On 23/02/2018 12:27, Martin Brown wrote:
On 21/02/2018 21:39, N_Cook wrote:
Updated corrected set of Jason-3 curve-fit results including the
latest data from 17 December 2017, previous data outputed up to 2017.911,
associated plot

http://diverse.4mg.com/jason1+2+3r.jpg

Linear
Y= cm of sea-level as per Aviso output and x=0 for year 2000
Y = 1.446098 + 0.331877*x
R^2= 0.978086
RMS Error = 0.244821
projecting into the future
year 2030, 11.402 cm SL rise
2050, 18.04 cm
2100, 34.63cm
Update for extra 6 weeks of data, to 17 Dec 2017
Y = 1.414689 + 0.335684*x
R^2= 0.976966
RMS Error = 0.254395
gradient gives the linear MSL rise of 3.357 mm / year
projecting into the future
year 2030, 11.485cm SL rise
2050, 18.199 cm
2100, 34.983cm

Exponential
Y = 1.948854 -6.880730*(1-Exp(0.033013*x))
R^2 = 0.981571
RMS Error = 0.227110
projections
2030, 13.593 cm
2050, 30.919 cm
2100, 1.819 metres
update
Y = 2.002894 -5.56543*(1-Exp(0.038595*x))
R^2 = 0.981615
RMS Error = 0.229845
projections
2030 , 14.153cm
2050 , 34.771cm
2100, 2.605 metres

Quadratic
Y = 2.023609 + 0.204265*x + 0.005656*x^2
R^2 = 0.981740
RMS Error = 0.226064
projections
2030, 13.242cm
2050, 26.377cm
2100, 79.010cm
Update
Y = 2.088926 + 0.187200*x + 0.006555*x^2
R^2 = 0.981759
RMS Error = 0.228941
projections
2030, 13.604cm
2050, 27.836cm
2100, 86.359cm


It is worth noting on physical grounds that since the coefficient of
expansion of water is not a constant but varies almost linearly with
temperature you would expect there to be some second order polynomial
like behaviour in the ocean expansion and sea level rise.

Temperature Density (0-100°C at 1 atm, 100 °C at saturation pressure)
Specific weight Thermal expansion coefficient of liquid water

[°C] [g/cm3] [*10- 4 K-1]
0.1 0.9998495 -0.68
1 0.9999017 -0.50
4 0.9999749 0.003
10 0.9997000 0.88
15 0.9991026 1.51
20 0.9982067 2.07
25 0.9970470 2.57
30 0.9956488 3.03
35 0.9940326 3.45
40 0.9922152 3.84
45 0.99021 4.20
50 0.98804 4.54
55 0.98569 4.86
60 0.98320 5.16

Taken from https:
//www.engineeringtoolbox.com/water-density-specific-weight-d_595.html

I get the best fit to its properties as a cubic (almost exact)

-0.671+0.17114*T-0.00192*T^2+0.0001*T^3



I decided to limit to 20 deg C tops, using your figures and got very similar
-0.691901 +0.186743*T -0.003414*T^2 +0.000049*T^3
(R^2= 0.999978 RMS Err= 0.008323)

So I'll try a cubic curve-fit on the J1+2+3 data

Best fit of cubic form to the Jasons returned to Y=0 (x=3 to 18) at
about x= 50 years, and offsetting (delaying) the x-scale, would not
converge .

Curvefits for latest Jason3 data update of 06 Jan 2018, outputed 26 Feb
2018 or a day or 2 before.

