VALIDATION OF A
NUMERICAL SIMULATION
OF THE NORTH ATLANTIC
OCEAN WITH ARGO DATA
Deg ee in Ma ine Sciences
Lau a Gómez Na a o
Tu o : Alonso He nández Gue a
Co- u o : Joaquim Ballab e a Poy
June 2014
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
Abs ac
The ou pu s o a nume ical simula ion ha e been alida ed wi h A go da a; wi h he idea
o being able o use hese ou pu s o assimila e SSS in o ma ion e ie ed by he sa elli e
SMOS. In addi ion o A go da a, he model is also compa ed agains he WOA09
clima ology. The e a e a conside able amoun o alid p o iles a ailable du ing yea
2012, and he co e age does no ha e empo al biases, as he amoun o da a is
homogeneous h oughou he yea . This demons a es ha choosing A go o alida e
he simula ion was a co ec choice. The esul s show ha he e exis s a easonable
ag eemen be ween model and A go da a in he open Ocean. Howe e , i has been
obse ed ha he simula ion s ill con ains spa ial and empo al e o s p obably due o a
bad ep esen a ion o ce ain oceanic p ocesses. Ne e heless, as he e o s a e easily
iden i ied and do no co e he open Ocean in he opical and sub opical egions, he
simula ion could s ill be conside ed as an app op ia e ool o imp o e he emo e
sensing o SSS da a.
Lau a Gómez Na a o
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Table o Con en s
1. In oduc ion ................................................................................................................. 6
2. Da a .............................................................................................................................. 8
2.1. Model and simula ion desc ip ion .......................................................................... 8
2.2. Wo ld Ocean A las ................................................................................................. 9
2.3. A go ..................................................................................................................... 10
3. Me hodology and esul s ............................................................................................ 14
3.1. P elimina y da a explo a ion. ............................................................................... 14
3.2. The 2012 No h A lan ic nume ical simula ion ................................................... 19
3.3. The 2012 No h A lan ic A go da a ..................................................................... 23
3.4. Simula ion-A go di e ences. ............................................................................... 28
4. Conclusions ............................................................................................................... 34
5. Re e ences ................................................................................................................. 37
6. Appendices ................................................................................................................. 39
6.1. Appendix 1 ........................................................................................................... 39
6.2. Appendix 2 .......................................................................................................... 40
6.3. Appendix 3 ........................................................................................................... 41
7. Addi ional in o ma ion abou he de elopmen o his s udy .................................... 42
7.1. De ailed desc ip ion o he ac i i ies ca ied ou . ................................................ 42
7.1.1. S a is ical cha ac e iza ion o he simula ion’s ou pu s. ................................ 42
7.1.2. Reco e y and analysis o he da a measu ed by A go loa s. ................... 43
7.1.3. Global compa ison o he nume ical model and A go da a. ......................... 44
7.2. T aining ecei ed ................................................................................................. 45
7.3. Le el o in eg a ion and in ol emen in he depa men and ela ionship wi h
pe sonnel. .................................................................................................................... 45
7.4. Posi i e and nega i e aspec s ela ed o he de elopmen o his s udy. ............. 46
7.5. Pe sonal app aisal o he lea ning achie ed du ing his s udy. ............................ 46
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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Lis o igu es
Figu e 1. Schema ic o he su ace ci cula ion o he A lan ic Ocean ......................................................... 6
Figu e 2. Wo ld dis ibu ion o he uppe Oceans’ wa e masses ................................................................ 7
Figu e 3. A go loa s dis ibu ion on 11 h o May 2014 ............................................................................ 10
Figu e 4. Desc ip ion o A go loa s’ cycle ............................................................................................... 11
Figu e 5. Di e en pa s o one o he A go loa s models ....................................................................... 12
Figu e 6. Sea Su ace Heigh (m) in 2008 ................................................................................................. 14
Figu e 7. His og ams o po en ial empe a u e da a a laye 1 ................................................................... 16
Figu e 8. Change o mean and median alues wi h dep h ......................................................................... 16
Figu e 9. Con ou map o salini y (psu) a laye 1 wi h he pe iphe al egions ma ked. ........................... 17
Figu e 10. T-S diag am o all he simula ion da a. .................................................................................... 18
Figu e 11. Loca ion o Sm alues lowe han 30 psu. ............................................................................... 19
Figu e 12. Maximum alues’ dis ibu ion in me es o SSH and MLD ................................................... 20
Figu e 13. SSH’s s a is ical pa ame e s a ia ion wi h ime. ..................................................................... 20
Figu e 14. MLD’s s a is ical pa ame e s a ia ion wi h ime. ................................................................... 21
Figu e 15. Salini y 1D plo a laye 15 o yea 2012. .............................................................................. 22
Figu e 16. Minimum salini y alues’ dis ibu ion o 2012 laye 5 and WOA09 salini y alues. ............. 23
Figu e 17. Minimum salini y alues’ dis ibu ion o 2012 laye 14 and WOA09 salini y alues . .......... 23
Figu e 18. A go p o iles o 2012 .............................................................................................................. 25
Figu e 19. Po en ial empe a u e (ºC) o he A go p o iles o laye 1 and laye 2. .................................. 26
Figu e 20. Salini y (psu) o he A go p o iles o laye 1 and laye 2. ...................................................... 27
Figu e 21. Ta and Sa s a iscal pa ame e s a ia ion wi h ime o laye 3 ................................................. 27
Figu e 22. Sca e map o he ΔT (ºC) o laye 2, laye 14 and laye 29. ................................................. 29
Figu e 23. Sca e map o ΔS (psu) o laye 2 .......................................................................................... 30
Figu e 24. ΔT and ΔS alues o e e y buoy. ............................................................................................ 30
Figu e 25. Loca ion o he A go buoys wi h an absolu e alue o ΔT and ΔS .......................................... 31
Figu e 26. Va ia ion o ΔT and ΔS wi h ime o laye 2. ......................................................................... 31
Figu e 27. Posi ions o he selec ed buoys o he T-S diag ams ............................................................... 32
Figu e 28. T-S diag ams o he A go and simula ion da a ......................................................................... 33
Figu e 29. Salini y alues a he Gul o Sain Law ence, o he 15 h o No embe a 30m.................... 35
Lis o ables
Table 1. Ad an ages o he A go da ase in compa ison o o he sou ces o da a ..................................... 12
Table 2. A go quali y con ol lag scale .................................................................................................... 13
Table 3. Desc ip ion o he s a is ical pa ame e s calcula ed. .................................................................... 15
Table 4. S a is ical pa ame e s o SSH and MLD ...................................................................................... 17
Table 5. Da es o he 12 p o iles selec ed o he T-S diag ams. ............................................................... 32

Lau a Gómez Na a o
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1. In oduc ion
The A lan ic Ocean has been ho oughly s udied o mo e han a cen u y, and i
is s ill a subjec o g ea ocus o a ious easons. I has a signi ican ole in egula ing
he clima e, o he no he n hemisphe e ( o example, he hea anspo ed by he wa e
o he Gul S eam con ibu es o wa m he Eu opean Subcon inen ). In addi ion, deep
con ec ion occu s in he No h A lan ic Ocean. Finally, i has a human impo ance in
ma i ime secu i y, economic, social and mili a y ac i i ies.
