prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the water depth in the Greater Ekofisk Area?

Read the following text and answer briefly.

Context 1: Filip Fremo Minge – Ekofisk
Author: Filip Fremo Minge
Posted on 1  October 2019 12  October 2019
— Sunset over Ekofisk  Photo: Husmo Foto/Norwegian Petroleum Museum
The three are operated by ConocoPhillips on behalf of the Ekofisk licensees  The area also embraces former producers Albuskjell, Cod, Edda, Tor, West Ekofisk and Tommeliten G 
These fields all lie within production licence 018 apart from Tommeliten G, which was operated by Statoil from 1976 to 2003 
In all, 31 installations have been positioned in the Greater Ekofisk Area 
First Norwegian offshore field
Ekofisk began production on 15 June 1971, following its discovery in the autumn of 1969  Development of the field has occurred in several phases 
Its central facilities were installed during the early 1970s, with oil initially being buoy-loaded into tankers  From 1975, it has been piped to Teesside in the UK  The gas has been landed by pipeline at Emden in Germany from 1977 
ekofisk i et nøtteskall, engelsk
Jacked up six metres
The water depth in the Greater Ekofisk Area is 70-75 metres  However, declining pressure in the Ekofisk reservoir over the years has caused the seabed to subside 
Efforts began as early as 1985 to safeguard the installations against the effects of this development, and the steel platforms in the Ekofisk Complex were jacked up by six metres in 1987 
In addition, a protective breakwater was installed around the Ekofisk tank in 1989  The rate of seabed subsidence has declined sharply in recent years 
Waterflooding improves recovery
The Ekofisk 2/4 K water injection platform became operational in December 1987 as part of efforts to improve Ekofisk’s recovery factor – the share of petroleum in place actually produced 
Waterflooding capacity on the field to help maintain reservoir pressure was later expanded several times, and had reached just over 500 000 barrels per day by 2019 
Measured in barrels of oil equivalent, the recovery factor on Ekofisk has risen from an original estimate of 17 per cent to over 50 per cent 
Ekofisk I and II plus licence extension
The first phase of development and production on Ekofisk began with initial oil output from the converted Gulftide jack-up rig in 1971 and ended with the start-up of Ekofisk II in 1998 
Large parts of the Greater Ekofisk Area were restructured in the latter year, leading to plans for removing 15 installations – 14 steel platforms and the process facilities on the Ekofisk tank 
plattformer, historie, 2004, driftsenter åpnet,
Embla 2/7 D  Photo: ConocoPhillips/Norwegian Petroleum Museum
Designated Ekofisk I, these redundant structures include Ekofisk 2/4 A, 2/4 B, 2/4 FTP, 2/4 Q, 2/4 H, 2/4 R, 2/4 P and 2/4 T 
In addition come the Edda 2/7 C, Albuskjell 1/6 A, Albuskjell 2/4 F, Cod 7/11 A, West Ekofisk 2/4 D, Norpipe 36/22 A and Norpipe 37/4 A installations 
The concrete part of the tank – Ekofisk 2/4 T – will remain  Gulftide was removed as far back as 1974  Two platforms owned by other companies – Ekofisk 2/4 G and 2/4 S – have also gone 
A new plan for development and operation (PDO) of the field (Ekofisk II) was approved in 1994, at the same time as the Ekofisk licence was extended to 2028 
This creates a new Ekofisk Complex with two structures – the Ekofisk 2/4 X wellhead unit installed in the autumn of 1996 and the Ekofisk 2/4 J processing and transport platform in 1997 
Ekofisk II became operational in August 1998 and is intended to produce until 2028  Ekofisk, Eldfisk and Embla are tied back to the new complex, as was Tor until it shut down in December 2015 
Ekofisk West
historie, forsidebilde, 2003, ekofisk vekst godkjent i statsråd
Ekofisk Growth  Illustration: Ståle Ådland
In December 2002, soon after the Conoco-Phillips merger had been announced, the Ekofisk West project was presented to improve oil and gas recovery  Process capacity and reliability on Ekofisk were also to be enhanced 
This development primarily involved the construction and installation of a new platform, Ekofisk 2/4 M, with processing facilities and 24 new wells drilled over five years 
The latter could contribute to improved recovery both because there were more wells and because they would tap new locations in the reservoir  On stream in 2005, 2/4 M was linked to the Ekofisk Complex with a bridge 
Process capacity for produced water was also to be increased through upgrading on Ekofisk 2/4 J and Eldfisk 2/7 E  A third measure concerned laying a power cable from the Ekofisk Complex to 2/4 K in order to make electricity supplies more efficient 
New developments: Eldfisk II and Ekofisk South
Eldfisk 2/7 S løft
The deck of Eldfisk 2/7 S being mated with the steel jacket  Foto: Øyvind Sætre/ConocoPhillips
The plan for development and operation (PDO) of Eldfisk II, approved by the Storting (parliament) on 9 June 2011, includes a new wellhead, process and accommodation platform – Eldfisk 2/7 S 
In addition come 42 new wells as well as upgrades to existing platforms which extend their commercial life 
The PDO for Ekofisk South involves the construction of a new wellhead platform – Ekofisk 2/4 Z – as well as a new subsea water injection facility and 44 additional wells 
ConocoPhillips Norge, 2004 
Ministry of Petroleum and Energy, press release, “Vekstprosjekt på Ekofisk godkjent”, 6 June 2003 
https://www stortinget no/no/Saker-og-publikasjoner/Saker/Sak/?p=50343
https://www stortinget no/globalassets/pdf/innstillinger/stortinget/2010-2011/inns-201011-398 pdf
https://www regjeringen no/no/aktuelt/klart-for-40-nye-ar-pa-ekofisk-feltet/id642376/)
Published 1  October 2019 • Updated 12  October 2019
— Gassterminalen i Emden  Foto: Husmo Foto/Norsk Oljemuseum
Oil terminal in Teesside
Olje- og gassterminalene, engelsk,
Teesside terminal  Brian Henderson Thynne takes samples of refrigerated propane  Photo: Husmo Foto/Norwegian Petroleum Museum
The terminal at Teesside in north-east England receives oil and natural gas liquids (NGL) by pipeline from the Ekofisk field  It comprises stabilisation, NGL fractionation, storage tanks for crude oil and an export port 
After arriving through the Norpipe Oil line, crude and NGL are separated and the oil goes through a stabilisation process before reaching the 10 storage tanks, which each hold 750 000 barrels 
The NGLs go to the fractionation facility, with a daily capacity of 64 000 barrels, for separation into methane, ethane, propane, and normal and iso butane 
While the methane (natural gas) is used to fuel the plant, the other products (now known as liquefied petroleum gases – LPG) are made liquid by cooling and stored for export by sea 
One reason for the choice of Teesside as the landfall for the Ekofisk pipeline was the opportunity it offered to install deepwater quays 
The terminal has four of these, with those for crude oil able to handle tankers up to 150 000 deadweight tonnes  The LPG quays can accept carriers loading as much as 60 000 cubic metres 
Two of the crude oil quays lie on the main channel of the River Tees, while the others have been installed in dredged docks 
Gas terminal in Emden
Gas arriving at the Emden terminal from the Ekofisk Complex enters nine parallel treatment trains for cleaning, metering and onward distribution to the buyers 
The North Sea gas is very clean, and needs only limited treatment to remove small amounts of sulphur compounds using an absorption process  Impure molecules from the gas accumulate on the surface of small particles, which act as filter spheres 
Each of the nine trains comprises four process columns and a process oven  The gas enters the top of a column and leaves through the base after passing through the filter spheres 
That leaves the gas ready for sale, and it is piped to the fiscal metering station before entering the buyer receiving pipelines and distribution network 
Three separate commercial pipeline systems connect to the terminal, operated by Ruhrgas, BEB and Gastransport Services (previously Gasunie) respectively  They pipe the gas away on behalf of the gas buyers 
The Norsea Gas Terminal in Emden was officially opened in September 1977 by Norwegian industry minister Bjartmar Gjerde and Phillips executive Gordon Goerin 
Ranking as the first gas sales deal for the Norwegian continental shelf, the Ekofisk agreement paved the way for later contracts covering other fields off Norway 
Regularity at the Emden terminal has been very high, with its own equipment never causing shutdowns  Maintenance takes place when other parts of the system are off line 
The terminal has a daily capacity of about 2 1 million cubic feet of gas per day 
Gas transport restructured
Norpipe AS owned the gas pipeline from Ekofisk to Emden until the transport system for the Norwegian offshore sector was restructured at 1 January 2003 
Norsea Gas A/S furthermore served as the formal owner of the Emden facility, with Phillips Petroleum and then ConocoPhillips as operator for both pipeline and terminal 
olje- og gassterminalene,
Teesside gas terminal  Photo: Husmo Foto/Norwegian Petroleum Museum
Since 2007, Norway’s state-owned Gassco company has been responsible for technical operation of the facilities on behalf of their owners 
That included operator responsibility for the H7 and B11 booster platforms along the gas pipeline, which were shut down in 2007 and 2013 respectively and have since been removed 
The Gassled partnership is a project collaboration embracing 10 companies which collective own large parts of the gas infrastructure on the Norwegian continental shelf (NCS) 
A substantial proportion of Norway’s gas deliveries to Germany continues to arrive at the Emden terminal, including the volumes piped from Ekofisk 
Preliminary planning for a new terminal in the German port began in 2011, with Gassled taking the investment decision for this development in the autumn of 2012 
Construction work began in the following year, with the new facility being built on an unused part of the existing terminal site 
The new terminal has not expanded export capacity  But its functionality is well adapted to future processing needs for fields in the Greater Ekofisk Area and other parts of the NCS sending gas through the Norpipe system 
