Volume Predictions.

[ChrisReynolds]

Its looks very similar to me to the ice age category maps.

The message I take from seeing the same story in melt and age categories is that the arctic doesn't have a way to regenerate MYI any more. The stuff that thickens in the summer is the stuff on the way out of the Fram, and the zone that bulks up during winter is next year's high melt zone.

Agreed,

The Beaufort Gyre used to be a 'flywheel' that would age ice, ice taking many years to go from the coast of the CAA and back into the transpolar drift. Since 2007 that ice has increasingly entered a 'killing zone' in the waters of the Beaufort/Chukchi/East Siberian Seas, as open water in those seas has opened up to a greater degree with thinning of ice.

Thanks for those plots Wipneus, doesn't really add anything to my understanding, but breaking down into categories on the scatter plot is something that hadn't occurred to me and it's really good to see the patterns I'd expected borne out by your calculations. I'm working on a related blog post, thinking before calculating and composing. I would like to reference the stuff you've done and your approach. If you want to do a guest blog post at Dosbat I'd be happy to host, although you'd get a bigger audience at Neven's.

If you don't want to do a blog post, do you mind if I write one up about what you've done as a 'placeholder' so there's a single place I can reference?

[Bob Wallace]

Nice.  How about making the key large enough to read? Looks like there's plenty of white room.

[Wipneus]

If you don't want to do a blog post, do you mind if I write one up about what you've done as a 'placeholder' so there's a single place I can reference?

I have no need to do a blog post right now. By all means use what I did, after all this was inspired from the graphs from yourself and crandles.

[Wipneus]

Nice.  How about making the key large enough to read? Looks like there's plenty of white room.

I am having fun hacking a quick extension with a software routine that was designed to plot an Arctic map plot of some field plus legend and transformed it to multiple maps and single legend.

Next version will probably do better.

[Vergent]

Nice.  How about making the key large enough to read? Looks like there's plenty of white room.

Try clicking on the image, then click on it again to magnify.

Vergent

[crandles]

I have done a bit more with my sorted by thickness thinning:

Code: [Select]

Thinning of ApplyTo2013 ApplyTo2012 ApplyTo2011 ApplyTo2010 ApplyTo2009 ApplyTo2008 ApplyTo2007 ApplyTo2006 ApplyTo2005
2012    2,772,007 3,342,706 2,932,211 4,676,027 5,155,666 5,659,778 5,052,518 6,672,333 7,645,561
2011      3,594,690 4,583,155 4,173,901 6,174,703 6,602,539 6,622,493 6,096,716 8,208,924 8,898,698
2010    2,771,167 3,254,986 2,932,270 4,551,958 5,026,581 5,747,883 5,017,106 6,491,297 7,559,923
2009    3,777,972 4,722,270 4,194,480 6,641,969 6,930,162 6,868,341 6,496,296 9,056,654 9,810,166
2008    4,152,712 5,233,423 4,757,053 7,142,888 7,449,184 7,199,295 6,832,828 9,504,776 10,089,763
2007    3,803,520 4,619,772 4,065,011 6,543,521 6,793,399 7,007,011 6,477,514 9,090,400 9,875,079
2006    3,983,900 4,808,303 4,376,289 6,567,405 7,225,387 7,522,309 6,921,367 9,055,172 10,212,246
2005    3,684,982 4,349,719 3,837,372 6,140,808 6,380,704 6,454,129 6,057,908 8,653,313 9,223,221

Avg 3 prev 3,045,955 4,186,803 3,961,268 6,776,126 7,155,990 6,994,483
Lin Extrap 2,935,011 4,181,114 3,589,190 7,080,660 7,655,531 7,547,366
Actual                3,342,706 4,173,901 4,548,128 6,930,162 7,199,295

These volume numbers are in m*km^2  units, so to get km^3 you need to divide by 1000.


So what to make of that lot?

The differences looking down a column from using different years thinning profiles makes least difference to the numbers.

Looking across a row, using different March Thicknesses makes more difference.

Both of those are less than the difference in actual September minimum.

So, the March volume makes more difference than the thinning profile.

This sounds encouraging for using this to estimate September volume when we have the March thickness distribution. Applying the different thinning profiles to 2013 thickness distribution gives the first column numbers a range from 2.77 to 4.15 K Km^3. We should expect towards the low end of that range because the thinning profiles are getting to melt more as time goes on. So probably estimate somewhere around 3 K Km^3.

