 and S2) have names like 2N2, Mu2, N2, L2, K2 etc. (and it will be seen that they all have their own wheels on the machine). They will all contribute to the hourly water levels to make the total tide plot of hourly values more complicated (Figure 3), and also to contribute individually to the X2 and Y2 values (Figure 6). The results of the filtering in the second step, by means of the daily and monthly multipliers, produces values that have contributions in different amounts from each semi-diurnal constituent. Therefore, the information from the second step needs to be recombined, using linear combinations of each parameter in a process that Doodson called ‘Correction’, in order to provide information specific to each constituent. This was also labour intensive but it was straightforward once a clearly-defined procedure could be explained to a ‘computer’. The method is described in great detail in his 1928 paper with a worked example for Vancouver (which has some mistakes that do not matter).
Doodson then had all the amplitudes and phases that he needed to programme the machine. (In fact, the long-period tides were treated differently but they are not important for this note.) All the amplitudes and phases were written down on a special ‘constants card’ for the port in question which the machine operator would use whenever the machine was required for setting up for that port and year in the future; Figure 7a,b shows the front and reverse of an example card.
The top part of the first card shows amplitudes for each constituent in the order they appear on the machine for Hilbre Island for 1987 and 1988. The amplitudes are in metres and the lunar ones are slightly different each year because of the nodal variations (the ‘f’ factors). You can see the amplitudes for the solar constituents such as S2 are the same both years. The lower part of the card shows the values of frequency * amplitude for each year, with an overall scaling factor, which represents the rate of change of the constituent.
The amplitudes are programmed onto the shafts (also called ‘amplitude blocks’) for each wheel on the front of the machine using a Vernier screw, and the values of frequency * amplitude are programed similarly on the back of the machine (i.e. the machine is a ‘double sided’ one, in effect two separate machines, one for the heights and one for the rates). Finally the phases for each constituent, shown on the reverse of the card for each year (e.g. Figure 7b), are programmed onto the wheels at the front of the machine using the rotating dials. In order to rotate the dials, it is first necessary to release the associated clutch, remembering to tighten it up again before running the machine, otherwise the contribution of the particular constituent would not be included in the total tide. These phases are not phase lags (or they would be the same for both years) but are values of V+u-g, where ‘V’ is the astronomical argument for the start of the year, ‘u’ is the nodal correction, and ‘g’ is the phase lag. These can all be readily computed for each year once one knows ‘h’ and ‘g’, as explained above.
Was this Method of Doodson the Best or Easiest for Finding the Amplitudes and Phases?
The Doodson method described above was by no means the first. You can find many papers from the 19th century which have lists of amplitudes and phases for the various constituents computed in different ways (e.g. see Baird and Darwin, 1885).
For example, in the 19th century there had been methods:
The British Association method, as used by Kelvin, Roberts and Darwin. This was the method used by Roberts to determine amplitudes and phases with which to programme his earlier tide prediction machines.
Darwin’s own later method.
A method used by the US Coast and Geodetic Survey.
Börgen’s method in Germany.
The methods differed in the amount of labour involved, in how well they could eliminate the overlap of information from different constituents in the derivation of the amplitudes and phases, and in the completeness of the analysis. The BA method was the most arduous for the ‘computer’ person.
Doodson’s method was largely an extension, and more complete version, of that employed by Darwin many years before. As always with Doodson, it was devised with more than one eye on subsequent application of the tidal information for use by the prediction machines. Doodson was excellent at handling numbers (and, as important, in showing how his ‘computers’ could handle the numbers) and the method he devised, although complicated and long-winded, was perfectly adapted to routine working by people with basic mathematical skills.
Finally, we can refer to the work of Lord Kelvin (William Thomson), who not only invented the ‘Kelvin Machines’, as the Tide Prediction Machines (TPMs) like the Doodson machine were called, but also invented a mechanical analyser which he thought should be capable of the numerical tidal analysis described above. His prototype ‘tidal analyser’ allowed for determining 5 constituents and a later one allowed for 11. The 5 constituent machine can be seen at Glasgow University, while the 11 constituent machine is in the Science Museum, and both are described by Hughes (2005).