Linear
Y= 1.41235 + 0.335950 * x
R^2 = 0.977652
RMS Error = 0.250962

Same ranking of curve-fit , by increasing R^2 and reducing RMS
Exponential
Y = 1.999281 -5.648695*(1-Exp(0.038178*x))
R^2 = 0.982207
RMS Error = 0.226405
Projections
2030, 14.107cm
2050, 34.456 (previous , Nov 2017 update 30.919 cm)
2100, 2.534m

Quadratic
Y = 2.084548 + 0.188390*x +0.006489 * x^2
R^2 = 0.982348
RMS Error = 0.225504
Projections
2030, 13.576cm
2050, 27.727cm (previous 26.377cm)
2100, 85.814cm

Indicial
Y = 2.314316 + 0.090372 * x^1.405678
R^2 = 0.982429
RMS Error = 0.224986
Projections
2030, 13.089cm
2050 , 24.406 (previous 23.13cm)
2100, 60.846

On 27/02/2018 19:13, N_Cook wrote:
Best fit of cubic form to the Jasons returned to Y=0 (x=3 to 18)Â* at
about x= 50 years, and offsetting (delaying) the x-scale, would not
convergeÂ* .

Whose curve fitting algorithm are you using? There is no excuse for a
cubic polynomial fit to diverge on a decent amount of data.
(although Excel may well still do and possibly Matlab as well)

If you send me the raw data as a CSV file with time, value I would be
interested in fitting it just to see what the fit looked like.
(I am certain that I have a solution that will converge)

I have posted about such problems on the Excel groups in the past.

If you want to DIY it then rescale your data so that time is symmetrical
around the mid point and with a range of -1 to 1 - this makes the matrix
condition as well behaved as possible so even a bad algorithm can cope.
(it won't be exact but it might be much closer to a real answer)

Ie given a dataset running from MinTime to MaxTime then for each t

MyTime = (2t-(Mintime+Maxtime))/(MaxTime-MinTime)

t = Mintime Mytime = -1
t = Maxtime Mytime = 1
(subject to typos)

Unless their fitting code is hopeless this should be stable.

--
Regards,
Martin Brown

On 27/02/2018 19:53, Martin Brown wrote:
On 27/02/2018 19:13, N_Cook wrote:
Best fit of cubic form to the Jasons returned to Y=0 (x=3 to 18) at
about x= 50 years, and offsetting (delaying) the x-scale, would not
converge .

Whose curve fitting algorithm are you using? There is no excuse for a
cubic polynomial fit to diverge on a decent amount of data.
(although Excel may well still do and possibly Matlab as well)

If you send me the raw data as a CSV file with time, value I would be
interested in fitting it just to see what the fit looked like.
(I am certain that I have a solution that will converge)

I have posted about such problems on the Excel groups in the past.

If you want to DIY it then rescale your data so that time is symmetrical
around the mid point and with a range of -1 to 1 - this makes the matrix
condition as well behaved as possible so even a bad algorithm can cope.
(it won't be exact but it might be much closer to a real answer)

Ie given a dataset running from MinTime to MaxTime then for each t

MyTime = (2t-(Mintime+Maxtime))/(MaxTime-MinTime)

t = Mintime Mytime = -1
t = Maxtime Mytime = 1
(subject to typos)

Unless their fitting code is hopeless this should be stable.


Y = 9.785195 + 0.250222*(x-25) -0.013375* (x-25)^2 -0.000468*(x-25)^3
converged with
R^2 = 0.982687
RMS Error = 0.225851
so larger R^2 than best Fractional power fit but also larger RMS Error
and also the awkward y=9.785195 intercept

On 27/02/2018 20:46, N_Cook wrote:
On 27/02/2018 19:53, Martin Brown wrote:
[snip]
Ie given a dataset running from MinTime to MaxTime then for each t

MyTime =Â* (2t-(Mintime+Maxtime))/(MaxTime-MinTime)

t = MintimeÂ*Â* Mytime = -1
t = MaxtimeÂ*Â* Mytime = 1
(subject to typos)

Unless their fitting code is hopeless this should be stable.


Y = 9.785195 + 0.250222*(x-25) -0.013375* (x-25)^2 -0.000468*(x-25)^3
converged with
R^2 =Â* 0.982687
RMS Error =Â* 0.225851
so larger R^2 than best Fractional power fit but also larger RMS Error
and also the awkward y=9.785195 intercept


That is because you have moved the origin. You can compute the
equivalent ordinary polynomial by expanding the terms again.