That seems as if there may not be that much uncertainty in this method. Unfortunately had we used this technique to estimate 2010 September volume we would have concluded a figure in the range 6.14 to 7.14 K Km^3 was likely and the actual turned out to be 4.55 K Km^3. That is rather bad compared to

http://www.arcus.org/search/seaiceoutlook/2010/june

So probably not a good technique. Never mind - you don't know unless you try and it still might provide useful information in some other way.

eg area with thickness less than 1.25m:
2013____ 2012____ 2011____ 2010____ 2009____ 2008____ 2007____ 2006____ 2005
53527138 54026206 53654765 53666713 52730476 52922816 53034398 52978849 52910004

[ChrisReynolds]

I think it's an interesting method. I've been pondering using the %melt per April thickness as a basis for a projection for this year, not got round to it yet.

Anyway:

• The issue of a failure in 2010 is not relevant in my opinion. 2010, like 2007 was an event that was hard to predict. Unless one had noticed the large export of MYI into the Siberian sector in that year one would not have expected the crash. Even having seen that export it would have been hard to tell how much of the volume would be lost.
• I think you can narrow down your range for 2013 still further because of the effect of 2010 on the seasonal cycle. The anomalous spring melts of 2010 - 2012 means that you range would be 3.59 to 2.77 k km^3. I would have been tempted to say discount 2010 as in this year there was an anomalous loss of MYI. However if you did that 2012 would still mean the range wouldn't change.

Those points noted, last year's volume was 3.261k km^3. So your method doesn't rule out either further loss or a gain.

Simple subtraction of the melt season losses for 2010 - 2012 from the 2013 maximum gives the following minima:

2010    2011    2013
2.849    3.879    3.161

Which is a similar pattern to your method, and again doesn't rule out either recovery or further loss.

However only 2001, 2005, and 2008 have seen gains on the previous year, with all other post 2000 years showing year on year losses. And the March ice state seems worse than last year, with the state of the pack from MODIS looking worse than previous years. So I'm still betting on a further loss this year.

[crandles]

Simple subtraction of the melt season losses for 2010 - 2012 from the 2013 maximum gives the following minima:
2010    2011    2013
2.849    3.879    3.161

Thank you for your comments.

Above quote is a nice simple method. I suggest it is biased slightly high because the melt volume is growing in recent year. My figures were a little lower so that puts the figures in even better agreement.

Just wondering if we should have a volume crowd sourced estimate. If we do, is it a bad idea to run through a few methods through fear of influencing people or would it help people formulate their ideas and explain why they are making the estimate they say.

Being able to say an estimate comes from method 14 but a bit lower because of the way the ice looks on MODIS allows a brief explanation.

So for example
Methods 1 to 6 would be graph what we are attempting to estimate and extrapolate in some way.



Link: https://sites.google.com/site/arctischepinguin/_/rsrc/1359893289843/home/piomas/grf/piomas-trnd1.png

Hum, how do I display Wipneus trend1.png graph? Edit Thank you Wipneus

So Method 1 Exponential fit volume = 1.88  K Km^3 +/- 2sd of 1.9
Method 2 Gompertz fit volume = 3.01  K Km^3 +/- 2sd of 1.96
Method 3 Log fit = x.x K Km^3 +/- 2sd of y.y
Method 4 2nd order polynomial fit = x.x  K Km^3 +/- 2sd of y.y
Method 5 First order Lag fit (time constant of x years) = x.x K Km^3 +/- 2sd of y.y
Method 6 Linear fit over last x years = x.x K Km^3 +/- 2sd of y.y

These methods basically use time. In reality time is unlikely to be the cause of the effect; it is more likely to be a combination of:

Volume at start of season,
GHG levels,
Upward heat flux from warmer water several meters below ice,
Weather,
Quantity of MYI,
Location distributions of volume,
Distribution of thicknesses volume,
Distribution of MYI (location and or thickness),
ENSO/NAO/AO/PDO and any other oscillations you think may have effect,
Extent to which ice looks fractured or other look or review of state of ice,
and probably others.

Several of these factors are likely to be difficult to judge let alone quantitatively estimate.

Volume may well deserve the top place in that list because the information is both easily available and can be used quantitatively quite easily.

Method 7: You can graph and extrapolate the volume reduction from the latest available data.

eg

Link


Method 8: Rather than extrapolate the thinning forward in time, it is likely that the less volume of ice there is, the more the volume reduction increases. So you might arrive at a formula like
Melt volume = 22.78 - .213 * Max volume

Method 9: Rather than just using time to extrapolate the volume reduction like method 7 or just using volume as in method 8, you might consider that there is both a volume influence on the volume reduction but also a time element (as a proxy for GHG levels).

Method 10 Use volume influence and a GHG measure to estimate the volume reduction.

[Wipneus]

Hum, how do I display Wipneus trend1.png graph?

The google sites software makes this not so straight-forward:
right-click on the image on the overview page (https://sites.google.com/site/arctischepinguin/home/piomas) and either choose "copy image location" or "copy link location" (or something similar).