All o hese oles o he A lan ic Ocean a e due by he wind and hea o cing, bu
egula ed by i s wa e masses and ci cula ion which can be seen in Figu e 1.
Figu e 1. Schema ic o he su ace ci cula ion o he A lan ic Ocean. Con inuous lines a e wa m and
discon inuous cold cu en s (B own e al., 1988).
The Ocean ci cula ion is modula ed by local densi y, which is a unc ion o
empe a u e and salini y. Acco ding o Eme y and Thomson (2004), in oceanog aphy,
wa e masses a e de ined in di e se ways; “ eal, objec i e physical en i ies, building
blocks om which he oceanic s a i ica ion ( e ical s uc u e) is cons uc ed”, “mainly
desc ip i e wo ds, summa y sho hand o poin ing o p ominen ea u es in p ope y
dis ibu ions” o “a single poin on a cha ac e is ic diag am such as a Tempe a u e-
Salini y (T-S) cu e”.
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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Thus, he analysis o he cha ac e is ics o he wa e masses p esen in a gi en
egion is impo an as i p o ides elemen s helping o desc ibe he Oceans’ ci cula ion.
In he A lan ic Ocean di e en wa e masses a e ound:
 No h A lan ic Cen al Wa e (NACW)
o No h A lan ic Eas Cen al Wa e (NAECW)
o No h A lan ic Wes Cen al Wa e (NAWCW)
 Sou h A lan ic Cen al Wa e (SACW)
 An a c ic In e media e Wa e (AIW)
 A c ic In e media e Wa e (AAIW)
 Medi e anean In e media e Wa e (MIW)
 No h A lan ic Deep Wa e (NADW)
o No h A lan ic Eas Deep Wa e (NAEDW)
o No h A lan ic Wes Deep Wa e (NAWDW)
 An a c ic Bo om Wa e (ABW)
Each one can be iden i ied by hei pa icula empe a u e, salini y and dep h.
Fo example he MIW, is iden i ied easily by i s high salini y and empe a u e. In
addi ion, di e en egions ha e di e en p ominen wa e masses (Figu e 2).
Figu e 2. Wo ld dis ibu ion o he uppe Oceans’ wa e masses (Chen, 2009).
Lau a Gómez Na a o
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In his s udy, he ou pu s o a nume ical simula ion o he No h A lan ic Ocean
a e alida ed. This wo k ep esen s a i s s ep owa ds he goal o p oducing a be e
Seas Su ace Salini y (SSS) emo ely-sensed p oduc in he No h A lan ic Ocean.
Indeed, since he launch o he Soil Mois u e/Ocean Salini y (SMOS) sa elli e, membe s
o he Ins i u de Ciències del Ma (CSIC) a e wo king o p oduce wo ldwide maps o
SSS om he in o ma ion ga he ed by he sa elli e. Howe e , he SSS e ie als a e
pa icula ly noisy in he A lan ic Ocean as he sa elli e measu es (in he L-band,
1.4GHz) a e con amina ed by many adio equency in e e ences in his egion. The
nume ical model used he e is going o be used o help educe he noise o signal a io o
he SMOS SSS in he No h A lan ic using a ious da a assimila ion echniques ha will
use he model as a dynamical in e pola o .
2. Da a
2.1. Model and simula ion desc ip ion
The nume ical model sol es he so called P imi i e Equa ions wi h a ee su ace
o mula ion (B odeau e al., 2010). The egional, eddy-pe mi ing con igu a ion
co esponds o he No h A lan ic Ocean. The sou he n and no he n open bounda ies
a e elaxed owa ds he Wo ld Ocean A las 09 (WOA09) clima ology da a, o ep esen
mo e ealis ically he Me idional O e u ning Ci cula ion (Hoa eau e al., 2014). This
con igu a ion couples he Nucleus o Eu opean Modelling o he Ocean (NEMO)
oceanic model and he Lou ain-la-Neu e Ice Model (LIM2.0), which is inside he
ORCA4 ¼º g id global model. The ho izon al domain o his g id includes he A lan ic
Ocean’s basin om 80ºN o 20ºS, he No dic Seas, he Denma k S ai and pa o he
Medi e anean Sea, up o 23ºE (Min ielle e al., 2011). The spa ial esolu ion is 0.25
deg ees. I s e ical domain is o 45 geopo en ial le els, and he g id spacing is 6 m a
he su ace and 250 m a he bo om (Hoa eau e al., 2014). Ri e uno is simula ed by
adding a lux o wa e wi h a salini y o 0 psu and wi h he same empe a u e as he one
a i s mou h. The solu ions a e ad anced in ime using a Leap-F og me hod, combined
wi h a Robe -Asselin il e o ensu e he s abili y o he scheme. The ime s ep o he
simula ion is 40 minu es.
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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A 2001-2012 simula ion using ealis ic o cing ields has been ca ied ou by
Ms. Nina Hoa eau. The simula ion equi es abou 6 hou s o each simula ed yea when
60 Cen al P ocessing Uni s (CPUs) a e used. Each ou pu ile o he simula ion was
sa ed in Ne CDF o ma , occupied 106 Mb, and includes he i e-day a e age o he
s o ed pa ame e s.