However, Kelvin’s analysers were not successful and so hand-calculated computations of the Darwin and Doodson type were needed for many years. Nowadays, all the tidal analysis of a year of data can be performed in a split second on a modern computer using methods that have many similarities to those of Doodson (see Pugh and Woodworth, 2014).
Valerie Doodson and Ian Vassie are thanked for comments on a first draft of this article.
Baird, A.W. and Darwin, G.H. 1885. Results of the harmonic analysis of tidal observations. Philosophical Transactions of the Royal Society, 34, 135-207.
Doodson, A.T. 1928. The analysis of tidal observations. Philosophical Transactions of the Royal Society, A 227, 223-279.
Doodson, A.T. 1951. New tide-prediction machines. International Hydrographic Review, 28(2), 88-91 and 6 plates.
Doodson, A.T. and Warburg, H.D. 1941. Admiralty Manual of Tides, His Majesty’s Stationery Office, 270pp.
Hughes, P. 2005. A study in the development of primitive and modern tide tables. PhD Thesis, Liverpool John Moores University.
Pugh, D.T. and Woodworth, P.L. 2014. Sea-level science: Understanding tides, surges, tsunamis and mean sea-level changes. Cambridge: Cambridge University Press. ISBN 9781107028197. 408pp.
The exhibition – at the National Oceanography Centre in Liverpool – showcases some of the fascinating achievements made in the Liverpool area in understanding and predicting the tides. The highlights of the exhibition are the rare Roberts-Légé and Doodson-Légé tide prediction machines, extraordinary analogue computers that calculate the rise and fall of the ocean tide. See these beautifully intricate machines up and running at the only place in the world where you can see two of them together.
Bidston Observatory was the home of the Roberts-Légé and Doodson-Légé tide prediction machines while they were still in use. The machines are now owned by National Museums Liverpool, who have carefully restored them to working condition.
Tide & Time is open to the public once a month (usually the first Tuesday of each month from 15:00 to 16:00) or by special arrangement for group visits and events. See this page for information on planning your visit and how to book.
The exhibition will also be open to the public during LightNight Liverpool on Friday 19th May 2017 from 17:00 to 22:00.
This article originally appeared in the newsletter of the Friends of Bidston Hill in February 2016. It is reproduced here with the permission of the author.
The role of Bidston Observatory has changed several times through the years. In its early decades, following the decision in the 1860s by the Mersey Docks and Harbour Board to move the Liverpool Observatory from Waterloo Dock to Bidston Hill, the focus was on astronomical measurements. These were required in order, amongst other things, to determine accurately the latitude and longitude of the site. Famous names involved included John Hartnup and his son (also John) and W.E. Plummer. Other areas of science undertaken by the Observatory included meteorology and seismology. In addition, it provided several local services, such as the calibration of accurate chronometers for port users and precise timing via the “One O’Clock Gun”.
By the 1920s, the Observatory had become ‘moribund’ (to quote from the excellent book by David Cartwright) and, after the death of its then Director Plummer, the decision was made to combine its work with that of the University of Liverpool Tidal Institute, with both to be located at Bidston. The latter had been founded in 1919 on the university campus in Liverpool with Joseph Proudman as Director and Arthur Doodson as Secretary, with funding from several sources including the major Liverpool shipping companies. The formal amalgamation of the Observatory and the Tidal Institute took place in 1929.
Proudman is another famous name, with Bidston Observatory later becoming known as the Proudman Oceanographic Laboratory. However, it is Arthur Doodson who is more relevant to this article. In the first year of the Tidal Institute, Doodson and Proudman began work on the problem of predicting tides, especially in shallow waters. They also undertook an evaluation of the benefits of mechanical tide prediction machines, which had been invented in the late 19th century by Lord Kelvin (William Thomson) and later developed by Edward Roberts. In effect, they were ‘analogue computers’. By 1924 Doodson had taken delivery of a brand new tide machine, the so-called ‘Bidston Kelvin machine’ thanks to the generosity of Liverpool ship-owners. Then in 1929, with all staff now installed at Bidston, he acquired and refurbished the so-called ‘Roberts machine’ which had been constructed by Roberts in 1906. The Roberts family had used this machine as part of a business of providing tidal predictions to the government but, due to the death of Roberts’ son, were no longer able to continue.