BTW here is an other, based on a Loess smooth:



Uncertainties of all these methods seem to be similar. Currently I am thinking of using a linear OLS fit of the last 10 years and using that value for a prediction of extent/area. That prediction is 2.6 +/- 1.6 [10^3 km3], simplest method and smallest uncertainty.

[crandles]

BTW here is an other, based on a Loess smooth:

Uncertainties of all these methods seem to be similar. Currently I am thinking of using a linear OLS fit of the last 10 years and using that value for a prediction of extent/area. That prediction is 2.6 +/- 1.6 [10^3 km3], simplest method and smallest uncertainty.

OK. Method 6.1 Loess smooth 2.6 +/- 1.6 [10^3 km3]

OK, 10 year OLS fit is simple. I am curious though: Looking at my orange melt curve that looks to me to curve up in a smoother way than the september minimum goes down. Do you think that might be because more up to date volume information from the previous maximum is available and used? Isn't it advisable to try to use the latest information?

I think you can probably tell that I think we can do better again by observing that the lower the most recent volume data is the more the melt is likely to be. I believe this allows a better fit that is not smooth with time. Other factors like area with thickness under 1m might also help.

Simple to avoid overfitting is good, but don't you think these improvements could be worthwhile?


Estimated volume, using that to convert to extent for search estimate seems a very sensible approach to estimating extent. Your thickness declines in same ratio as extent seems sensible and I don't see any improvement on that.

[Wipneus]

I am curious though: Looking at my orange melt curve that looks to me to curve up in a smoother way than the september minimum goes down. Do you think that might be because more up to date volume information from the previous maximum is available and used? Isn't it advisable to try to use the latest information?

Yes, but it depends how much smoother (closer to the data) your melt curve is. Eyeballing there are still some +/- 1.5 deviations from a possible curve. In all my calculations I include a 2*sd as +/- 1.35 10^3 km3, estimated by the modelers.
Here the sd must be calculated from the data: once you have a nice fit calculate the sd of the residuals.

[crandles]

Est 2003 using straight line through 1993 to 2002 =10.887
Est 2004 using straight line through 1994 to 2003 =10.227
Est 2005 using straight line through 1995 to 2004 = 9.923
Est 2006 using straight line through 1996 to 2005 = 8.933
Est 2007 using straight line through 1997 to 2006 = 8.686
Est 2008 using straight line through 1998 to 2007 = 7.596
Est 2009 using straight line through 1999 to 2008 = 6.756
Est 2010 using straight line through 2000 to 2009 = 5.981
Est 2011 using straight line through 2001 to 2010 = 4.499
Est 2012 using straight line through 2002 to 2011 = 3.613

Est 2013 using straight line through 2003 to 2012 = 2.600 K Km^3

errors in last 10 years:
0.647
0.346
0.764
-0.060
2.228
0.524
-0.137
1.553
0.482
0.352

So RMSE = 0.957
Therefore, it appears to me that two standard deviation of this method in forecast mode is circa 1.92

So 1.6 (or 1.35) for 2sd looks optimistically low for that method. 1.92 looks similar to 1.9 and 1.96 that I gave for exponential and gompertz methods though I estimated those numbers rather than calculating them.

(Properly calculating the numbers for all the methods may take a little longer.)

Should the 2.6K Km^3 estimate be reduced by the average of those 10 error numbers (=0.67 making the estimate 1.93 K Km^3. Method 6.51) ?

[Wipneus]

Well, well. What I can say (as a defense)  is that I am just interested in an estimate for 2013. The errors for other years, using the similar method do not have to be same.

Actually a better method might be to calculate the 2013-2012 estimates, but with std deviation and use those deviations as weights to calculate the mean and error of those estimates.

I will do that tomorrow, if you don't.

[ChrisReynolds]

I still think you should be leaving out the years 2007 and 2011 as outliers.

We cannot rule out 2007 weather for 2013, but we can rule out a large MYI export into the Siberian sector. This leaves a re-run of 2007, but no prediction method can be expected to account for such 'black swan' events.

By doing that you reduce your sample size by two (I've not got the time to re-run your linear fits to add the two preceding years), but your RMSE drops to 0.47, with a Stdev (Excel's Stdevp function) of 0.188.

I'm not a fan of 'crowd sourced' predictions. I really don't think they tell us anything of use. I'm also still not a fan of statistical extrapolation, except as an exercise in answering the question 'what if current behaviour continues.

The reason I've been pondering a method is that it would add to this year being a test of my impression* that MYI is playing a large role, and the geographically confined nature of this year's MYI means a further large drop is to be expected. That is why I find prediction interesting - as a test of 'theory'.

* I say impression because I feel it arrogant to attach the term 'theory' to my amateur bumblings.