2.2. Wo ld Ocean A las
Clima ology is, as de ined by he Na ional Wea he Se ice o he Na ional
Oceanic and A mosphe ic Adminis a ion (NOAA), “a quan i a i e desc ip ion o
clima e showing he cha ac e is ic alues o clima e a iables o e a egion”
(www.nws.noaa.go /clima e/help/glossa y.php). Clima ologies a e s a ic, a e aged
alues o a gi en pa ame e , calcula ed du ing a ce ain pe iod o ime. The e o e, i
does no p o ide, in gene al, an obse able alue, bu i should gi e an idea o he mos
p obable alue o be measu ed.
In pa icula , empe a u e and salini y clima ologies in a gi en a ea a e usually
es ima ed by he a e age alues o all obse a ions e e measu ed a ha a ea, and hus,
hey would de ine he mean s a e o he Ocean i he sampling we e app op ia e. The T
and S obse a ions a e collec ed om measu ed su ace da a (as he mosalinog aphe s)
and/o p o iles. Howe e , as measu emen s a e nei he cons an in ime no space, hey
equi e ime-spa ial a e aging and smoo hing (Higgison e al., 2009).
The Wo ld Ocean A las, WOA (An ono e al., 2010; Loca nini e al., 2010) is
he mos widely used clima ology o oceanic empe a u e and salini y. This Wo ld
Ocean clima ology was i s published by Le i us in 1982, and h oughou he yea s i
has been imp o ed by including mo e da a and using new me hods o calcula e
clima ology, and he e o e eleasing new a lases e e y ew yea s (Cha e jee e al.,
2012).
The uses o hese clima ology a lases a e di e se: bounda y and/o ini ial
condi ions in Ocean ci cula ion nume ical models and A mosphe e-Ocean models,
e i ica ion o nume ical simula ions o he ocean, as a o m o “sea u h” o sa elli e
Lau a Gómez Na a o
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empe a u e, i is in e es ing o no ice he educ ion o he di e ences be ween he mean
and he median. The la ge he di e ences be ween he mean and he median, he la ge
is he impac o he ails o he dis ibu ion, and he less signi ican a e he alues o he
s a is ical pa ame e s lis ed in Table 3. On he o he hand, he esul s shown in Table 4
indica e ha he s a is ical desc ip ion o SSH is li le a ec ed by he la i ude limi s, bu
he MLD is.
Figu e 7. His og ams o po en ial empe a u e da a a laye 1 (3m). Le : O iginal da a. Righ : Wi h
la i ude limi s and land alues changed.
Figu e 8. Change o mean ( ed line) and median (blue line) alues wi h dep h o he o iginal da a (le )
and wi h la i ude limi s and land alues changed ( igh ).
-5 0 5 10 15 20 25 30 35
0
0.5
1
1.5
2
2.5
3x 104
Po en ial empe a u e (ºC)
F equency
-5 0 5 10 15 20 25 30 35
0
0.5
1
1.5
2
2.5
3x 104
Po en ial empe a u e (ºC)
F equency
010 20 30
0
1000
2000
3000
4000
5000
6000
Po en ial empe a u e (ºC)
Dep h (m)
010 20 30
0
1000
2000
3000
4000
5000
6000
Po en ial empe a u e (ºC)
Dep h (m)

Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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Table 4. S a is ical pa ame e s o SSH and MLD o he o iginal da a and columns ma ked wi h* a e o
da a be ween 60ºN and -5ºN.
Pa ame e s
SSH (m)
SSH* (m)
MLD (m)
MLD* (m)
Maximum
0,79
0,79
393,84
249,92
Minimum
-0,94
-0,94
12,84
12,84
Range
1,74
1,74
381,00
237,08
Pe cen ile 99
0,53
0,55
189,89
139,91
Pe cen ile 1
-0,87
-0,89
12,84
12,84
Robus ange
1,40
1,44
177,05
127,08
Pe cen ile 75 (Uppe qua ile)
0,13
0,17
74,89
77,93
Pe cen ile 25 (Lowe qua ile)
-0,38
-0,24
27,09
28,89
In e qua ile ange
0,51
0,41
47,79
49,04
Median
-0,12
0,03
45,82
59,89
Mean
-0,12
-0,04
56,28
57,62
Va iance
0,11
0,10
1530,65
1010,65
S anda d de ia ion
0,33
0,32
39,12
31,79
E en hough he p esence o ex eme alues has been educed, some his og ams
emain bimodal (see Figu e 7). Wi h he help o salini y maps (like he one in Figu e 9),
some egions could be iden i ied, whe e he e was s ill anomalous da a. These
co esponded o pe iphe al egions, speci ically he Medi e anean Sea, he No h and
Bal ic Seas and he Hudson Bay. Figu e 9 shows one o he con ou maps used o
es ablish he pe iphe al egions, pa icula ly in his case he Bal ic Sea and Hudson Bay
due o hei low salini y.
Figu e 9. Con ou map o salini y (psu) a laye 1 (3m) wi h he pe iphe al egions ma ked.
Longi ude (ºE)
La i ude (ºN)
-100 -80 -60 -40 -20 0 20
-10
0
10
20
30
40
50
60
5
10
15
20
25
30
35
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Figu e 10 shows a T-S diag am o he whole egion (including pe iphe al
egions). Some cha ac e is ic wa e masses s and ou om he es o he sys em. The
Medi e anean wa e (12-20ºC S>35), he Bal ic and No h seas’ wa e (T<10, wide
ange o S), and he Hudson Bay wa e (T<3ºC, 15<S<33), all appea concen a ed
a ound localized egions o he TS plo . Mo eo e , he ci cles ci cled in pu ple
co espond o he Chesapeake Bay (15<T<18ºC, 25<S<35) ha also s ands ou om he
es o he sys em.
Figu e 10. T-S diag am o all he simula ion da a.