It was officially opened on 24 May 2016 by Elisabeth Aspaker, the Norwegian government minister for the EU and the European Economic Area  That closed a chapter in Ekofisk’s history 
Source: ConocoPhillips Norge
— Gas pipes at Ekofisk  Photo: Husmo Foto/Norwegian Petroleum Museum
In addition to ConocoPhillips’ own production from Ekofisk, these pipelines carry gas and oil from the company’s fields in the UK sector and from other fields on the Norwegian and British continental shelves 
The three fields in the Greater Ekofisk Area are also tied together by pipelines 
Oil pipeline to Teesside
rørledningene, engelsk,
Pipes and oil tanks at the Teesside plant  Photo: ConocoPhillips/Norwegian Petroleum Museum
The pipeline linking Ekofisk with the terminal for oil and natural gas liquids (NGL) at Teesside on the north-east English coast became operational in October 1975 
Pumps raise the pressure of the oil and NGL before they start their journey to land  Two pumping stations – 37/4 A and 36/22 A ­– originally stood along the pipeline to maintain this pressure, but have now been disconnected and removed 
The pipeline was installed with the ability to carry a million barrels per day  However, that much capacity has never been required 
In the UK sector, a 24-inch pipeline has been tied in with a Y connection to receive input from several British fields – including the J block developments operated by ConocoPhillips 
Output from the Greater Ekofisk Area is supplemented by crude from Valhall, Hod, Ula and Gyda heading for Teesside, optimising pipeline utilisation and thereby boosting value creation 
The pipeline is owned by Norpipe Oil AS and operated by ConocoPhillips 
Gas pipeline to Emden
Sandbags and gravel were used to cover Norpipe to Emden  Photo: Unknown/Norwegian Petroleum Museum
This pipeline became operational in September 1977  The starting pressure of around 132 bar is provided by compressors on the Ekofisk Complex 
The 443-kilometre distance to Emden was split into three equal sections, with platforms B11 and H7 located at the intermediate points to provide boosting if required 
However, additional compression was seldom needed on the final stage to Emden  H7 was shut down in 2007 and B11 in 2013, and both have since been removed 
These two booster platforms were located in the German sector of the North Sea, while the pipeline also crosses the Danish sector 
The pipeline has been trenched or covered with sand  Its final section passes the island of Juist before making landfall on the coast of East Friesland to the north of Emden 
Its daily capacity is roughly 59 4 million standard cubic metres (2 1 billion cubic feet)  In addition to gas from the Greater Ekofisk Area, it carries output from Valhall, Hod, Ula, Gyda and the Statpipe system (primarily Statfjord and Gullfaks) 
Posted on 24  June 2017 25  October 2019
Embla 2/7 D
This unmanned wellhead facility is remotely controlled from Eldfisk 2/7 S located 5 2 kilometres to the north, where oil and gas output from the platform is also processed 
Unmanned and remotely operated wellhead platform
On stream 12 May 1993
— Embla 2/7 D  Photo: ConocoPhillips
sokkelkart, illustrasjon, blokker, lisens, forsidebilde, engelsk,
Hand-colored map of the licenses of the first licensing round on the Norwegian continental shelf  Norwegian Continental Shelf Map, 1965 
The Phillips group was awarded block 2/7 as early as 1965, and the Embla reservoir lies in the southern part of this acreage  Drilling began there in 1974 to depths of 4 500-5 000 metres, but pressure and temperature in the wells were too high for testing with the available equipment 
The first production well was not drilled and tested until 1988, followed by a second in 1990  Both yielded very promising results, and the field came on stream in May 1993 
Embla comprises a sandstone reservoir at least 250 million years old  The other fields in the Greater Ekofisk Area comprise fine-grained carbonate rocks deposited about 70 million years ago 
The Embla reservoir has a temperature of 160°C compared with the 125°C normally found in the chalk formations 1 000 metres higher up, and its pressure is almost twice as high 
Fabricated by Heerema in the Netherlands, the Embla 2/7 D jacket (support structure) was installed by the M 7000 crane vessel  It stands 84 metres high and weighs 2 300 tonnes 
A 5 2-kilometre subsea umbilical from Eldfisk comprises three power cables for electricity supply and eight fibreoptic lines handling data transmission and telecommunication 
Eldfisk 2/7 S, embla,
Eldfisk 2/7 S  Photo: ConocoPhillips
The platform has six production wells and an average daily output of roughly 7 000 barrels of oil  All processing and metering took place on Eldfisk 2/7 FTP until 2015, and has now been switched to Eldfisk 2/7 S 
A 14-inch flowline linked 2/7 D with 2/7 FTP and runs today to 2/7 S  Produced at Wick in Scotland, this line was floated out to the field in one piece 
Topside equipment includes the wellhead area, helideck (built by Vindholmen Services in Arendal), crane, control room, workshop, test separator and glycol pump 
Normally unmanned, the platform is maintained as and when required and therefore incorporates a simplified accommodation module with lounge, mess, coffee room, galley, changing room, WC and 12 emergency beds 
More about platforms
Ekofisk 2/4 Z
This installation is a wellhead platform in the Ekofisk Complex 
Gulftide
This four-leg jack-up drilling rig was built in Glasgow during 1967 for Ocean Drilling & Exploration Co 
Posted on 1  September 2019 8  October 2019
— Gulftide with Ekofisk 2/4 A in the background  Photo: Aker Mek  Verksted/Norwegian Petroleum Museum
Gulftide was converted to cope with conditions on Ekofisk in the Åmøy Fjord near Stavanger  This jack-up drilling rig was equipped with process equipment and its derrick, helideck, hangar and legs were reinforced 
To win time, it was decided that the discovery well and three appraisals drilled on Ekofisk by Ocean Viking would be completed for production 
Principles for producing from Gulftide were relatively simple  Output flowed from the subsea wellheads to the platform, where it went through two-stage separation to remove gas and water 
With pressure also reduced, the gas was flared off and the oil sent on by flowlines to two loading buoys where shuttle tankers moored to take on cargo 
utbyggingen,
Tankskipet Donovania laster olje fra lastebøyen på Ekofisk  I bakgrunnen skimtes så vidt Gulftide  Foto: ConocoPhillips/Norsk Oljemuseum
Production could only continue while ships were loading  As soon as one tanker had been filled, the oil stream was diverted to the vessel waiting at the other loading buoy 
The problem with this approach was manifested when weather conditions ­– strong winds and/or high waves – forced the tankers to leave the buoys 
If that happened, production from the wellheads had to be suspended immediately  Given the prevailing weather on Ekofisk, that happened regularly  Output was halted for 20 per cent of the time during the first year 
https://ekofisk industriminne no/wp-content/uploads/sites/2/2019/09/Building-Ekofisk mp4
Gulftide was replaced as the temporary production installation in 1974 by the permanent Ekofisk 2/4 A (Alpha) and 2/4 B (Bravo) platforms for production, drilling and quarters 
In addition came the Ekofisk 2/4 C (Charlie) production, drilling and compression facility, the Ekofisk 2/4 FTP (field terminal platform) for production and risers, and Ekofisk 2/4 Q for accommodation 
Oil and gas were produced by 2/4 A, B and C through their own wells for processing in their separation plants and piping on the 2/4 FTP for a three-stage separation process 
At the same time, the tanker loading buoys were moved further from the platforms and the Ekofisk 2/4 T oil storage tank became operational 
This facility was extremely advantageous, because it allowed production to continue virtually regardless of whether bad weather prevented tankers from connecting to the buoys 
Ekofisktanken ble satt i drift i 1974  Foto: ConocoPhillips/Norsk Oljemuseum
The 2/4 FTP platform, where oil and gas from the three producing facilities was processed, had been planned to handle the level of output estimated for the main field 
Clear restrictions had been imposed by the Norwegian government on the amount of gas Phillips was allowed to flare  That also set a ceiling for oil production, since gas accompanies it up from the reservoir 
The solution was to install two powerful compression packages on 2/4 C in order to inject the gas under pressure back into the producing formation 
Accommodation facilities had to be provided on the two first platforms, 2/4 A and B  Where 2/4 C and FTP were concerned, however, they were tied together with bridges and to 2/4 Q 
Published 1  September 2019 • Updated 8  October 2019
Posted on 9  April 2019 25  October 2019
Jack-up drilling rig
Built 1967 in Glasgow for Ocean Drilling & Exploration Co 
Began test production on Ekofisk 15 June 1971
Produced on Ekofisk until 1974
— Gulftide at theEkofisk field  Photo: Terje Tveit/Norwegian Petroleum Museum
gulftide,
Gulftide  Photo: Unknown/Norwegian Petroleum Museum
A mere 17 months after the Ekofisk discovery was announced in December 1969, Gulftide was ready to come on stream as a temporary production platform 
Its official inauguration took place on 9 June, with initial test output commencing on 15 June  Full production began on 8 July 
The rig was chosen because it was available on the market  Established equipment for processing oil and gas was tailored to the limited space on board  Separate flowlines carried wellstreams from four subsea wells  Oil, gas and water were separated on board, with the gas flared and the oil piped to two buoys for loading into shuttle tankers 
Work on the process equipment was relatively simple  The problem was to tailor it to the rig  The subsea wellheads had to be reinforced to meet the demands posed by the North Sea, and a buoy loading system needed to be developed for waters where this technology had never been used before 
To gain time, it was decided that the three appraisal wells drilled by Ocean Viking to map the extent of the field – in addition to the discovery well – would be completed for production 
Første testflamme tent på Ekofisk  På Gulftide