[crandles]

Why stdevp? That is for an entire population. Even when I calculate all 24 easily calculable 10 year linear extrapolation errors I still view that as a sample of 24 numbers drawn from a population equal to age of planet in years. The recent 24 are obviously more relevant but it is still drawn from a sample. So it seems more appropriate to use Stdev rather than stdevp.

Stdev of last 10 years is 0.72 (RMSE 0.957)
Stdev of 24 readily calculable 10 year period extrapolation errors is 1.04 (RMSE 1.11)

It would seem the error level is trending downwards as the volume numbers get smaller. I thought I might get a lower error level for years before the downward acceleration really took hold, but it seems that acceleration creates bias in this simple method but the level of variability is trending downwards.

Reducing the error estimate for the downward trend may well be reasonable as recent ones are more relevant. However, if there are two outliers in a sample of just 10 then that seems to tell me that sort of thing happens not infrequently and removing outliers to reduce an error estimate seems a 'bit dodgy'* to me. So I am inclined to leave them in.

* bit dodgy - in other words I am not an expert and certainly not expert enough to give a definitive opinion.

Using last 10 seems a simple easy approach rather than using 24 and weighting more recent ones more heavily. Hope that seems OK. Seems sensible to try to use same error of last 10 years for each method and compare that rather than trying to compare uncertainty bands calculated in different ways for different methods.

[ChrisReynolds]

Yep, Stdev would have been better. Sorry, was posting while doing something else.

Here's what I was doing:



I've used the thinning profile (March to Sept) calculated from the post 2007 years of the PIOMAS data, i.e. thinning as a function of initial thickness in bands of 5cm. I've used each of the thinning profiles of the years 2007 to 2012 as 'scenarios', and have applied these to the March thickness profiles from 2007 to 2013 on a per grid box basis.

2013 projection is in red. Columns are years (March thickness) to which the thinning profile is applied. Rows are thinning profiles for the given years (March to Sept thinning).

The % errors are included below for the 'hindcasts'. Note that the method tends to over predict as shown when the thinning profile is applied to the same year. Note also the pattern of % errors. There is obviously a shift in the thinning profiles as time proceeds such that applying the profile for years before the year of profile predicts too much volume at the end of the season, applying it after predicts less volume than actually occurred. This is because the melt profiles are proving more aggressive as time proceeds.

So whilst all of the 2013 values are above 2012's record, I don't think this can be read as a recovery this year.

[crandles]

Note that the method tends to over predict as shown when the thinning profile is applied to the same year.

That sounds similar to what I did and posted above except for the sorting where I assumed thickest remaining ice came from thickest ice in Sept. You have laid out table more logically than I did.

I got a slight error when applying 2010 thinning to 2010 but the other years were spot on despite my added complexity of different areas in the cells I matched.

Your numbers seem higher. A systematic error choosing the appropriate thinning band or something similar perhaps? (I managed to do that before finding that error.)

Did you work the thinning of a band on %volume decline? If you average the thickness declines ignoring the different areas, there might be systematic bias by larger cells are at lower latitudes and have a greater % thickness decline.

[ChrisReynolds]

For each band I worked out the total area and total volume loss to September. Average thinning is then VolumeLoss(n)/TotalAprilArea(n), where n is the reference of the thickness band. It's done that way to give area weighted thinning.

I'll look at whether there's a mismatch in bands, but the codes are very similar, the projection is a modified copy of the thinning code. I'll look at it tomorrow after work.

[ChrisReynolds]

I had the chance to check this morning, there's no mismatch in the bands.

Bear in mind I'm using the average thinning for 5cm bands and applying to the native thickness for each grid cell. So a small mismatch will be expected.

[Wipneus]

I have looked a bit more at the results from my proposed 10 year linear prediction. Crandles is completely right, it is crappy. The errors are larger than what the statistics predict, and are mainly postive. The latter is of course caused by the non linearity: the accelerating decline that is even visible on the 10 year scale. Reducing the period to 7 years helps a bit, but at an expense of increasing uncertainty in the individual predictions.

So I tried kwadratic fits, much better for positive/negative prediction errors. Unfortunately the prediction uncertainty is also increasing and not giving any better results as other methods.

Finally I looked at the possibily that absolute errors are not the best way to measure how good the prediction is. September ice has gone down more than 80% and a 1000km^3 error used to be small, but is now almost as big as the volume itself. So it seems reasonable to look at relative errors.

Attached is the relative error, computed from the residual divided by the estimate itself. Estimate  is a quadratic OLS fit of the 12 previous years.

The relative errors have of sd of 0.124, max relative error for the last 21 years is about 0.25.

A relative error of 0.25  would lead to a prediction for 2013 of 2.23 +/- 0.55 [1000 km^3]

A much smaller error than before, or did I miss something?