The T-S diag am also allows o iden i y wa e om pa icula egions: Eas coas
o Canada ( om he Gul o Sain Law ence up o Unga a Bay), Mississippi-Alabama
Shel and Ma shland Island, Gul o Pa ia (T inidad Island), Amazon Ri e del a, Gul
o Guinea, pa icula ly a he Bigh o Bonny and Chesapeake Bay al eady iden i ied as
a egion o anomalous salini y (Figu e 11).
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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Figu e 11. Loca ion o Sm alues lowe han 30 psu.
3.2. The 2012 No h A lan ic nume ical simula ion
A e he i s se o p elimina y expe imen s was done, all he iles
co esponding o he yea 2012 we e p ocessed. This co esponds o a o al o 73 iles
in Ne CDF o ma . The p ocedu e ca ied ou was he same as he p e ious one wi h
he excep ion ha he s a is ical pa ame e s we e calcula ed bo h in space (2D) and ime
(1D). Fo example, a each g id-poin , he maximum alue eached du ing he whole
2012 can be es ima ed, and a (2D) map o he maximum alues can be cons uc ed.
Al e na i ely, he maximum alue a each snapsho can be calcula ed o e he egion o
in e es , and a (1D) ime se ies o egional maximum alues can be c ea ed.
Figu e 12 displays he map showing he maximum 2012 alues o he sea le el
and he mixed laye dep h. Al hough he g aph does no include in o ma ion abou he
ime o occu ence o he maximum alue, i clea ly is a ool o delinea e he di e en
dynamical p o inces o he No h A lan ic Ocean. I also helps o illus a e he main
mechanisms a wo k in he egion as he 1 m sea le el nea he wes e n bounda y due o
he wind d i en oceanic ci cula ion (combined wi h he mal e ec s), as well as he
ela i e size o he esul ing me idional and zonal g adien s a he egion. On he o he
hand, he la ges alues o he mixed laye dep h a e ound a he wes e n A lan ic
Ocean, nea he pa h o he Gul S eam. The lowes alues a e concen a ed a he
100oW 80oW 60oW 40oW 20oW 0o 20oE 40oE
0o
15oN
30oN
45oN
60oN
Lau a Gómez Na a o
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no h-eas e n coas o Canada and a he Gul o Guinea, wo egions s ongly
in luenced by he p esence o su ace esh wa e luxes.
Figu e 12. Maximum alues’ dis ibu ion in me es o SSH (le ) and MLD ( igh ).
Figu e 13. SSH’s s a is ical pa ame e s a ia ion wi h ime.
Longi ude (ºE)
La i ude (ºN)
-100 -80 -60 -40 -20 0 20
-10
0
10
20
30
40
50
60
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Longi ude (ºE)
La i ude (ºN)
-100 -80 -60 -40 -20 0 20
-10
0
10
20
30
40
50
60
20 40 60 80 100 120 140 160 180 200 220 240
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
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Figu e 14. MLD’s s a is ical pa ame e s a ia ion wi h ime.
An example o ime se ies o he s a is ical p ope ies o he sys em is gi en in
Figu e 13, whe e he ime e olu ion o he minimal, cen al, and la ges sea le el alues
is shown. The mean alue emains nea o ze o as he model uses a olume conse a ion
cons ain in he pa ame e iza ion o he open bounda ies. The depa u es o ze o a e due
o he non-accoun ed con ibu ion o he pe iphe al seas. The ime e olu ion o he
obus maximum ( he 99 pe cen ile) eaches i s la ges alue du ing he mon h o
Augus , indica ing he seasonal wa ming o he uppe Ocean. Con a ily, he ime
e olu ion o he absolu e maximum has wo maximum alues (a he beginning o he
yea and owa ds he end o he mon h o Oc obe ). The mo e easonable e olu ion o
he obus maximum illus a es he need o emo ing ex eme alues be o e he
s a is ical explo a ion o he da a.
Figu e 14 helps o isualize he s ong seasonal a ia ion o he MLD. As i
would be expec ed, he maximum alues occu du ing win e due o he combined e ec
o low s a i ica ion and s ong wind s i ing, and he lowes alues occu du ing he
summe season due o he s abilizing e ec o he su ace seasonal wa ming.
Figu e 15 shows he co esponding ime se ies o he salini y a laye 15
(cen ed on 228 m dep h). The mos s iking ea u e is he anomalous beha iou o he
minimum salini y owa ds he end o he yea . Fu he analysis ha e shown ha such

Lau a Gómez Na a o
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sudden educ ion o salini y happens in some isola ed g id-poin s, and ha a e no
linked o any physical p ocess. The analysis was epea ed o he yea 2011, and i was
ound ha some g id-poin s, loca ed a di e en dep hs, had simila e o s. All poin s
a ec ed by his beha iou a e loca ed a he ba hyme y, indica ing a p oblem in he
pa ame e iza ion o he ace s in he bo om mixed laye .
Figu e 15. Salini y 1D plo a laye 15 (228m) o yea 2012.
The spa ial dis ibu ion o he minimum salini y du ing he yea 2012 in he Gul
o Sain Law ence a e shown in Figu es 16 (32 m dep h) and 17 (191 m dep h). The
model alues a e compa ed wi h he WOA09 salini y da a. The model su ace salini y
can be much lowe han he clima ological salini y (mo e han 10 psu eshe ). A dep h,
he di e ences a e o abou 1 psu. Such a 10 psu su ace salini y di e ences be ween
he model and clima ology indica es de iciencies o he model in ob aining ealis ic
esh-wa e balances in such egion domina ed by local wa e uno and shallow
ba hyme y.
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
23
Figu e 16. Le : Minimum salini y alues’ dis ibu ion o 2012 laye 5 (32m). Righ : WOA09 Salini y
alues o Decembe 2012 laye 4 (30m).
Figu e 17. Le : Minimum salini y alues’ dis ibu ion o 2012 laye 14 (191m). The pink poin is whe e
he lowes alue is loca ed. Righ : WOA09 Salini y alues o Decembe 2012 laye 10 (200m).
3.3. The 2012 No h A lan ic A go da a
The A go da a co esponding in ime and space o he simula ion was ob ained
om he Ba celona Expe Cen e (BEC) da abase which is a mi o o he Co iolis
da abase. Only he p o iles which we e ound in he A lan ic Ocean, be ween 60ºN and
5ºS and o he yea 2012, we e e ie ed om he da abase.