1 x x^2 x^3
a a
+b(x-25) -25b b
+c(x-25)^2 625c -50c c
+d(x-25)^3 -15625d 1875d -75d d

And sum up the terms (subject to typos).

I suggest you also try it with fitting to [(x-25)/25]^N

The coefficients will then be 25^n bigger for each polynomial but the
numerical solution will be significantly more stable.

Iff your data are uniformly spaced you could also get the right answer
by taking their dot product with the Tchebyshev polynomials. That is
another way to make the problem even better conditioned |x| 1.

They are of the form
t0(x) = 1
t1(x) = x
t2(x) = 2x^2-1
t3(x) = 4x^3-3x
(subject to typos)
tn+1(x) = 2xTn(x)-Tn-1(x)

A naughty way to compute them is cos(n arcos(x)) which is obviously
capable of going badly wrong if |x|1


--
Regards,
Martin Brown

On 27/02/2018 20:46, N_Cook wrote:
On 27/02/2018 19:53, Martin Brown wrote:
On 27/02/2018 19:13, N_Cook wrote:
Best fit of cubic form to the Jasons returned to Y=0 (x=3 to 18) at
about x= 50 years, and offsetting (delaying) the x-scale, would not
converge .

Whose curve fitting algorithm are you using? There is no excuse for a
cubic polynomial fit to diverge on a decent amount of data.
(although Excel may well still do and possibly Matlab as well)

If you send me the raw data as a CSV file with time, value I would be
interested in fitting it just to see what the fit looked like.
(I am certain that I have a solution that will converge)

I have posted about such problems on the Excel groups in the past.

If you want to DIY it then rescale your data so that time is symmetrical
around the mid point and with a range of -1 to 1 - this makes the matrix
condition as well behaved as possible so even a bad algorithm can cope.
(it won't be exact but it might be much closer to a real answer)

Ie given a dataset running from MinTime to MaxTime then for each t

MyTime = (2t-(Mintime+Maxtime))/(MaxTime-MinTime)

t = Mintime Mytime = -1
t = Maxtime Mytime = 1
(subject to typos)

Unless their fitting code is hopeless this should be stable.


Y = 9.785195 + 0.250222*(x-25) -0.013375* (x-25)^2 -0.000468*(x-25)^3
converged with
R^2 = 0.982687
RMS Error = 0.225851
so larger R^2 than best Fractional power fit but also larger RMS Error
and also the awkward y=9.785195 intercept

The postulation is that if you adjust the Jason global sea level curve,
for the mass-loss and gain of Greenland via the GRACE curve, then the
result should look like the ENSO multivariate strength curve.
It is possible to convert Greenland mass-loss to global sea level, via
458 GigaTons to 1.45mm of global sea level rise
The average annnual loss is about 45GT so about 0.14mm of sea level rise
on average, from that.
The grey linear trace trace is the GRACE curve , in annual simplified
terms, wrt to this 45GT average, so above or below avearage about the
centre line, expressed as cm sea-level equivalent. Also no account taken
of likely different lag between melted ice lost to the ocean and mass
loss from the oceans as snow fall on Greenland.
Also the ENSO curve is just a curve, unrelated directly to sea-level and
no reason to treat the red and green sections equally for this purpose.
This is just a first draft, to see if there is any correlation that
could then be improved on.
The blue trace is the Jason1+2+3 curve , above and below the best fit
indicial power curve detailed before, ie the yellow line is flattened
out, to become the common centre-line. The orange curve is Jason curve
with the "Greenland" effect removed.
The red and green curve is the ENSO mulivariate curve , horizontally
scaled to match , but x shifted and vertical scaling adjusted for best
visual curve match to the orange curve. I've not seen any GRACE
mass-loss data for 2017, but ice loss was about half the previous year
and apparently more than average snow fall in the gain part of the year.
if that is so then the modified Jason, the orange curve, extended on
when data emerges, will probably resemble the positive excursion of the
ENSO curve. Not bad first go.
http://diverse.4mg.com/jason1+2+3+grace+enso.jpg
I've not yet added the later Jason data beyond decimal year 2017.91,
may as well wait until the next GRACE update.
Lag of Jason behind ENSO about 3 months and lag of Jason behind GRACE
very approximately 3 months after the midpoint of the mass-loss year
(-45GT point on average), having somewhat arbitrarily chosen the month
of maximum Greenland ice-loss , per year , as it is the more obvious
part of the repeating annual part-curve for the rectilinear plot.