[crandles]

A much smaller error than before, or did I miss something?

Well I think the errors are of a similar size but the way you are predicting them to get smaller means that your estimate of the error is now smaller.

The distribution of your relative errors does look reasonably normal so it does seem a sensible approach to make a realistic estimate of the expected error size.

(Shame it increases my workload of calculating this for all the methods I have mentioned but if it looks to be a big improvement then it is worth doing at least on some of the possible methods. Slightly worried it might need to change again and/or turn out not to give a normal distribution for relative errors for some of the methods making comparison of estimates less easy.)

[SATire]

A much smaller error than before, or did I miss something?

Wipneus - the proper error depends a bit on what you are looking for. If the error of the predicted value is of your interest, then you have to go with the errors of the data - that is PIOMAS errors. They use to give absolute errors, but perhaps they are getting wrong with that as volume approaches zero. Relative errors are not good in the vicinity of zero - but errors should be asymetrical of course, since volume is a positive number. So - you should do fiting with that asymetric errors, if you could get them, to calculate the errors of the fitting parameters from which you can determine errors for the predicted value.

If the "right" function is of interest, you know that there is no chance to distinguish which functions must be refused, if the errors of the data are large. Likelyhood will be very similar for a lot of different functions (linear, exponential, Gompertz are all about the same for next year). Maybe you just try them all and do averaging the value predicted by different functions? Anyway they will all be much closer together than the error of the data. So guessing is maybe the best trick to forecast the weather of next summer ;-)

On more test on errors: If the Chi^2 is about number of data - number of fitting parameters, then the errors are probably proper guessed. If Chi^2 is much smaller (as in your case now) the errors of the data are guessed to large for some reasons. It would be OK to adjust them, because a large portion of the error are probably systematic errors inside PIOMAS and not statistical errors, which you are looking for.

 

[crandles]

http://farm8.staticflickr.com/7288/8737347747_d1aabd4938_b.jpg

This gives estimate of what PIOMAS will report as 1.88 +/- 2sd of 2.77 K Km^3

or using your relative errors the estimate is

1.88 +/-2sd relative error of 0.28.
0.28*1.88 = 0.52 K Km^3

0.52 K Km^3 looks a little better than your quadratic fit's relative error size of 0.55 K Km^3.



Satire,

I think it is clear that what we are trying to predict is what the PIOMAS system will say the volume will be.

Yes, if we were interested in the possible error of the actual volume of ice then we would have to increase the estimates for the fact that PIOMAS numbers are only estimates.

I am interested in what methods best project forward what PIOMAS will say the volume is.

[SATire]

Satire,

I think it is clear that what we are trying to predict is what the PIOMAS system will say the volume will be.

Yes, indeed. So because Chi^2 is so large - just call the large part systematic error of PIOMAS (which also would have the asymetrical part) and use the smaller statistical error for prediction - the systematic error of PIOMAS probably will not change much the next month. You may also get the statistical errors easily from the residuals (if you know how the units of your fitting programm like Gaussian, FWHM or what)

[crandles]

>"your fitting program"

I am just using excel. While I see functions for:

CHIDIST Returns the one-tailed probability of the chi-squared distribution
CHIINV Returns the inverse of the one-tailed probability of the chi-squared distribution
CHITEST Returns the test for independence

Unfortunately I have no idea how to calculate Chi^2 let alone use such functions to get at the "smaller statistical error for prediction".

[Wipneus]

SATire:

The aim is to reduce the interval in which the minimum of the PIOMAS numbers in 2013 will fall. The PIOMAS people estimate an uncertainty of 1.35 [1000 km^3], but also state that the estimate is "conservative". IMO they say, that the real uncertainty may be lower.

I know about the CHI2 test, and I don't know if it is too small. But it is not directly applicable yet: I'm using an un-weighted OLS fit for the prediction and std error and at a later step calculate the deviations in proportion to the absolute value. 

[ChrisReynolds]

Correction,

I did make a mistake - I was comparing monthly PIOMAS gridded data to actual daily minima.

Here's the same table but with monthly PIOMAS figures as the Actual data, and the percentage difference table reflecting that.

[crandles]

Correction,

I did make a mistake - I was comparing monthly PIOMAS gridded data to actual daily minima.

Here's the same table but with monthly PIOMAS figures as the Actual data, and the percentage difference table reflecting that.

Now I am confused. I thought you had correctly reported 2012 gridded average September data as 3342.706KM^3 I reported that number as both the correct number and what I got by applying 2012 thinning to 2012. So your actual numbers looked correct to me but now you have gone and changed those numbers.

[SATire]

I know about the CHI2 test, and I don't know if it is too small. But it is not directly applicable yet: I'm using an un-weighted OLS fit for the prediction and std error and at a later step calculate the deviations in proportion to the absolute value.