Only Delayed mode da a and p o iles ou side he pe iphe al egions ha e been
conside ed in his pa o he wo k. The hyd og aphic p o iles whe e hen p ocessed
ollowing hese s eps:
Longi ude (ºE)
La i ude (ºN)
-65 -60 -55 -50 -45 -40
42
44
46
48
50
52
54
56
58
60
18
20
22
24
26
28
30
32
34
36
Longi ude (ºE)
La i ude (ºN)
-65 -60 -55 -50 -45 -40
42
44
46
48
50
52
54
56
58
60
18
20
22
24
26
28
30
32
34
36
Longi ude (ºE)
La i ude (ºN)
-65 -60 -55 -50 -45 -40
42
44
46
48
50
52
54
56
58
60
33
33.5
34
34.5
35
35.5
36
Longi ude (ºE)
La i ude (ºN)
-65 -60 -55 -50 -45 -40
42
44
46
48
50
52
54
56
58
60
33
33.5
34
34.5
35
35.5
36
Lau a Gómez Na a o
24
a. Adjus ed p essu e, empe a u e and salini y a iables we e ead.
b. I he numbe o measu emen le els is less han 20, he whole p o ile is
dismissed
c. I Z_QC a a poin was no equal o 1, he co esponding p essu e, empe a u e
and salini y alues we e se o 99999 (missing alue).
d. I T_QC o a poin was no equal o 1, he co esponding empe a u e and
salini y alues we e se o missing.
e. I S_QC o a poin was no equal o 1, he co esponding salini y alue was se
o missing. This is done as he salini y alues depend on bo h empe a u e and
p essu e, while empe a u e alue depends only on p essu e.
. Fo each e ained A go p o ile, a p o ile is c ea ed in such a way ha only alid
salini y da a a e s o ed.
g. The numbe o da a in each p o ile o hese new a iables was coun ed. I he e
we e no a leas 20 alues o i mo e han 40% o he da a in he o iginal p o ile
was disca ded, he whole p o ile was disca ded. Also, o his da a il e , all he
p o iles which hei i s alid salini y measu e was a dep h lowe han 100m
we e disca ded oo.
h. As he empe a u e measu ed by he buoys is in si u he po en ial empe a u e
was calcula ed. See Appendix 1 o see he p og amme used o i s calcula ion.
i. The po en ial empe a u e alues we e e ically in e pola ed.
j. The da a was in e pola ed using h ee in e pola ion me hods:
 Akima splines
 Cubic splines
 Thi d-o de polynomial i ing
k. When he A go da a was in e pola ed o he simula ion’s e ical le els, any
le el o he simula ion deepe han he las dep h o he p o ile and shallowe
han he i s alid dep h o he p o ile was changed o missing.
l. A e he in e pola ion he mean was calcula ed.
m. I he absolu e di e ence be ween each indi idual in e pola ion and he mean
was g ea e han 5% o he mean, he co esponding po en ial empe a u e alue
was changed o missing.
n. S eps g) o k) we e epea ed o salini y.
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
25
The inal esul s o po en ial empe a u e and salini y o he A go da a he ea e
e e ed o as Ta and Sa espec i ely, we e sa ed in a Ne CDF ile. In his ile,
in o ma ion abou each p o ile was sa ed oo: i s longi ude, la i ude, da e and he ID o
he p o ile (Did) in he BEC da abase. In addi ion, he addi ional pa ame e s we e
calcula ed and included in his ile. The indices ipos and jpos a e he alues o he x and
y dimension espec i ely, o he poin in he model’s g id which is closes o he A go
p o ile’s posi ion. The pos a iable poin s o he model snapsho ha co esponds o
he A go measu emen .
In addi ion, he numbe o alid Ta and Sa alues o 3 di e en su ace dep hs;
10 m, 25 m and 50 m, and in di e en poin s o he da a p ocessing (a e ha ing
applied di e en il e s). This was done o know how much da a was eally a ailable
and alid, as pa icula ly in he su ace, he e a e many A go da a p oblems, bu on he
o he hand, i ‘s he su ace da a which is o g ea e in e es o alida e, o i s use o
SMOS SSS.
F om a o al o 4121 A go p o iles, only 3117 p o iles ha e passed he Quali y
Con ol desc ibed abo e (Figu e 18). Some egions, as he No h A lan ic sub opical
gy e, show a clea lack o A go da a, due o he ee ad ec i e na u e o he buoys.
Figu e 18. A go p o iles o 2012. The alid p o iles a e ma ked in blue and he in alid in ed.
The spa ial dis ibu ion o he Ta (Figu e 19) and Sa (Figu e 20) is shown o
laye s 1 (3 m) and 2 (9 m). No ice he lack o A go da a o he i s laye o he model.
100oW 80oW 60oW 40oW 20oW 0o 20oE 40oE
0o
15oN
30oN
45oN
60oN
Lau a Gómez Na a o
32
These 4 egions a e he ones shown in Figu e 2 (Chen, 2009). The da es o he 12
andomly selec ed p o iles a e shown in Table 5.
Table 5. Da es o he 12 p o iles selec ed o he T-S diag ams.
Buoy numbe
Day
Mon h
Yea
1
12
10
2012
3
6
1
2012
2
11
10
2012
4
6
2
2012
5
1
2
2012
6
3
7
2012
7
23
11
2012
8
25
3
2012
9
4
4
2012
10
23
7
2012
11
20
2
2012
12
8
10
2012
Figu e 27. Posi ions o he selec ed buoys o he T-S diag ams. The blue ci cles indica e he egion.
The T-S diag ams in Figu e 28 show ha , o egion 4 (bo om), bo h T-S
diag ams a e simila . Simila ly, in egion 2 (second ow) he model and A go ag ee
wi h he excep ion o he su ace laye s. On he o he , o egions 1 ( op) and 3 ( hi d
ow) he T-S diag ams o some o he p o iles a e qui e di e en , especially nea he
su ace. Again, hese T-S diag ams p o ide u he e idence ha he nume ical
simula ion is less accu a e a high la i udes, as he ones wi h he g ea es di e ences a e
hose o egions 1 and 3, which a e he egions wi h buoys selec ed a highe la i udes.