The Roberts machine was in many ways superior to the Kelvin machine, being capable of predicting 40 ‘constituents’ of the tide instead of 29. Such machines can only have a decent stab at simulating the tide at all thanks to the fact that the tide is capable of being described as the sum of individual harmonic constituents. Constituents can be thought of as cosines with particular frequencies (or periods) that are known from astronomy. So, for example, two of the most important constituents are called M2 and S2. These come from the Moon and Sun respectively with periods of 12 hours 25 minutes for M2 and 12 hours exactly for S2. These two terms are responsible for the regular twice-daily tide we have at Liverpool. However, many more constituents than these two are required to do a decent job of simulating the real tide to the accuracy required, and a machine with as many constituents as possible is highly desirable.
These two machines were responsible for many important achievements in the Observatory’s history. Bidston had become the undoubted centre of excellence in tidal research, both from theoretical perspectives (primarily Proudman) and on more practical bases such as the provision of tidal predictions worldwide using these machines (primarily Doodson). Doodson was excellent at devising techniques for handling numbers within complicated scientific calculations that nowadays would be undertaken by digital computers. He also became an expert in the technical design and construction of the tide prediction machines.
Although important individual machines were constructed in Germany and the USA, the majority of the 33 ever made (24 machines) were designed and manufactured in the UK, in either London, Glasgow or Liverpool. The UK was the only country to export machines to other countries. The construction of the majority of the machines made after 1920 was supervised, one way or another, by Arthur Doodson. These included a series of machines made after World War II, of which one (called locally the “Doodson-Légé machine”) was to be found in the lobby of the main POL building for many years until the move of the laboratory to the Liverpool campus in 2004.
Two of the three machines at Bidston have an importance in a notable period in the Observatory’s history, in providing tidal predictions during World War II and, in particular, for the D-Day landings and in other military operations around the world. These were the Kelvin and Roberts machines, which were located in separate buildings at the Observatory during the 1940s in case of bomb damage. The Kelvin machine, Doodson’s first, is now to be found in good condition at the headquarters of the French Hydrographic Service in Brest. Its disposal by Bidston after the war was a financial requirement in order to obtain funding for the Doodson-Légé machine.
The Roberts and Doodson-Légé machines are still located in Liverpool and are now owned by the Liverpool Museum. Recently, they have both been refurbished excellently and are capable of working as well as they can in order to show how things were done at Bidston, before the advent of digital computers in the 1960s saw their demise as the Observatory’s main technical assets.
Both machines are now on long-term loan from the Museum to the National Oceanography Centre in Brownlow Street on the Liverpool University campus, NOC being the successor to POL and therefore the ‘spiritual home’ of the machines. They are available for viewing by the public but arrangements must be made beforehand with the NOC Administration.
For anyone interested in Bidston Observatory and these machines, there is more to read. For an excellent introduction to tidal science, see Cartwight (1999), while histories of the Observatory and the people who worked there are given by LOTI (1945), Jones (1999) and Scoffield (2006). Aspects of Doodson’s career have been described by Carlsson-Hislop (2015). An ‘inventory’ (or overview) of tide prediction machines can be obtained from me, while the story of the use of the Kelvin and Roberts machines in World War II is given by Parker (2011).
Philip L. Woodworth National Oceanography Centre, 6 Brownlow Street, Liverpool L3 5DA December 2015
Carlsson-Hislop, A. 2015. Human computing practices and patronage: anti-aircraft ballistics and tidal calcuations in First World War Britain. Information and Culture: A Journal of History, 50, 70-109, doi:10.1353/lac.2015.0004.
Cartwright, D.E. Tides: a scientific history. Cambridge University Press: Cambridge, 1999. 292pp.
LOTI. 1945. Liverpool Observatory and Tidal Institute. Centenary Report and Annual Reports for 1944-5. Available from P.L. Woodworth.