Wipneus - the Chi^2 in your fit is way to small, so there is plenty of room to reduce your statistical error. If Chi^2 is typical (=number of degrees of freedom = number of data - number of fitting parameters), then about 68% of data is within 1 std dev. and ~32% are out, but not to far. So it is possible to reduce the statistical error by thinking of a systematic error (error=sqrt(err_stst^2 + err_sys^2)/1.41), in which you are not interested for your prediction. Therefore, you will obtain a prediction with smaller statistical error.
If you use least square fit - the sum of least squares is the Chi^2, if you have data with errors. If you don't you can then compute the statistical error from that sum of least squares.

The errors are not proportional to the value - only for linear fit. You have to take that errors of the fitting parameters from the covariance matrix - you use that matrix allready for the least square fitting in the very last iteration. 

[ChrisReynolds]

Crandles,

Sorry, I got confused.

Daily volume minima for 2007 to 2012
6.458
7.072
6.893   
4.428   
4.017   
3.261

Monthly volume minima for 2007 to 2012
6.5277
7.2457
6.9771
4.5864
4.2104
3.3723

These are both from the main series of PIOMAS. Which left me confused as to what I had done.

However looking at the sum of volume from my thickness breakdowns it's obvious what I did. I took the September average volume from that data, which is calculated from the same grid areas as the data used to calculate that table.

6.477513015
7.199295787
6.930161004
4.548127589
4.173901184
3.342706195

So this is probably the best series to use technically, although I still prefer comparison with the PIOMAS main series data as that's what most people use, and what we will judge the minimum by.

[crandles]

6.477513015
7.199295787
6.930161004
4.548127589
4.173901184
3.342706195

So this is probably the best series to use technically, although I still prefer comparison with the PIOMAS main series data as that's what most people use, and what we will judge the minimum by.

Yes, for prediction you may want to predict the daily series volume number either minimum day or monthly average. For checking that you haven't made mistakes, the only series that you should be able to predict exactly is the Sept monthly gridded numbers.

[Wipneus]

Wipneus - the Chi^2 in your fit is way to small, so there is plenty of room to reduce your statistical error.

I don't think we are discussing the same thing. My method, extrapolating using linear fit gives a relatively small statistical error. Until crandles showed that the real spread of the results is much larger than that. In my books that is a large chi^2

This is partly explained by the insufficiency of the linear fit: a quadratic fit helps but still the spread in the results are too larger than expected (an also what I would like it to be).

So now I have arrived at the idea that maybe it is the way that we are judging the results by looking at the relative size of the deviation.  I am not quite out of it yet, but with the statement "relative error was less than 0.25 for the last 20 years", I would be tempted to bet on it to be true for 2013 as well.

Today is a prepare and packing day for a week vacation, I will look at this again after that.

[SATire]

I don't think we are discussing the same thing. My method, extrapolating using linear fit gives a relatively small statistical error.

You are right. I thought you would do something similar to your exponential fit with a quite large symetric error bar you show on arktischepinguin - I meant that kind of fit. Sorry for confusing. For linear fitting relative errors are ok.

[Richard Rathbone]

Yep, Stdev would have been better. Sorry, was posting while doing something else.

Here's what I was doing:



I've used the thinning profile (March to Sept) calculated from the post 2007 years of the PIOMAS data, i.e. thinning as a function of initial thickness in bands of 5cm. I've used each of the thinning profiles of the years 2007 to 2012 as 'scenarios', and have applied these to the March thickness profiles from 2007 to 2013 on a per grid box basis.

2013 projection is in red. Columns are years (March thickness) to which the thinning profile is applied. Rows are thinning profiles for the given years (March to Sept thinning).

The % errors are included below for the 'hindcasts'. Note that the method tends to over predict as shown when the thinning profile is applied to the same year. Note also the pattern of % errors. There is obviously a shift in the thinning profiles as time proceeds such that applying the profile for years before the year of profile predicts too much volume at the end of the season, applying it after predicts less volume than actually occurred. This is because the melt profiles are proving more aggressive as time proceeds.

So whilst all of the 2013 values are above 2012's record, I don't think this can be read as a recovery this year.

I think this shows there's a factor missing. The changing profile of the ice thickness isn't sufficient to explain accelerating volume loss.

One thing I think is missing is that thick FYI melts out a lot easier than thick MYI. Wipheus' maps upthread looked to me like the areas which had recently bulked up, were the areas which melted above the projected rate.

Is there some way to account for this? Maybe project a cell based on the lower of its last two Spring values. Say 2m would be projected to 0.5m and 3m would be projected to 1,8m, then project a cell with 2m in Spring 2112 and 3m this year to 0.5m rather than 1.8m.