100oW 80oW 60oW 40oW 20oW 0o 20oE 40oE
0o
15oN
30oN
45oN
60oN
1
2
3
4
5
6
7
8
9
10
11
12
3
4
2
1

Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
33
Figu e 28. T-S diag ams on he le co espond o he A go da a, and on he igh o he simula ion’s
ou pu s. F om bo om o op hey co espond o poin s om egion 4 o 1, and ollowing he colou o de
blue, ed, black, hey co espond o he poin s 1 o 12, as labelled on Figu e 27. The g een, dashed lines
a e he isopycnals.
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
22
23 24
25
26
27
28
29
Salini y (psu)
Tempe a u e (C)
32 34 36 38
0
5
10
15
20
25
30
· .
'
'"
... 1//
Lau a Gómez Na a o
34
4. Conclusions
The analysis o he ou pu s o he nume ical simula ion has indica ed ha no h
and sou h egions, close o he open bounda ies o he model, oge he wi h he
pe iphe al egions a e iden i ied as p oblema ic. As desc ibed in Sec ion 2.1, he open
bounda ies o he simula ion a e elaxed owa ds clima ology da a. The e o e, i s
a iabili y is no e lec ed accu a ely. In addi ion, a high la i udes, an impo an
p ocess is no ep esen ed a all by he simula ion: he in e annual a ia ion o he cold,
eshwa e inpu om he A c ic Ocean.
In he pe iphe al egions, he cha ac e is ics o he wa e masses a e e y
di e en han he cha ac e is ics o he pelagic Ocean ha is he main objec i e o he
simula ion. The cha ac e is ics o he pe iphe al egions a ise om he complex
in e ac ion be ween la ge i e uno , semi-closed domain, shallow ba hyme y and
a mosphe ic o cing. E o s in any o hese ac o s ( o example, he i e uno ) may
s ongly a ec he ealism o he simula ion. As he e a e e y ew, o none, A go
p o ile sampling hese egions, hey a e elimina ed om ou analysis.
The eas e n Canadian coas could ha e also been conside ed a pe iphe al egion
as i clea ly ollows a di e en da a dis ibu ion, han he es o he sys em. The
simula ion a he Gul o Sain Law ence has been ound excessi ely esh by espec o
he clima ology da a and by espec o published da a om he egion (Pe ie e al.,
2006). The minimum salini y obse ed in he A las a a dep h o 30 m du ing he win e
is o 29 psu (Figu e 29), whils he co esponding alue o he simula ion is 18 psu.
This is a e y impo an di e ence in salini y alues, which could no be de ec ed wi h
he A go da a as his is one o he egions which a e poo ly ep esen ed by he A go
ne wo k (Figu e 17).
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
35
Figu e 29. Salini y alues a he Gul o Sain Law ence, o he 15 h o No embe a 30m (Pe ie e al.,
2010).
The egion o he Gul o Sain Law ence is a pa icula case, as eshwa e
seems o accumula e he e. This could be due o he combined e ec o he i e uno ,
he p esence o ice du ing ce ain imes o he yea and due o ba hyme y e o s.
As s a ed in Buongio no (2012), empe a u e and salini y alues, especially he
su ace ones, a e sensible o ex e nal o ces, like eshwa e inpu s om i e s o
p ecipi a ion, and o he Ocean’s in e nal dynamics, like con ec ion, ad ec ion and
mixing p ocesses. Consequen ly, i hese in lows o eshwa e and in e nal p ocesses
a e no accu a ely ep esen ed by he model, i s e ec can be clea ly obse ed in he
su ace laye s. As is he case o igu e 10 whe e low alues o salini y can be de ec ed
a he mou h o g ea i e s, as he model a ibu es a salini y o 0 psu o hem, which is
qui e an unde es ima e.
The anomalous esul s ob ained om he simula ion, as al eady discussed abo e,
could ha e a ious easons and explana ions. In Tho pe e al., 2005 i is sugges ed ha
possible sou ces o e o a e low e ical esolu ion, poo ep esen a ion o he
ba hyme y, and issues wi h he e ical mixing schemes (as desc ibed by Buongio no,
2012), which lead o an un ealis ic oceanic ci cula ion, which can cause ce ain wa e
masses o no be de ec ed in simula ions. The impo ance o he ba hyme y used in
simula ions, is also highligh ed in Min ielle e al. (2011) and Gi on and San o d
(2003). They s a e ha i s impo ance is due o ha i a ec s bo om laye p ocesses
which con ol he deep oceanic ci cula ion.
Lau a Gómez Na a o
36
The T-S diag ams (Figu e 28) and he di e ence be ween he model and A go
da a (Figu es 22, 23 and 24), show how he empe a u e o he model has a posi i e bias
and he salini y has a nega i e one. The nega i e salini y endency is much mo e
ma ked han he posi i e one seen o he empe a u e alues. This means ha he
model is oo ‘ esh’, and could be due o all he easons exposed abo e. They also show
ha he simula ion is mo e inaccu a e nea he su ace, bu also i should be kep in
mind ha he e is whe e he oceanic pa ame e s show a g ea e a iabili y.
The shallowes dep h a which he simula ion could be alida ed was 9 m, due o
he lack o alid A go da a nea he su ace, and also he e we e egions like he No h
A lan ic Sub opical gy e and he Gul o S . Law ence be ween o he s we e he e was a
lack o da a oo. In he la e egion, ano he sou ce o da a had o be used o alida e
he ou pu s o he simula ion he e. This lack o alid A go da a a he su ace was
expec ed, as in many occasions when he buoys a e nea he su ace he hyd aulic pump
is s opped and so he salini y senso s ops measu ing.
Mo e A go da a a he su ace da a could ha e been ob ained i no only he
p o iles lagged wi h a QC = 1 ha e been conside ed, bu also he ones lagged wi h a
QC o 2, 5 o 8 (Table 2). Due o he limi ed amoun o ime o his s udy, only he
da a wi h QC=1 we e selec ed. Howe e , i has been no iced ha he A go obse ing
sys em is able o ep esen seasonal a iabili y o he Ocean (Figu e 21). The e is a
conside able amoun o alid p o iles (3117) o yea 2012, wi h he excep ion o some
poin s and buoys he da a hey measu e is e y easonable (Figu e 20) and he e a e no
empo al biases as hey p o ide da a h oughou he yea . This demons a es ha
choosing A go o alida e he simula ion was a co ec choice.