1973, Teddy Broadhurst, gulftide,
arbeidsliv, hjelpearbeider
Gulftide, separator – på bildet kan man se at det er fire brønner 
arbeidsliv, gulftide, pionerkultur, arbeid, dekk, Norges første havbunnsbrønner, historie, 1971,
The producers would be topped with hydraulically controlled wellheads  Such equipment had been tried out on the seabed earlier, but on a limited scale and not in the deep and rough waters found on Ekofisk  This challenge was overcome by having the wellheads manufactured and then reinforced at the Phillips base in Dusavik outside Stavanger  Flowlines and control cables would also be laid from each well to Gulftide, with production comingled in a single riser to the topsides 
Weather conditions also represented a major problem when designing the loading buoys  Phillips itself had experience with such facilities, but the concept had only been used before in harbour-like conditions and waters no deeper than 27 metres  They were now to stand in 70 metres in the middle of the North Sea 
Gulftide was converted in the Åmøy Fjord outside Stavanger to cope with conditions on Ekofisk  The processing facilities were installed and reinforcements made to the derrick, helideck, hangar and leg structures 
Gulftide, Ekofisk 2/4 A, boretårn, flare, 1971, utbygging,
Gulftide with Ekofisk 2/4 A in the background  Photo: Aker Mek  Verksted/Norwegian Petroleum Museum
Planning began in late 1970, when Phillips received approval to begin laying the flowlines between wellheads and rig  Brown & Root won this contract, with the first oil pipelines on the Norwegian continental shelf laid by the Hugh W Gordon laybarge 
The production principle on Gulftide was relatively simple  Output flowed from the subsea wellheads to the rig, where it passed through two separation levels to be split into oil and gas while the huge pressure was reduced 
Gas was flared off and the oil was piped to one of the loading buoys where a shuttle tanker was moored  Production could only take place when a ship was present 
Offisiell åpning av norsk oljeproduksjon,
The Greek tanker, Theogennitor, unloads crude oil from loading buoys on the Ekofisk field  Gulftide in the background  Photo: ConocoPhillips/Norwegian Petroleum Museum
As soon as one tanker had become fully laden, the oil flow was switched to the other buoy where another ship was waiting to take on cargo 
The problem with this approach arose when weather conditions meant the tankers had to cast off from the buoys because of strong winds or high waves  The rig then had to shut down production from the wellheads immediately 
Given the weather conditions found on Ekofisk, output regularly had to cease  Production was suspended for 20 per cent of the first year for this reason 
Output began cautiously on 8 July 1971 from a single well  The second producer came on stream that September, the third was ready the following month and all four were producing by February 1972  They each flowed 10 000 barrels of oil per day 
Source: Kvendseth, Stig, Giant discovery, 1988 
Published 9  April 2019 • Updated 25  October 2019
Norpipe H-7
This platform served as a pumping/compressor station to maintain pressure in the 443-kilometre Norpipe gas pipeline from Ekofisk to Emden in Germany, which became operational in September 1977 
Kjappe fakta::
Compressor platform on Ekofisk-Emden gas pipeline
Installed 1976
Operational 1977
Shut down 29 October 2007
Removed 2013
— Norpipe GNSC-H7  Photo: Husmo Foto/Norwegian Petroleum Museum
Gas received initial compression to 132 bar at the Ekofisk Complex  The pipeline was divided into three equal lengths, with Norpipe GNSC B11 positioned at the end of the first third to maintain pressure as and when required 
From there, the gas then travelled the next third of the distance to the second and virtually identical compressor platform, H7 
This was also responsible for maintaining pressure, but additional compression was seldom required on this final leg of the journey to Emden 
Both platforms stood on the German continental shelf, but 48 kilometres of the pipeline also ran across the Danish North Sea sector 
The pipeline is trenched or covered with sand  On its final approach to the coast of East Friesland, it passes beneath the island of Juist before making landfall north of Emden 
Capacity in Norpipe is about 60 million standard cubic metres (scm) or 2 1 billion cubic feet per day  In addition to output from the Ekofisk-area fields, it carries gas from Valhall, Ula and the Statpipe system – primarily Statfjord and Gullfaks  Gas was also transported for a time from Hod and Gyda, but that has ceased 
fritid, Norpipe GNSC-H7,
Magnus Refsland and Werner Hein have pulled the crab trap (full of starfish) on the Norpipe H-7 platform  Photo: Husmo Foto/Norwegian Petroleum Museum
Built in 1976, the B11 platform had six decks  Its permanent staffing totalled 14 people, but various service personnel were also often on board  The regular crew included three in catering 
The 11 Phillips employees comprised the offshore installation manager, the nurse/radio operator, eight operators and a roustabout 
In addition to their direct function, the operators covered various other trades which meant the crew was self-sufficient in most circumstances 
Both platforms obtained a satellite antenna in 1986 which allowed them to received Norwegian TV, while the 24-bed accommodation were redecorated in 1981 and upgraded in the summer of 1990 
Work on the upgrading largely comprised converting all cabins to doubles with shower and WC  The galley and changing rooms were renewed and changing facilities for women provided 
A new module with a lounge for non-smokers, a smoking room, gym and pool room was also installed  During this work, the West Gamma accommodation rig was positioned alongside 
Upgrading equipment on the platform was also initiated in 1990  While the pipeline’s original daily capacity had been estimated at 2 100 million standard cubic feet, this was found to have declined after a number of years to 1 975 million 
To return to the original capacity, the compressors needed to be upgraded and power supply from the turbines increased  This was done both on the Ekofisk tank and on the H7 and B11 platforms  Gas coolers on the tank were replaced as well 
Norpipe GNSC-H7, yrker, radiooperatør,
Radio operator Torleif Førland on the platform Norpipe H-7, with his amateur radio  Photo: Husmo Foto/Norwegian Petroleum Museum
The control systems were also upgraded in parallel  Control panels on turbines and compressors were replaced and metering instruments installed to conduct measurements in this equipment 
While the nearest neighbour to B11 was a Danish oil field, H7 stood in the middle of the shipping channel  M/S Hero broke down 15 nautical miles west of the latter platform at around 13 00 on 12 November 1977 
By 21 00, the ship was still adrift and heading directly for H7, and all 14 crew on the platform made ready to evacuate by helicopter – the waves were too high for the lifeboats  The wreck passed at 21 40 with a clearance of 400 metres 
German cargo carrier Reint collided with H7 on 30 September 1995, despite efforts by the standby ship to avert the threat  Production was halted as a safety measure, but the platform luckily suffered only minor damage  The collision was caused by inadequate watchkeeping on the ship’s bridge 
Operator responsibility for B11 and H7 was transferred at the beginning of 2003 to Norway’s state-owned Gassco company, which runs the Norwegian gas transport network 
This change had little significance for operation of the platforms, since the actual work was still carried out by ConocoPhillips as a technical service provider to Gassco 
H7 was shut down in 2007, and removal had been completed in 2013  In connection with preparations to remove the structure, operator responsibility was transferred to Statoil as the company in charge of the project on Gassco’s behalf 
Published 24  August 2016 • Updated 22  October 2019
Phillips inundates Sola with oil revenues
person by Kristin Øye Gjerde
Stavanger and neighbouring Sola were the first Norwegian local authorities to experience fantastic oil-related growth after the award of the first exploration licences in 1965 
— Phillips er i ferd med å etablere seg på Norscobasen nederst til høyre Ca 1972 Foto: Norsk fly og flyfoto/Norsk Oljemuseum
The Shell refinery at Risavika in Sola was completed two years later, while the Norsco base in Tananger became operational as early as 1966 
But things really took off once the Ekofisk field had been discovered in the autumn of 1969 and started trial production on 14 July 1971 
Operator Phillips Petroleum Company moved its offices from the Dusavik base outside Stavanger to Tananger in Sola, and Shell could finally start refining Norwegian rather than imported crude 
Sola’s population now rose steadily from 8 400 in 1965 to 15 000 two decades later, and jobs grew even faster – from about 2 000 in 1970 to almost 8 000 in 1985  That averages 10 per cent annually 
Phillips and Shell became cornerstone companies  A large part of their workforce, particularly in Phillips, worked offshore  In addition came newly established oil supply firms 
More jobs were also created in retail, public administration, education, health and social care, personal services and so forth 
Although traditional agriculture remained important for the local authority, the number of farmers gradually declined as a result of mechanisation [REMOVE]Fotnote: This article is based on the chapter “Elverket i Oljealderen” in I det regionale spenningsfelt  Sola Energi 1913-1999, Kristin Øye Gjerde 
Boreskipet Drillship ligger ved kai på Norscobasen i Tananger (1968)  Foto: NOM/Norsk Fly og Flyfoto
Boreskipet Drillship ligger ved kai på Norscobasen i Tananger (1968)  Foto: Norsk Fly og Flyfoto/Norsk Oljemuseum
The “agio tax”
The sharp rise in Sola’s revenues was attributable entirely to the oil industry, and it found itself in an enviable position during this period  Tax revenues rose even faster than population and jobs 
To give an indication, the local authority’s overall income from wealth and income taxes rose from NOK 9 3 million in 1966 to NOK 198 million in 1990  The biggest growth came in 1978-82, when it averaged 39 per cent a year [REMOVE]Fotnote: Sola local authority, plans 