Last year seemed to be pretty much on trend, and unless a projection method would have worked in 2012, I don't think its worth considering for this year.

[crandles]

I think this shows there's a factor missing. The changing profile of the ice thickness isn't sufficient to explain accelerating volume loss.

One thing I think is missing is that thick FYI melts out a lot easier than thick MYI. Wipheus' maps upthread looked to me like the areas which had recently bulked up, were the areas which melted above the projected rate.

Is there some way to account for this? Maybe project a cell based on the lower of its last two Spring values. Say 2m would be projected to 0.5m and 3m would be projected to 1,8m, then project a cell with 2m in Spring 2112 and 3m this year to 0.5m rather than 1.8m.

Last year seemed to be pretty much on trend, and unless a projection method would have worked in 2012, I don't think its worth considering for this year.

Starting volume is the major factor looking across rows while more aggressive thinning profiles in later years effect up and down column is a less significant factor. Combining these two gives the full effect but that is because full effect has only been split into 2 factors.

Effect of albedo differences between FYI and MYI and between thin ice and thick ice almost certainly has an effect, which if separated out would leave less of an effect for maximum volume reduction effect and for thinning profile effect. Is is possible to split that out using CT area data? Not sure I could.

[ChrisReynolds]

Richard,

I think this shows there's a factor missing. The changing profile of the ice thickness isn't sufficient to explain accelerating volume loss....One thing I think is missing is that thick FYI melts out a lot easier than thick MYI.

The whole approach is based on my increasing conviction that loss of MYI is key to the changes in the pack over the last decade. The approach implicitly includes this tendency of FYI to melt out more than FYI, in terms of the thickness distributions applied to the March thickness profiles and in terms of separating out post 2007 years from all other years.

Is there some way to account for this? Maybe project a cell based on the lower of its last two Spring values. Say 2m would be projected to 0.5m and 3m would be projected to 1,8m, then project a cell with 2m in Spring 2112 and 3m this year to 0.5m rather than 1.8m.

I think that could reasonably (if uncharitably) be viewed as cherry-picking designed to get the lowest result. By applying various years as scenarios and giving a matrix of thinning profiles and initial conditions, I think I've covered all I can with this method.

Last year seemed to be pretty much on trend, and unless a projection method would have worked in 2012, I don't think its worth considering for this year.

I agree. Indeed in my blog post on this method and the results I've closed by saying I'm not convinced by it and still expect another record this year, both in volume and CT Area.

Although I did forget to note in that post that 2012 was on-trend.

[Richard Rathbone]

I think that could reasonably (if uncharitably) be viewed as cherry-picking designed to get the lowest result. By applying various years as scenarios and giving a matrix of thinning profiles and initial conditions, I think I've covered all I can with this method.

Its designed to get a lower result. When a method is projecting high, there's not much point in adjusting it in a way that doesn't lower the the result.

I agree with you that FYI/MYI is important. This a simple way of assigning which is ridged up FYI thats going to melt, and which is MYI that isn't. Consider that thick ice near Siberia. Do you reckon this spring's thickness in those cells, or last spring's thickness in those cells, provides a better indicator as to how much ice there is going to be in those cells come September?

Its only cherry picking if it was reverse engineered to get the fit exact. I don't know if it would end up still biassed high, or fit, or end up biassed low. If 2011/2012 did give a decent projection of 2012, I'd reckon it was doing a decent job of assigning thick FYI and 2012/2013 would give a decent projection of 2013.

[ChrisReynolds]

All reasonable points. Thanks.

Perhaps the method needs to incorporate Wipneus's work on where the regions in the scatter plot are (geographically speaking).

I'll think about it.

[Richard Rathbone]

I look forward to reading the results of your thoughts.

Something else to think about for a method. When the arctic goes seasonally ice free, is it going to be able to predict 0 for summer based off what you'd expect the spring coverage to look like?

And if by some fluke of geoengineering a previously seasonally ice-free arctic was not going to melt out, it ought to be able to predict that based off what the spring coverage looked like.

Winter ridging is going to lead to some thick ice in the spring, and some way is needed to aggressively project the melting of it to successfully project a seasonally ice free arctic. Conceptually it ought to work for recovery from a seasonally ice free state too, even if you don't expect the conditions ever to arise in practice.

How about a melt curve based off the amount of thick ice as well as its thickness?
If (total volume of ice thicker than x is greater than y) then project x to x - delta else project x to 0.

 

[crandles]

Something else to think about for a method. When the arctic goes seasonally ice free, is it going to be able to predict 0 for summer based off what you'd expect the spring coverage to look like?

And if by some fluke of geoengineering a previously seasonally ice-free arctic was not going to melt out, it ought to be able to predict that based off what the spring coverage looked like.