To conclude, in his wo k I ha e con ibu ed o he i s e o s o he
Depa men o Physical and Technological Oceanog aphy o he Ins i u de Ciències del
Ma (CSIC) o he alida ion o hei No h A lan ic Ocean simula ion. This simula ion
couples wo o he s a e-o -a oceanic models: NEMO (oceanic dynamics) and LIM-2
(sea-ice dynamics). By compa ing he ou pu s o he model agains he A go
hyd og aphic p o iles, I ha e ound ha he pelagic ci cula ion o he model has some
easonable simila i ies wi h he obse a ions. Al hough impo an de ia ions be ween
he model and da a exis nea he me idional bo de s and o he Canadian Coas , ou
esul s indica e ha he pelagic ci cula ion o he model has some easonable simila i ies
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
37
wi h obse a ions. This is impo an because he abili y o e ie ing SSS om emo e
sensing is ex emely low in cold wa e s and nea he coas (due o land sea ab up
emissi i y ansi ion). Tha is, he model p o ides he la ges compa ibili y wi h da a
igh in he egion whe e i would be mo e use ul o p o ide a cohe en map o salini y
in he No h A lan ic Ocean.
5. Re e ences
An ono , J. I., Seido , D., Boye , T. P., Loca nini, R. A., Mishono , A. V., Ga cia, H.
E., Ba ano a, O. K., Zweng, M. M. and Johnson, D. R. (2010): Wo ld Ocean A las
2009 Volume 2: Salini y. S. Le i us Ed. NOAA A las NESDIS 69, U.S. Go .
P in ing O ice, Washing on, D.C., 184 pp.
B odeau, L., Ba nie , B., T eguie , A. M., Pendu , T. and Gule , S. (2010): An
ERA40-based a mosphe ic o cing o global ocean ci cula ion models. Ocean
Modelling 31, 88–104.
B own, E., Colling, A., Pa k, D., Phillips, J. and Ro he y, D. (1988): Ocean
Ci cula ion. The Open Uni e si y, 287 pp.
Buongio no Na delli, B. (2012): A No el App oach o he High-Resolu ion
In e pola ion o In Si u Sea Su ace Salini y. J. A mos. Oceanic Technol., 29, 867–
879.
Ca al, T., Keeley, B., Taka suki, Y., Yoshida, T., Loch, S., Schmid, C., Goldsmi h, R.,
Wong, A., McC eadie, R., Th eshe , A. and T an, A. (2012): ARGO USER’S
MANUALVe sion 2.4, 85 pp.
Cha e jee, A., Shanka 1, D., Shenoi, S. S. C., Reddy, G. V., Michael, G. S.,
Ra ichand an, M., Gopalk ishna, V. V., Rama Rao, E. P., Udaya Bhaska , T. V. S.,
and Sanjee an, V. N. (2012): A new a las o empe a u e and salini y o he No h
Indian Ocean. J. Ea h Sys . Sci.121, No. 3, 559–593.
Chen, C. (2009): Ocean Wa e Masses-T-S Diag ams and Uppe Ocean Wa e s.
Gene al Physical Oceanog aphy, MAR 555, School o Ma ine Sciences and
Technology, Umass-Da mou h, 18 pp.
Eme y, W. J., and Thomson, R. E. (2004): Da a analysis me hods in physical
oceanog aphy 2nd edi ion. Else ie Ed., 654 pp.
Gi on, J.B. and San o d, T. B. (2003): Descen and modi ica ion o he o e low plume
in he Denma k S ai . J. Phys. Oceanog ., 33, 1351–1364.
Loca nini, R. A., Mishono , A. V., An ono , J.I., Boye , T. P., Ga cia, H. E., Ba ano a,
O. K., Zweng, M. M. and Johnson, D. R. (2010): Wo ld Ocean A las 2009, Volume
1: Tempe a u e. S. Le i us, Ed.,NOAA A las NESDIS 68, U.S. Go e nmen P in ing
O ice, Washing on, D.C., 184 pp.

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38
Higgison, S., Thompson, K. R. and Liu, Y. (2009): Es ima ing ocean clima ologies o
sho pe iods: A simple echnique o emo ing he e ec o eddies om empe a u e
and salini y p o iles. GEOPHYSICAL RESEARCH LETTERS, 36, 4 pp.
Hoa eau, N., Po abella, M., Ga cia-Ladona, E., Tu iel, A. and Ballab e a-Poy, J.
Me idional a iabili y o No h-A lan ic Sea Su ace Salini y: Wa enumbe spec a
de i ed om clima ology, ocean model and sa elli e obse a ions. Submi ed o J.
Geophys. Res.
Me O ice, Uni ed Kingdom:
h p://www.me o ice.go .uk/wea he / ma ine/obse a ions/ga he ing_da a/a go.h ml
Min ielle, M., Cassou, C., Bou dallé-Badie, R., Te ay, L. and Najac, J. (2011): A
s a is ical–dynamical scheme o econs uc ing ocean o cing in he A lan ic. Pa II:
me hodology, alida ion and applica ion o high- esolu ion ocean models. J. Clima e
Dynamics, 36, 401-417.
Na ional Wea he Se ice, NOAA: www.nws.noaa.go /clima e/help/glossa y.php
Pe ie, B., D inkwa e , K., Sands öm, A., Pe ipas, R., G ego y, D., Gilbe , D. and
Sekhon, P. (1996): Tempe a u e, salini y and sigma- A las o he Gul o S .
Law ence. Canadian Technical Repo o Hyd og aphy and Ocean Sciences 178,
263 pp.
Tho pe, S. E., S e ens, D. P. and Heywood, K. J. (2005): Compa ison o wo ime-
a ian o ced eddy-pe mi ing global ocean ci cula ion models wi h hyd og aphy o he
Sco ia Sea. Ocean modeling, 9, 105-132.