The secret behind this sharp increase was the tax paid by the oil companies – primarily Phillips – on agio, or the percentage fee charged when exchanging one currency for another 
Under Norwegian law at the time, the companies paid tax on their interest income to the local authority where they had their head office  In making this rule, however, the government had failed to take account of the considerable sums involved 
As operator of the Greater Ekofisk Area, Phillips had placed capital to be used for new investment in banks around the world – particularly the UK 
These deposits yielded substantial interest payments, and tax was payable on converting this income into Norwegian kroner [REMOVE]Fotnote: Toralv Torstenbø, former chief executive officer in Sola local authority, interviewed by Kristin Øye Gjerde, 22 February 2001 
Sola council is said to have almost gone into shock the first time Phillips paid this agio tax  It suddenly had more money than it could spend 
During the 1970s and early 1980s, Sola’s municipal income always exceeded the budgeted amount  Large sums could be transferred every year to a capital fund 
Since the local authority was in a growth phase, additional funding was needed for the big developments it faced  While the rest of Norway experienced a slump in the late 1970s, Sola continued in top gear without a sign of unemployment 
Net income tax revenues came to NOK 55 5 million in 1978, while net spending was NOK 31 9 million  And these fantastic results went on improving 
By 1982, wealth and income taxes yielded NOK 203 4 million – compared with a budget of NOK 146 million, which was upgraded to NOK 190 million during the year 
According to Toralv Torstensbø, the financial controller, agio tax accounted for almost half this amount – in other words, as much as the tax paid by all other enterprises, private individuals and industry in Sola 
Its chief executive officer became a little overweening  In his comments on the 1982 budget, he declared that it would be “natural for Sola local authority to feel a strong regional responsibility and not to be too strict about the traditional division of costs between state, county and local authority ”
In line with this open-handed policy, Sola paid for both road projects and an upper secondary modern school which the county council was supposed to fund [REMOVE]Fotnote: Chief executive officer’s budget proposal for Sola local authority covering 1974-85 
Tightening up petroleum tax
This unexpected prosperity undoubtedly created some jealously in the neighbouring local authorities, and the media began to show an interest in the issue 
Local daily Stavanger Aftenblad interviewed Sola’s chief executive and controller in 1981, when its photographer took a shot which illustrated the boundless wealth – Torstensbø stood showering hundred-krone notes over his colleague 
This story was not only read by the paper’s regular subscribers  The following day, 150 copies were distributed to members of the Storting (parliament) 
That in turn prompted Centre Party representative Lars Velsand to make a passionate speech in which he described the position as a misuse of tax revenues 
He called on the government to intervene so that individual local authorities were unable to benefit in this way  Nor was he alone in finding it unreasonable that a small community like Sola should get so much money 
The result was an amendment to the Petroleum Tax Act on 11 June 1982, which specified that the proceeds from the agio tax should be transferred in future to central government 
Løfteskipet Uglen i aksjon ved Norscobasen i juli 1980  Foto: NOM/Norsk Fly og Flyfoto
Løfteskipet Uglen i aksjon ved Norscobasen i juli 1980  Foto: Norsk Fly og Flyfoto/Norsk Oljemuseum
Unfortunately, however, Sola had got used to consuming these revenues  It is easy to learn expensive habits, but not so straightforward to shrug them off again 
Matters had become a little unusual when the council’s executive board adopted the style of the oil company chiefs and took a helicopter outing during an ordinary budget meeting [REMOVE]Fotnote: Oskar Goa, former chief technical officer in Sola local authority, interviewed by Kristin Øye Gjerde, 23 October 2000 
However, most of the tax money benefitted the general public  Paying for Sola upper secondary school and new national and county highways is an example of this 
The council also invested on local authority school buildings and community facilities such as the big sports complex at Åsen, with an outdoor athletics ground and two modern indoor arenas  Dysjaland and Tananger also acquired new sports arenas 
A new cultural centre built in central Sola has a distinctive architecture in brick and glass, with a grassed roof to blend with the surrounding Jæren landscape  With two stages and a public library, this became the community’s main venue for events and so forth 
The local authority thereby built up a very good infrastructure  Power cables were laid in the same trenches as water and sewage pipes, a network of cycle lanes was built and street lighting installed 
On the downside, virtually all these investments boosted operating expenses  The council’s running costs rose by an annual average of 30 per cent in 1978-84, with the biggest growth in the last three years of the period 
So the calls by Storting representatives to transfer agio tax receipts from councils to central government represented a real threat to local politicians 
Sola joined forces with other local authorities in the same position, including Stavanger, Oslo and Bærum as well as Rogaland county council 
A delegation met the Storting’s standing committee on finance to present their case, and secured a commitment to accept a phased reduction in revenues over four years 
The local authorities would receive 80 per cent of agio tax receipts during the first year, then 60 per cent, 40 per cent and finally 20 per cent [REMOVE]Fotnote: Amendment to the Petroleum Tax Act adopted on 14 May 1982 
In reality, however, the run-down percentages were adjusted to extend over five years in annual steps of 80, 60, 20, 20 and 20 per cent  The total amount going to the local authorities was the same 
The arrangement was controversial to the last, and also uncertain because it had to be approved in each annual government budget 
Living within its means
After the tax change, Sola’s chief executive officer saw the writing on the wall  It seemed “to be unquestionable that [Sola] has seen its best days in purely financial terms and must return to setting tougher priorities for various assignments,” he asserted in connection with the budget process for 1983 [REMOVE]Fotnote: Chief executive officer’s budget proposal for Sola local authority, 1983 
It took the politicians a little longer to accept this reality, but they were forced to reduce investment and operating expenditures in the years which followed 
Cutting back on the new sports arenas and cultural centre was not very desirable  Nor was it pleasant to have to slow down  But savings had to be made, and long-terms spending plans were removed from the budget for possible reintroduction later 
A raft of measures were stripped from the budget in 1985, such as extensions to and modernisation of schools, sports arenas and swimming pools, a new somatic nursing home, housing for the intellectually disabled and sheltered housing  Grants for national and county roads were reduced [REMOVE]Fotnote: Chief executive officer’s budget proposal for Sola local authority, 1985 
Once the government’s compensation scheme had ended, Torstensbø – now chief executive officer – told Stavanger Aftenblad that he did not want to paint too gloomy a picture 
“But it’s clear that we must set much more moderate financial priorities than we’ve been used to  To sum up the position, we were previously flush with cash and poor in facilities  We’re now flush with facilities and poor in cash ”[REMOVE]Fotnote: Stavanger Aftenblad, ”Alt blir dyrere i det rike Sola”, 19 May 1987 
Sola kulturhus fotografert vinteren 2004
Rogaland county council also raised the question of whether it would be possible to establish a permanent arrangement which allowed local authorities and counties to benefit from some of the tax revenues paid by local oil companies 
The council pointed out that it was otherwise normal practice for Norwegian companies to pay taxes to the local communities they were based in 
This request was turned by Labour finance minister Gunnar Berge because the councils concerned still benefitted from bigger tax payments by oil company employees and on property [REMOVE]Fotnote: Stavanger Aftenblad, “Rogaland reiser skattekrav på ny”, 16 January 1988 
According to Torstensbø, this was only partly true  The big oil companies were not so significant for Sola’s income once the agio tax was excluded 
About NOK 2 million was received annually from Phillips, primarily in property tax  The most important taxpayers in the local authority were the roughly 90 companies at Aker Base  These were service providers such as Halliburton, Schlumberger and Baker Hughes 
At the same time, Sola acquired a steadily growing number of affluent residents and a growing share of its revenue came from income tax  Despite the cut-backs, it remained prosperous 
Published 29  July 2019 • Updated 29  July 2019
More about economy
Participants in Ekofisk
The question of who “owns” Ekofisk is not straightforward  In simple terms, however, the field and the rest of Norway’s continental shelf (NCS) belongs to the Norwegian state  This was determined on 14 June 1963, when the Storting (parliament) passed the Act Relating to Exploration for and Exploitation of Submarine Natural Resources  This permits licences to be awarded on certain terms 
Riding out the oil crisis
The greatest-ever oil bonanza, with oil prices hitting USD 130 per barrel, came to an abrupt end in 2014, when the cost of a barrel of crude slumped to less than USD 50 from June to December  And the bottom had still not been reached – this was only the start of a new oil crisis which lasted several years  What effect did this have on ConocoPhillips’ financial position off Norway?
Context 2: \section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}  Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation 