Second draft will differentiate the lag on the rectilinear
curve,relating to ice melt and snow fall in Greenland, decrease the lag
a couple of months for points below the mid-line and increase the lag 4
or even 6 months for points above the mid-line.

On 22/02/2018 10:39, N_Cook wrote:
The second down image shows
http://diverse.4mg.com/jason2+enso_overlay2.jpg
the overshoot end of 2016, artifact of the filter or whatever, no longer
present in that period of the J3 plot.
Also a graphical interpretation of shifting and comparing the ENSO plot.

Jason2 , some spot data as outputted 24 Apr 2017, up to Jan 19, 2017, as
in that above image
2016.5, 7.32cm
2016.75, 7.17
2016.997, 7.61
2017.052, 7.71

Jason3, revisiting the same period
2016.5, 7.13cm, -0.19
2016.75, 6.97 , -0.2
2016.997, 6.95, -0.66
2017.052, 6.86, -0.85
note the 2mm long-term apparent offset betwen J2
and J3

So curvefitting on J3 adjusted downwards for end of 2017, the same same
degree as end of 2016, nullifying the "recent" sharp upswing
Y= 2.211603 + 0.114799*x^1.324879
2030, 12.609cm
2050, 22.669cm
2100, 54.462cm

compared with
as-is without reducing the perhaps overshoot end of 2017
Y = 2.317755 + 0.089566*x^1.408787
2030 , 13.106cm
2050, 24.481cm
2100 , 61.164cm

So best guess projection to 2100 is between 54cm and 61cm global
sea-level rise. So little difference in the fis of the different
curve-types, maybe the next J3 output, the indicial curve will be
surplanted.




For Jason3 data output of 05 Feb 2018, publically available about 10 Apr
2018 fitted to concattenated Jason1+2+3 data back to 2003.
Best fit exponential, R^2 = 0.983876
1.995164 -5.721914*(1-Exp( 0.037837*x))
2020 8.468cm
2050 34.218cm
2100 2.479m
Best fit linear
1.407961 + 0.336498*x , R^2 = 0.97938
year Sea Level
2010 4.772cm
2050 18.232cm
2100 35.057cm
Best fit quadratic, R^2= 0.984018
2.080588 + 0.189329*x + 0.006446*x^2
year Sea Level
2020 8.445cm
2050 27.662cm
2100 85.473cm
Best fit indicial power, best R^2= 0.984105
y= 2.311576 +0.090898*x^1.403844
year Sea Level
2020 8.406cm
2050 24.373cm
2100 60.688 cm


For anyone in or near Hampshire, public talk series i run
Tuesday 17 April 2018, Prof Ivan Haigh, NOC Southampton : Sea level rise
and coastal flooding: past, present and future
further details
http://www.diverse.ip3.co.uk/scicaf.htm

On 12/04/2018 11:06, N_Cook wrote:
On 22/02/2018 10:39, N_Cook wrote:
The second down image shows
http://diverse.4mg.com/jason2+enso_overlay2.jpg
the overshoot end of 2016, artifact of the filter or whatever, no longer
present in that period of the J3 plot.
Also a graphical interpretation of shifting and comparing the ENSO plot.