Uni e si y o Cali o nia, San Diego: h p://www.a go.ucsd.edu/
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
39
6. Appendices
6.1. Appendix 1
SUBROUTINE sw_p mp(n,S,T,P,PR,TP)
IMPLICIT NONE
INTEGER, INTENT(in) :: n
eal(kind=8), dimension(n), in en (in) :: T,P
eal(kind=8), in en (in) :: PR
eal(kind=8), dimension(n), in en (in) :: S
eal(kind=8), dimension(n), in en (ou ) :: TP
! ... Local a iables
eal(kind=8), dimension(n)
:: ADGT,del_P,del_ h,q, h,PP
! ... he a1
del_P = PR - P
CALL sw_ad g(n,S,T,P,ADGT)
del_ h = del_P * ADGT
h = T + 0.5D0 * del_ h
q = del_ h
! ... he a2
PP = P+0.5D0*del_P
CALL sw_ad g(n,S, h,PP,ADGT)
del_ h = del_P * ADGT
h = h + (1.0D0 -1.0D0/sq (2.0D0))*(del_ h -q)
q = (2.0D0-sq (2.0D0)) * del_ h +
(-2.0D0+3.0D0/sq (2.0D0))* q
! ... he a3
PP = P+0.5D0*del_P
CALL sw_ad g(n,S, h,PP,ADGT)
del_ h = del_P * ADGT
h = h + (1.0D0 +1.0D0/sq (2.0D0))*(del_ h -q)
q = (2.0D0+sq (2.0D0)) * del_ h
+ (-2.0D0-3.0D0/sq (2.0D0))* q
! ... he a4
PP = P+del_P
CALL sw_ad g(n,S, h,PP,ADGT)
del_ h = del_P * ADGT
TP = h + (del_ h -2.0D0*q)/6.0D0
END SUBROUTINE sw_p mp
!%%%%%%%%%%%%%%%%%%
! Adiaba ic empe a u e g adien
SUBROUTINE sw_ad g(n,S,T,P,ADGT)
IMPLICIT NONE
in ege , in en (in) :: n
eal(kind=8), dimension(n), in en (in) :: S,T,P
eal(kind=8), dimension(n), in en (ou ) :: ADGT
! ... Pa ame e s:
eal(kind=8), pa ame e :: a0 = 3.5803D-5
eal(kind=8), pa ame e :: a1 = 8.5258D-6
eal(kind=8), pa ame e :: a2 = -6.836D-8
eal(kind=8), pa ame e :: a3 = 6.6228D-10
eal(kind=8), pa ame e :: b0 = 1.8932D-6
eal(kind=8), pa ame e :: b1 = -4.2393D-8
eal(kind=8), pa ame e :: c0 = 1.8741D-8
eal(kind=8), pa ame e :: c1 = -6.7795D-10
eal(kind=8), pa ame e :: c2 = 8.733D-12
eal(kind=8), pa ame e :: c3 = -5.4481D-14
eal(kind=8), pa ame e :: d0 = -1.1351D-10
eal(kind=8), pa ame e :: d1 = 2.7759D-12
eal(kind=8), pa ame e :: e0 = -4.6206D-13
eal(kind=8), pa ame e :: e1 = 1.8676D-14
Lau a Gómez Na a o
40
eal(kind=8), pa ame e :: e2 = -2.1687D-16
ADGT = a0 + (a1 + (a2 + a3 * T)*T)*T + (b0 + b1
*T)
* (S -35.0D0) + ((c0 + (c1 + (c2 + c3 *T) *T)*T) +
(d0 + d1 *T)*(S-35.0D0))*P + ( e0 + (e1 + e2 *T)*T)
*P *P
END SUBROUTINE
6.2. Appendix 2
SUBROUTINE pe cen iles (n,x,np,p,xp)
IMPLICIT NONE
in ege , in en (in) :: n,np
eal(kind=8), dimension(np), in en (in) :: p
eal(kind=8), dimension(n), in en (in) :: x
eal(kind=8), dimension(np), in en (ou ) :: xp
in ege i,kk
eal(kind=8) ,dd
in ege indx(n)
CALL indexx(n,x,indx)
DO i=1,np
= p(i)*(n-1.0D0)/100.0D0 + 1.0D0
kk = FLOOR( )
dd = - kk
IF (kk.EQ.0) THEN
xp(i) = x(indx(1))
ELSE IF (kk.EQ.n) THEN
xp(i) = x(indx(n))
ELSE
xp(i) = x(indx(kk)) + dd*(x(indx(kk+1))-x(indx(kk)))
ENDIF
ENDDO
RETURN
END
Valida ion o a nume ical simula ion o he No h A lan ic Ocean wi h A go da a.
6.3. Appendix 3
SUBROUTINE a e a (da a,n,a e, a )
IMPLICIT NONE
in ege , in en (in) :: n
eal(kind=8), dimension(n), in en (in) :: da a
eal(kind=8), in en (ou ) :: a e, a
! ... Local a iables:
in ege j
eal(kind=8) s,ep
a e = SUM(da a(1:n))/n
a = 0.0d0
ep = 0.0d0
do j=1,n
s = da a(j) - a e
ep = ep + s
a = a + s*s
ENDDO
a = ( a -ep**2/n)/(n-1)
RETURN
END
! (C) Cop . 1986-92 Nume ical Recipes So wa e
*5sV1.
! ...
! =====================================
! ...
SUBROUTINE a e a 4 (da a,n,a e, a )
IMPLICIT NONE
in ege , in en (in) :: n
eal(kind=4), dimension(n), in en (in) :: da a
eal(kind=4), in en (ou ) :: a e, a
! ... Local a iables:
in ege j
eal(kind=8) s,ep, 8
a e = SUM(da a(1:n))/n
8 = 0.0d0
ep = 0.0d0
do j=1,n
s = DBLE(da a(j) - a e)
ep = ep + s
8 = 8 + s*s
ENDDO
8 = ( 8-ep**2/n)/(n-1)
a = SNGL( 8)
RETURN
END
! (C) Cop . 1986-92 Nume ical Recipes So wa e
*5sV1.