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary  This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency} 

Although these two approaches share some underlying relations, there are very few connections in the literature  The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i e  Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al } \cite{haykin1997adaptive}  The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model  This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes 

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation  The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step  However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm 

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS)  Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution  We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS  Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}  Here, we show the usefulness of this probabilistic approach for adaptive filtering 

The probabilistic perspective we adopt throughout this work presents two main advantages  Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments  The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w r t  these algorithms and standard LMS  Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications 

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works  However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering \\


\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors 


In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process  In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:


\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix  In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}) \nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$ 
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner  The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i e  the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k  
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive} 

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$  Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful  In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution  We also show that this approximation leads to an LMS-like estimation 
 


\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution 

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ) 
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i e , 

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}  \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A  In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M} 
\label{eq:sigma_hat}
\end{equation}


We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule  First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$  Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}), 
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber 
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch  4]{murphy2012machine})  Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where 
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k   
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq ~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size 


Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2) 
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k, 	
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} \nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$ 

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}  In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$  

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}  More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases  We will comment more about this in the next section  
\end{itemize}



\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments  In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$  The entries of the vector are independently and uniformly chosen in the range $[-1,1]$  Then, the vector is normalized so that $\|{\bf w}^o\|=1$  Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix  The additive noise variance is such that the SNR is $20$ dB
Context 3: Paper Info

Title: Bistability between π-diradical open-shell and closed-shell states in indeno[1,2-a]fluorene
Publish Date: Unkown
Author List: Shantanu Mishra (from IBM Research Europe -Zurich), Manuel Vilas-Varela (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leonard-Alexander Lieske (from IBM Research Europe -Zurich), Ricardo Ortiz (from Donostia International Physics Center (DIPC)), Igor Rončević (from Department of Chemistry, University of Oxford), Florian Albrecht (from IBM Research Europe -Zurich), Diego Peña (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leo Gross (from IBM Research Europe -Zurich)

Figure

Fig  1 | Non-benzenoid non-alternant polycyclic conjugated hydrocarbons a, Classical nonbenzenoid non-alternant polycyclic conjugated hydrocarbons: pentalene, azulene and heptalene b, Generation of indacenes and indenoindenes through benzinterposition and benzannelation of pentalene, respectively Gray filled rings represent Clar sextets c, Closed-shell Kekulé (left) and openshell non-Kekulé (right) resonance structures of QDMs Note that meta-QDM is a non-Kekulé molecule All indenofluorene isomers, being derived through benzannelation of indacenes, contain a central QDM moiety d, Closed-shell Kekulé (top) and open-shell non-Kekulé (bottom) resonance structures of indenofluorenes Compared to their closed-shell structures, 1 and 5 gain two Clar sextets in the openshell structure, while 2-4 gain only one Clar sextet in the open-shell structure Colored bonds in d highlight the ortho-and para-QDM moieties in the two closed-shell Kekulé structures of 5  e, Scheme of on-surface generation of 5 by voltage pulse-induced dehydrogenation of 6 (C20H14) Structures 7 and 8 represent the two monoradical species (C20H13) 
Fig  2 | Characterization of open-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of 5OS in the triplet configuration for the spin up (occupied) level (isovalue: 0 002 e -Å -3 ) Blue and red colors represent opposite phases of the wave function b, Corresponding DFT-calculated spin density of 5OS (isovalue: 0 01 e -Å -3) Blue and orange colors represent spin up and spin down densities, respectively c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) d, DFT-calculated bond lengths of 5OS e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig 7 f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible Scanning parameters: I = 0 3 pA (V = -1 2V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively Scale bars: 10 Å (f) and 5 Å (g) 
Fig  3 | Characterization of closed-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of closed-shell 5 0 (isovalue: 0 002 e -Å -3 ) The wave functions shown here are calculated for the 5para geometry b, DFT-calculated bond lengths of 5ortho (top) and 5para (bottom) c, Constant-height I(V) spectra acquired on a species of 5 assigned as 5para, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 15 pA (negative bias side) and V = 2 2 V, I = 0 15 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig  7  d, Scheme of many-body transitions associated to the measured ionic resonances of 5para Also shown are STM images of assigned 5para at biases where the corresponding transitions become accessible Scanning parameters: I = 0 15 pA (V = -1 5 V) and 0 2 pA (V = 1 7 V)  e, Laplace-filtered AFM image of assigned 5para STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 7 Å  f, Selected bonds labeled for highlighting bond order differences between 5para and 5ortho For the bond pairs a/b, c/d and e/f, the bonds labeled in bold exhibit a higher bond order than their neighboring labeled bonds in 5para g, Laplace-filtered AFM images of 5 on bilayer NaCl/Cu(111) showing switching between 5OS and 5para as the molecule changes its adsorption position The faint protrusion adjacent to 5 is a defect that stabilizes the adsorption of 5  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å  STM and STS data in c and d are acquired on the same species, while the AFM data in e is acquired on a different species Scale bars: 10 Å (d) and 5 Å (e,g) 
NMR (300 MHz, CDCl3) δ: 7 51 (m, 2H), 7 40 -7 28 (m, 5H), 7 27 -7 20 (m, 2H), 7 13 (d, J = 7 7 Hz, 1H), 2 07 (s, 3H), 1 77 (s, 3H) ppm  13C NMR-DEPT (75 MHz, CDCl3, 1:1 mixture of atropisomers) δ: 141 2 (C), 141 1 (C), 140 0 (C), 139 4 (2C), 137 5 (C), 137 4 (C), 136 0 (3C), 134 8 (C), 134 5 (C), 134 1 (C), 134 0 (C), 133 7 (C), 133 6 (C), 131 6 (CH), 131 2 (CH), 131 1 (CH), 130 7 (CH), 129 8 (CH), 129 7 (CH), 129 5 (CH), 129 4 (CH), 129 0 (CH), 128 9 (CH), 128 7 (2CH), 128 6 (2CH), 127 2 (CH), 127 1 (CH), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 20 6 (CH3), 20 5 (CH3), 17 7 (CH3), 17 5 (CH3) ppm MS (APCI) m/z (%): 327 (M+1, 100) HRMS: C20H16Cl2; calculated: 327 0702, found: 327 0709 
NMR (500 MHz, CDCl3) δ: 7 93 (d, J = 7 6 Hz, 1H), 7 85 (d, J = 7 5 Hz, 1H), 7 78 (d, J = 7 7 Hz, 1H), 7 65 (d, J = 7 4 Hz, 1H), 7 61 (d, J = 7 5 Hz, 1H), 7 59 (d, J = 7 7 Hz, 1H), 7 47 (ddd, J = 8 4, 7 2, 1 1 Hz, 1H), 7 42 (dd, J = 8 1, 7 0 Hz, 1H), 7 35 (m, 2H), 4 22 (s, 3H), 4 02 (s, 3H) ppm  13C NMR-DEPT (125 MHz, CDCl3) δ: 144 1 (C), 143 3 (C), 142



ground_truth: The water depth in the Greater Ekofisk Area is 70-75 meters.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the proposed approach in this research paper?