Jason2 , some spot data as outputted 24 Apr 2017, up to Jan 19, 2017, as
in that above image
2016.5, 7.32cm
2016.75, 7.17
2016.997, 7.61
2017.052, 7.71

Jason3, revisiting the same period
2016.5, 7.13cm, -0.19
2016.75, 6.97 , -0.2
2016.997, 6.95, -0.66
2017.052, 6.86, -0.85
note the 2mm long-term apparent offset betwen J2
and J3

So curvefitting on J3 adjusted downwards for end of 2017, the same same
degree as end of 2016, nullifying the "recent" sharp upswing
Y= 2.211603 + 0.114799*x^1.324879
2030, 12.609cm
2050, 22.669cm
2100, 54.462cm

compared with
as-is without reducing the perhaps overshoot end of 2017
Y = 2.317755 + 0.089566*x^1.408787
2030 , 13.106cm
2050, 24.481cm
2100 , 61.164cm

So best guess projection to 2100 is between 54cm and 61cm global
sea-level rise. So little difference in the fis of the different
curve-types, maybe the next J3 output, the indicial curve will be
surplanted.




For Jason3 data output of 05 Feb 2018, publically available about 10 Apr
2018 fitted to concattenated Jason1+2+3 data back to 2003.
Best fit exponential, R^2 = 0.983876
1.995164 -5.721914*(1-Exp( 0.037837*x))
2020 8.468cm
2050 34.218cm
2100 2.479m
Best fit linear
1.407961 + 0.336498*x , R^2 = 0.97938
year Sea Level
2010 4.772cm
2050 18.232cm
2100 35.057cm
Best fit quadratic, R^2= 0.984018
2.080588 + 0.189329*x + 0.006446*x^2
year Sea Level
2020 8.445cm
2050 27.662cm
2100 85.473cm
Best fit indicial power, best R^2= 0.984105
y= 2.311576 +0.090898*x^1.403844
year Sea Level
2020 8.406cm
2050 24.373cm
2100 60.688 cm


For anyone in or near Hampshire, public talk series i run
Tuesday 17 April 2018, Prof Ivan Haigh, NOC Southampton : Sea level rise
and coastal flooding: past, present and future
further details
http://www.diverse.ip3.co.uk/scicaf.htm


6-weekly update of Jason1+2+3 data from aviso.altimetry.fr data to 16
Mar 2018, publically accessible 13 May 2018. x= year minus 2000, Y= cm
height by Aviso assessment.
Various curve-fit types ranked by R^2 quality of fit, best fit still the
indicial power curve and best estimate so far , of 57cm global mean sea
level rise to year 2100. Officialdom is still showing linear "fits" to
the Jason data, downplaying to about 35cm rise to year 2100
Determinations still falling , but exceedingly unlikely to return to
linear as best fit of curve type. The linear rate here (0.335159
cm/year) does near enough agree with the Aviso reference assessment in
3.32 mm per year considering only subset of 51 datapoints used by me to
cover 2003 to 2018.
Sequence of best-fits of the 4 types, all indicial power curves falling
indices, for the 6-weekly asessments this year, out to 2100 61.2cm,
60.7cm and this latest 57.1cm

linear
Y = 1.419263 + 0.335159*x
R^2= 0.981084
2030 11.474cm
2050 18.177
2100 34.935


exponential
Y = 1.952271 -6.730993*(1-e^(0.033595*x))
R^2=0.984702
2030 13.662 cm
2100 1.889 m


quadratic
Y = 2.029890 + 0.202368*x + 0.005775*x^2
R^2 = 0.984857

2030 13.298 cm
2050 26.585
2100 80.016


Indicial power
Y = 2.263276 + 0.101848*x^1.365590
R^2 = 0.985011

2030 12.858cm
2050 23.547
2100 57.107

Global sea-ice new on-the-day record for 16 April, 296,000 sq km less
than previous record on-the-day 12017