Read the following text and answer briefly.

Context 2:  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}

Context 3: 3 (C), 141 9 (C), 141 8 (C), 141 2 (C), 138 2 (C), 136 5 (C), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 125 3 (CH), 125 2 (CH), 123 6 (CH), 122 2 (CH), 119 9 (CH), 118 4 (CH), 37 4 (CH2), 36 3 (CH2) ppm MS (APCI) m/z (%): 254 (M+, 88) HRMS: C20H14; calculated: 254 1090, found: 254 1090 

abstract

Indenofluorenes are non-benzenoid conjugated hydrocarbons that have received great interest owing to their unusual electronic structure and potential applications in nonlinear optics and photovoltaics  Here, we report the generation of unsubstituted indeno[1,2-a]fluorene, the final and yet unreported parent indenofluorene regioisomer, on various surfaces by cleavage of two C-H bonds in 7,12-dihydro indeno[1,2-a]fluorene through voltage pulses applied by the tip of a combined scanning tunneling microscope and atomic force microscope 
On bilayer NaCl on Au(111), indeno[1,2a]fluorene is in the neutral charge state, while it exhibits charge bistability between neutral and anionic states on the lower work function surfaces of bilayer NaCl on Ag(111) and Cu(111)  In the neutral state, indeno[1,2-a]fluorene exhibits either of two ground states: an open-shell π-diradical state, predicted to be a triplet by density functional and multireference many-body perturbation theory calculations, or a closedshell state with a para-quinodimethane moiety in the as-indacene core 
Switching between open-and closed-shell states of a single molecule is observed by changing its adsorption site on NaCl  The inclusion of non-benzenoid carbocyclic rings is a viable route to tune the physicochemical properties of polycyclic conjugated hydrocarbons (PCHs)   Non-benzenoid polycycles may lead to local changes in strain, conjugation, aromaticity, and, relevant to the context of the present work, induce an open-shell ground state of the corresponding PCHs  
Many nonbenzenoid PCHs are also non-alternant, where the presence of odd-membered polycycles breaks the bipartite symmetry of the molecular network   Figure shows classical examples of non-benzenoid non-alternant PCHs, namely, pentalene, azulene and heptalene  Whereas azulene is a stable PCH exhibiting Hückel aromaticity ([4n+2] π-electrons, n = 2), pentalene and heptalene are unstable Hückel antiaromatic compounds with [4n] π-electrons, n = 2 (pentalene) and n = 3 (heptalene) 
Benzinterposition of pentalene generates indacenes, consisting of two isomers s-indacene and as-indacene (Fig  )  Apart from being antiaromatic, indacenes also contain proaromatic quinodimethane (QDM) moieties (Fig  ) , which endows them with potential open-shell character  While the parent s-indacene and asindacene have never been isolated, stable derivatives of s-indacene bearing bulky substituents have been synthesized  
A feasible strategy to isolate congeners of otherwise unstable non-benzenoid non-alternant PCHs is through fusion of benzenoid rings at the ends of the π-system, that is, benzannelation  For example, while the parent pentalene is unstable, the benzannelated congener indeno[2,1-a]indene is stable under ambient conditions (Fig  )  
However, the position of benzannelation is crucial for stability: although indeno[2,1a]indene is stable, its regioisomer indeno[1,2-a]indene (Fig  ) oxidizes under ambient conditions   Similarly, benzannelation of indacenes gives rise to the family of PCHs known as indenofluorenes (Fig  ), which constitute the topic of the present work 
Depending on the benzannelation position and the indacene core, five regioisomers can be constructed, namely, indeno [ Practical interest in indenofluorenes stems from their low frontier orbital gap and excellent electrochemical characteristics that render them as useful components in organic electronic devices  
The potential open-shell character of indenofluorenes has led to several theoretical studies on their use as non-linear optical materials and as candidates for singlet fission in organic photovoltaics   Recent theoretical work has also shown that indenofluorene-based ladder polymers may exhibit fractionalized excitations 
Fundamentally, indenofluorenes represent model systems to study the interplay between aromaticity and magnetism at the molecular scale   Motivated by many of these prospects, the last decade has witnessed intensive synthetic efforts toward the realization of indenofluorenes  Derivatives of 1-4 have been realized in solution , while 1-3 have also been synthesized on surfaces and characterized using scanning tunneling microscopy (STM) and atomic force microscopy (AFM), which provide information on molecular orbital densities , molecular structure and oxidation state  
With regards to the open-shell character of indenofluorenes, 2-4 are theoretically and experimentally interpreted to be closed-shell, while calculations indicate that 1 and 5 should exhibit open-shell ground states   Bulk characterization of mesitylsubstituted 1, including X-ray crystallography, temperature-dependent NMR, and electron spin resonance spectroscopy, provided indications of its open-shell ground state  
Electronic characterization of 1 on Au(111) surface using scanning tunneling spectroscopy (STS) revealed a low electronic gap of 0 4 eV (ref  )  However, no experimental proof of an openshell ground state of 1 on Au(111), such as detection of singly occupied molecular orbitals (SOMOs) or spin excitations and correlations due to unpaired electrons , was shown 
In this work, we report the generation and characterization of unsubstituted 5  Our research is motivated by theoretical calculations that indicate 5 to exhibit the largest diradical character among all indenofluorene isomers   The same calculations also predict that 5 should possess a triplet ground state 
Therefore, 5 would qualify as a Kekulé triplet, of which only a handful of examples exist   However, definitive synthesis of 5 has never been reported so far  Previously, Dressler et al  reported transient isolation of mesityl-substituted 5, where it decomposed both in the solution and in solid state , and only the structural proof of the corresponding dianion was obtained 
On-surface generation of a derivative of 5, starting from truxene as a precursor, was recently reported   STM data on this compound, containing the indeno[1,2-a]fluorene moiety as part of a larger PCH, was interpreted to indicate its open-shell ground state  However, the results did not imply the ground state of unsubstituted 5  Here, we show that on insulating surfaces 5 can exhibit either of two ground states: an open-shell or a closed-shell 
We infer the existence of these two ground states based on high-resolution AFM imaging with bond-order discrimination and STM imaging of molecular orbital densities   AFM imaging reveals molecules with two different geometries  Characteristic bond-order differences in the two geometries concur with the geometry of either an open-or a closed-shell state 
Concurrently, STM images at ionic resonances show molecular orbital densities corresponding to SOMOs for the open-shell geometry, but orbital densities of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the closed-shell geometry  Our experimental results are in good agreement with density functional theory (DFT) and multireference perturbation theory calculations 
Finally, we observe switching between open-and closed-shell states of a single molecule by changing its adsorption site on the surface  Synthetic strategy toward indeno[1,2-a]fluorene  The generation of 5 relies on the solution-phase synthesis of the precursor 7,12-dihydro indeno[1,2-a]fluorene (6)  Details on synthesis and characterization of 6 are reported in Supplementary Figs 
  Single molecules of 6 are deposited on coinage metal (Au(111), Ag(111) and Cu(111)) or insulator surfaces  In our work, insulating surfaces correspond to two monolayer-thick (denoted as bilayer) NaCl on coinage metal surfaces  Voltage pulses ranging between 4-6 V are applied by the tip of a combined STM/AFM system, which result in cleavage of one C-H bond at each of the pentagonal apices of 6, thereby leading to the generation of 5 (Fig  ) 
In the main text, we focus on the generation and characterization of 5 on insulating surfaces  Generation and characterization of 5 on coinage metal surfaces is shown in Supplementary Fig    )  Blue and orange colors represent spin up and spin down densities, respectively  c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) 
d, DFT-calculated bond lengths of 5OS  e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra  Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side)  Acquisition position of the spectra is shown in Supplementary Fig    f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS 
Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible  Scanning parameters: I = 0 3 pA (V = -1 2 V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3
Å  The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint  f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island  The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively 
Scale bars: 10 Å (f) and 5 Å (g)  To experimentally explore the electronic structure of 5, we used bilayer NaCl films on coinage metal surfaces to electronically decouple the molecule from the metal surfaces  Before presenting the experimental findings, we summarize the results of our theoretical calculations performed on 5 in the neutral charge state (denoted as 5 0 ) 
We start by performing DFT calculations on 5 0 in the gas phase  Geometry optimization performed at the spin-unrestricted UB3LYP/6-31G level of theory leads to one local minimum, 5OS, the geometry of which corresponds to the open-shell resonance structure of 5 (Fig  , the label OS denotes open-shell) 
The triplet electronic configuration of 5OS is the lowest-energy state, with the openshell singlet configuration 90 meV higher in energy  Geometry optimization performed at the restricted closed-shell RB3LYP/6-31G level reveals two local minima, 5para and 5ortho, the geometries of which (Fig  ) exhibit bond length alternations in line with the presence of a para-or an ortho-QDM moiety, respectively, in the as-indacene core of the closed-shell resonance structures of 5 (Fig  )  
Relative to 5OS in the triplet configuration, 5para and 5ortho are 0 40 and 0 43 eV higher in energy, respectively  Additional DFT results are shown in Supplementary Fig    To gain more accurate insights into the theoretical electronic structure of 5, we performed multireference perturbation theory calculations (Supplementary Fig  ) based on quasi-degenerate second-order n-electron valence state perturbation theory (QD-NEVPT2) 
In so far as the order of the ground and excited states are concerned, the results of QD-NEVPT2 calculations qualitatively match with DFT calculations  For 5OS, the triplet configuration remains the lowest-energy state, with the open-shell singlet configuration 60 meV higher in energy  The energy differences between the open-and closed-shell states are substantially reduced in QD-NEVPT2 calculations, with 5para and 5ortho only 0 11 and 0 21 eV higher in energy, respectively, compared to 5OS in the triplet configuration 
We also performed nucleus-independent chemical shift calculations to probe local aromaticity of 5 in the openand closed-shell states  While 5OS in the triplet configuration exhibits local aromaticity at the terminal benzenoid rings, 5OS in the open-shell singlet configuration, 5para and 5ortho all display antiaromaticity (Supplementary Fig  ) 
The choice of the insulating surface determines the charge state of 5: while 5 adopts neutral charge state on the high work function bilayer NaCl/Au(111) surface (irrespective of its openor closed-shell state, Supplementary Fig  ), 5 exhibits charge bistability between 5 0 and the anionic state 5 -1 on the lower work function bilayer NaCl/Ag(111) and Cu(111) surfaces (Supplementary Figs  ) 
In the main text, we focus on the characterization of 5 on bilayer NaCl/Au(111)  Characterization of charge bistable 5 is reported in Supplementary Figs    We first describe experiments on 5 on bilayer NaCl/Au(111), where 5 exhibits a geometry corresponding to the calculated 5OS geometry, and an open-shell electronic configuration 
We compare the experimental data on this species to calculations on 5OS with a triplet configuration, as theory predicts a triplet ground state for 5OS  For 5OS, the calculated frontier orbitals correspond to the SOMOs ψ1 and ψ2 (Fig  ), whose spin up levels are occupied and the spin down levels are empty 
Figure shows the DFT-calculated bond lengths of 5OS, where the two salient features, namely, the small difference in the bond lengths within each ring and the notably longer bond lengths in the pentagonal rings, agree with the open-shell resonance structure of 5 (Fig  )  Figure shows an AFM image of 5 adsorbed on bilayer NaCl/Au(111) that we assign as 5OS, where the bond-order differences qualitatively correspond to the calculated 5OS geometry (discussed and compared to the closed-shell state below) 
Differential conductance spectra (dI/dV(V), where I and V denote the tunneling current and bias voltage, respectively) acquired on assigned 5OS exhibit two peaks centered at -1 5 V and 1 6 V (Fig  ), which we assign to the positive and negative ion resonances (PIR and NIR), respectively  Figure shows the corresponding STM images acquired at the onset (V = -1 2
V/1 3 V) and the peak (V = -1 5 V/1 6 V) of the ionic resonances



ground_truth: This research paper proposed an approach based on approximating the posterior distribution with an isotropic Gaussian distribution.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the two ground states observed for indeno[1,2-a]fluorene on NaCl surfaces?

Read the following text and answer briefly.

Context 3:  To draw a correspondence between the STM images and the molecular orbital densities, we consider tunneling events as many-body electronic transitions between different charge states of 5OS (Fig  )  Within this framework, the PIR corresponds to transitions between 5 0 and the cationic state 5  
At the onset of the PIR at -1 2 V, an electron can only be detached from the SOMO ψ1 and the corresponding STM image at -1 2 V shows the orbital density of ψ1  Increasing the bias to the peak of the PIR at -1 5 V, it becomes possible to also empty the SOMO ψ2, such that the corresponding STM image shows the superposition of ψ1 and ψ2, that is, |ψ1| 2 + |ψ2| 2 (ref 
)  Similarly, the NIR corresponds to transitions between 5 0 and 5 -1   At the NIR onset of 1 3 V, only electron attachment to ψ2 is energetically possible  At 1 6 V, electron attachment to ψ1 also becomes possible, and the corresponding STM image shows the superposition of ψ1 and ψ2  The observation of the orbital densities of SOMOs, and not the hybridized HOMO and LUMO, proves the open-shell ground state of assigned 5OS 
Measurements of the monoradical species with a doublet ground state are shown in Supplementary Fig    Unexpectedly, another species of 5 was also experimentally observed that exhibited a closedshell ground state  In contrast to 5OS, where the frontier orbitals correspond to the SOMOs ψ1 and ψ2, DFT calculations predict orbitals of different shapes and symmetries for 5para and 5ortho, denoted as α and β and shown in Fig   
For 5ortho, α and β correspond to HOMO and LUMO, respectively  The orbitals are inverted in energy and occupation for 5para, where β is the HOMO and α is the LUMO  Fig  shows an AFM image of 5 that we assign as 5para  We experimentally infer its closed-shell state first by using qualitative bond order discrimination by AFM 
In high-resolution AFM imaging, chemical bonds with higher bond order are imaged brighter (that is, with higher frequency shift Δf) due to stronger repulsive forces, and they appear shorter   In Fig  , we label seven bonds whose bond orders show significant qualitative differences in the calculated 5ortho, 5para (Fig  ) and 5OS (Fig  ) geometries 
In 5para, the bonds b and d exhibit a higher bond order than a and c, respectively  This pattern is reversed for 5ortho, while the bond orders of the bonds a-d are all similar and small for 5OS  Furthermore, in 5para bond f exhibits a higher bond order than e, while in 5ortho and 5OS bonds e and f exhibit similar bond order (because they belong to Clar sextets) 
Finally, the bond labeled g shows a higher bond order in 5para than in 5ortho and 5OS  The AFM image of assigned 5para shown in Fig  indicates higher bond orders of the bonds b, d and f compared to a, c and e, respectively  In addition, the bond g appears almost point-like and with enhanced Δf contrast compared to its neighboring bonds, indicative of a high bond order (see Supplementary Fig  for height-dependent measurements) 
These observations concur with the calculated 5para geometry (Fig  )  Importantly, all these distinguishing bond-order differences are distinctly different in the AFM image of 5OS shown in Fig  , which is consistent with the calculated 5OS geometry (Fig  )  In the AFM images of 5OS (Fig  and Supplementary Fig  ), the bonds a-d at the pentagon apices appear with similar contrast and apparent bond length 
The bonds e and f at one of the terminal benzenoid rings also exhibit similar contrast and apparent bond length, while the central bond g appears longer compared to assigned 5para  Further compelling evidence for the closed-shell state of assigned 5para is obtained by STM and STS  dI/dV(V) spectra acquired on an assigned 5para species exhibit two peaks centered at -1 4 V (PIR) and 1 6 V (NIR) (Fig  ) 
STM images acquired at these biases (Fig  ) show the orbital densities of β (-1 4 V) and α (1 6 V)  First, the observation of α and β as the frontier orbitals of this species, and not the SOMOs, strongly indicates its closed-shell state  Second, consistent with AFM measurements that indicate good correspondence to the calculated 5para geometry, we observe β as the HOMO and α as the LUMO 
For 5ortho, α should be observed as the HOMO and β as the LUMO  We did not observe molecules with the signatures of 5ortho in our experiments  We observed molecules in open-(5OS, Fig  ) and closed-shell (5para, Fig  ) states in similar occurrence after their generation from 6 on the surface  We could also switch individual molecules between open-and closed-shell states as shown in Fig  and Supplementary Fig   
To this end, a change in the adsorption site of a molecule was induced by STM imaging at ionic resonances, which often resulted in movement of the molecule  The example presented in Fig  shows a molecule that was switched from 5para to 5OS and back to 5para  The switching is not directed, that is, we cannot choose which of the two species will be formed when changing the adsorption site, and we observed 5OS and 5para in approximately equal yields upon changing the adsorption site 
The molecule in Fig  is adsorbed on top of a defect that stabilizes its adsorption geometry on bilayer NaCl  At defect-free adsorption sites on bilayer NaCl, that is, without a third layer NaCl island or atomic defects in the vicinity of the molecule, 5 could be stably imaged neither by AFM nor by STM at ionic resonances (Supplementary Fig  ) 
Without changing the adsorption site, the state of 5 (open-or closedshell) never changed, including the experiments on bilayer NaCl/Ag(111) and Cu(111), on which the charge state of 5 could be switched (Supplementary Figs  )  Also on these lower work function surfaces, both open-and closed-shell species were observed for 5 0 and both showed charge bistability between 5 0 (5OS or 5para) and 5 -1 (Supplementary Figs  ) 
The geometrical structure of 5 -1 probed by AFM, and its electronic structure probed by STM imaging at the NIR (corresponding to transitions between 5 -1 and the dianionic state 5 -2 ), are identical within the measurement accuracy for the charged species of both 5OS and 5para  When cycling the charge state of 5 between 5 0 and 5 -1 several times, we always observed the same state (5OS or 5para) when returning to 5 0 , provided the molecule did not move during the charging/discharging process 
Based on our experimental observations we conclude that indeno[1,2-a]fluorene (5), the last unknown indenofluorene isomer, can be stabilized in and switched between an open-shell (5OS) and a closed-shell (5para) state on NaCl  For the former, both DFT and QD-NEVPT2 calculations predict a triplet electronic configuration 
Therefore, 5 can be considered to exhibit the spin-crossover effect, involving magnetic switching between high-spin (5OS) and low-spin (5para) states, coupled with a reversible structural transformation  So far, the spin-crossover effect has mainly only been observed in transition-metal-based coordination compounds with a near-octahedral geometry  
The observation that the switching between open-and closedshell states is related to changes in the adsorption site but is not achieved by charge-state cycling alone, indicates that the NaCl surface and local defects facilitate different electronic configurations of 5 depending on the adsorption site 
Gas-phase QD-NEVPT2 calculations predict that 5OS is the ground state, and the closed-shell 5para and 5ortho states are 0 11 and 0 21 eV higher in energy  The experiments, showing bidirectional switching between 5OS and 5para, indicate that a change in the adsorption site can induce sufficient change in the geometry of 5 (leading to a corresponding change in the ground state electronic configuration) and thus induce switching 
Switching between open-and closed-shell states in 5 does not require the breaking or formation of covalent bonds , but a change of adsorption site on NaCl where the molecule is physisorbed  Our results should have implications for single-molecule devices, capitalizing on the altered electronic and chemical properties of a system in π-diradical open-shell and closed-shell states such as frontier orbital and singlet-triplet gaps, and chemical reactivity 
For possible future applications as a single-molecule switch, it might be possible to also switch between open-and closed-shell states by changing the local electric field, such as by using chargeable adsorbates   Scanning probe microscopy measurements and sample preparation  STM and AFM measurements were performed in a home-built system operating at base pressures below 1×10 -10 mbar and a base temperature of 5 K  Bias voltages are provided with respect to the sample 
All STM, AFM and spectroscopy measurements were performed with carbon monoxide (CO) functionalized tips  AFM measurements were performed in non-contact mode with a qPlus sensor   The sensor was operated in frequency modulation mode with a constant oscillation amplitude of 0 5 Å  STM measurements were performed in constantcurrent mode, AFM measurements were performed in constant-height mode with V = 0 V, and I(V) and Δf(V) spectra were acquired in constant-height mode 
Positive (negative) values of the tip-height offset Δz represent tip approach (retraction) from the STM setpoint  All dI/dV(V) spectra are obtained by numerical differentiation of the corresponding I(V) spectra  STM and AFM images, and spectroscopy curves, were post-processed using Gaussian low-pass filters 
Au(111), Ag(111) and Cu(111) surfaces were cleaned by iterative cycles of sputtering with Ne + ions and annealing up to 800 K  NaCl was thermally evaporated on Au(111), Ag(111) and Cu(111) surfaces held at 323 K, 303 K and 283 K, respectively  This protocol results in the growth of predominantly bilayer (100)-terminated islands, with a minority of trilayer islands 
Sub-monolayer coverage of 6 on surfaces was obtained by flashing an oxidized silicon wafer containing the precursor molecules in front of the cold sample in the microscope  CO molecules for tip functionalization were dosed from the gas phase on the cold sample  Density functional theory calculations  DFT was employed using the PSI4 program package  
All molecules with different charge (neutral and anionic) and electronic (open-and closed-shell) states were independently investigated in the gas phase  The B3LYP exchangecorrelation functional with 6-31G basis set was employed for structural relaxation and singlepoint energy calculations  The convergence criteria were set to 10 −4 eV Å −1 for the total forces and 10 −6 eV for the total energies 
Multireference calculations  Multireference calculations were performed on the DFToptimized geometries using the QD-NEVPT2 level of theory , with three singlet roots and one triplet root included in the state-averaged calculation  A (10,10) active space (that is, 10 electrons in 10 orbitals) was used along with the def2-TZVP basis set  
Increasing either the active space size or expanding the basis set resulted in changes of about 50 meV for relative energies of the singlet and triplet states  These calculations were performed using the ORCA program package   Nucleus-independent chemical shift (NICS) calculations  Isotropic nucleus-independent chemical shift values were evaluated at the centre of each ring using the B3LYP exchangecorrelation functional with def2-TZVP basis set using the Gaussian 16 software package  
Starting materials (reagent grade) were purchased from TCI and Sigma-Aldrich and used without further purification  Reactions were carried out in flame-dried glassware and under an inert atmosphere of purified Ar using Schlenk techniques  Thin-layer chromatography (TLC) was performed on Silica Gel 60 F-254 plates (Merck) 
Column chromatography was performed on silica gel (40-60 µm)  Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Varian Mercury 300 or Bruker Varian Inova 500 spectrometers  Mass spectrometry (MS) data were recorded in a Bruker Micro-TOF spectrometer  The synthesis of compound 6 was developed following the two-step synthetic route shown in Supplementary Fig  , which is based on the preparation of methylene-bridge polyarenes by means of Pd-catalyzed activation of benzylic C-H bonds  
Supplementary Figure | Synthetic route to obtain compound 6  The complex Pd2(dba)3 (20 mg, 0 02 mmol) was added over a deoxygenated mixture of 1,3-dibromo-2,4-dimethylbenzene (9, 100 mg, 0 38 mmol), boronic acid 10 (178 mg, 1 14 mmol), K2CO3 (314 mg, 2 28 mmol) and XPhos (35 mg, 0 08 mmol) in toluene (1:1, 10 mL), and the resulting mixture was heated at 90 °C for 2 h 
After cooling to room temperature, the solvents were evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording 11 (94 mg, 76%) as a colorless oil  The complex Pd(OAc)2 (7 mg, 0 03 mmol) was added over a deoxygenated mixture of terphenyl 11 (90 mg, 0 27 mmol), K2CO3 (114 mg, 0 83 mmol) and ligand L (26 mg, 0 06 mmol) in NMP (2 mL) 
The resulting mixture was heated at 160 °C for 4 h  After cooling to room temperature, H2O (30 mL) was added, and the mixture was extracted with EtOAc (3x15 mL)  The combined organic extracts were dried over anhydrous Na2SO4, filtered, and evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording compound 6 (8 mg, 11%) as a white solid  in AFM imaging due to their reduced adsorption height compared to the rest of the carbon atoms 
We attribute this observation to the significantly different lattice parameter of Cu(111) (2 57 Å) compared to Au(111) and Ag(111) (2 95 Å and 2 94 Å, respectively) , such that the apical carbon atoms of the pentagonal rings of 5 adsorb on the on-top atomic sites on Au(111) and Ag(111), but not on Cu(111) 
Our speculation is based on a previous study of polymers of 1 on Au(111) by Di Giovannantonio et al  , where both tilted and planar individual units of 1 were observed depending on whether the apical carbon atoms of the pentagonal rings in 1 adsorbed on the on-top or hollow sites of the surface, respectively 
Given the strong molecule-metal interaction, we found no electronic state signatures of 5 on all three metal surfaces  STM set point for AFM images: V = 0  e, Frontier orbital spectrum of 5 -1   In the anionic state, ψ2 becomes doubly occupied and ψ1 is the SOMO  Filled and empty circles denote occupied and empty orbitals, respectively 
For each panel, zero of the energy axis has been aligned to the respective highest-energy occupied orbital 



ground_truth: Open-shell π-diradical state and closed-shell state with a para-quinodimethane moiety.