From 31 May to minimum the volume reduction seems remarkably linear. Now approaching 15K Km^3 as shown in the first graph below. I still have a little work to do to see how this works in a forecast mode.



http://farm6.staticflickr.com/5341/8751120270_e1c3f54a36_h.jpg

So it seems to me that this sort of method should provide a good guide to whether future years are going to melt out again or bounce back.

2013 is .295 below 2012 at day 120. So there doesn't seem much reason to expect 2013 minimum to be much below last years minimum volume - 0.3 unless there is really rapid volume loss in May 2013.

[ChrisReynolds]

Richard

I'll have to think of something first, been pondering all day but have drawn a blank. Maybe all of this will trigger something in Wipneus's brain, he's far more capable than I.

As for a rapid transition and seasonally sea ice free state. I'm having doubts again about whether we face a rapid crash by 2020, or whether thickness/growth feedback will bolster winter ice sufficient to retard summer volume loss and keep a residual of summer ice well into the 2020s. If we don't see a further summer loss of volume from 2012's low, I'll see that as indicative that this is the case, but only 'indicative'.

Not sure I follow your final paragraph. My calculations are area weighted and any thickness calculation (March Thick minus Thinning for Thickness Band) that gives less than zero is already set to zero.

Crandles,

The increase in Spring volume loss is what seems to have been driving the melt season volume loss increase.

However as of the latest PIOMAS data it is too early to tell how aggressive this year's spring volume loss will be.


The anomalies above show that after the spring volume loss anomalies rise, indicating less loss than baseline, before levelling, indicating identical loss as baseline. Dr Zhang has suggested this may be because there's less ice from which to lose volume so loss of further volume is more difficult. I'm not sure about that.

It is worth pointing out that similar behaviour can be seen in CT Area anomalies for some regions, for example Baffin/Newfoundland.
http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/recent365.anom.region.4.html

Here an earlier start to melt leads to a drop in anomalies (imagine the downward slope of a sine with the 2012 data leading the baseline). But then the baseline flooring to zero catches up with 2012 and the anomalies first rise, then level. I suspect that something similar is happening in PIOMAS. The spring anomaly crash is indicative of earlier melt, but after the summer solstice the baseline melt rate is so high it 'catches up' with the post 2010 years, leading first to an increase in anomalies then a levelling. I suspect this might be happening because the limiting factor post solstice is the ongoing decline in insolation.

However the main doubt I have about PIOMAS is the lack of increasing summer melt, I suspect more melt is happening than PIOMAS shows during recent summers. But this is little more than a WAG.

By the way, this graphic of PIOMAS loss during months March to August is relevant.

For a given month it's the volume on the 1st of the following month minus the volume on the first of that month.

[Jim Pettit]

@crandles: your first graph in #139 is very similar to one I created last year, though mine covers total melt from maximum to minimum:



As was noted last year when I posted that graph, it's true that the annual melt hasn't been increasing much each year; in fact, 1981 saw a total volume loss of 18,259 km3, while 2012 saw a loss of a nearly-identical 18,662 km3. The real dropoff, then, is coming from an increasingly lower annual maximum. At first glance, that seemed almost counter-intuitive to me; after all, if the oceans are warming, I thought, shouldn't volume be disappearing at a more rapid rate? But then I realized that it's the overall percentage of the ice melting each year that's making a difference and that truly matters, not the absolute amount that's melting:



And that's the way it has to be, of course. As both your and my graphs show, in not too many years hence, the rapidly falling maximum line will intersect with the slowly rising melt line, and both will then fall in unison. That is, when the annual maximum in 10 years is just, say, 11,000 km3, only 11,000 km3 will be available to melt, so there'll be far less melt then than there is now (a fact which lunkheaded denialists will no doubt trumpet as "evidence" of a "recovery").

[Richard Rathbone]

Taking the May value and subtracting 15 seems like it will be hard to beat as a predictor.

[crandles]

Taking the May value and subtracting 15 seems like it will be hard to beat as a predictor.

Well maybe I am not explaining well because I think the green line in the second of the four graphs is better than the straight red line in the first graph.

Melt from max to min is curving upwards but melt from 30 June to min is curving downwards.
So it seems that 31 May seems a sweet spot where melt after 31 May is linear:



http://farm9.staticflickr.com/8413/8750936961_a9c3f19ccb_z.jpg

There must be some chance that melt from 31 May will start curving one way or the other and we just haven't seen it yet.

[Vergent]

Interesting discussion, and nice graphics all. I just want to point out that, since PIOMAS overestimates the thickness of thin ice, it will overestimate the early melt. Further, since it underestimates the thickness of thicker ice, it will underestimate the later melt. To this degree, this discussion is about the behavior of the model, and not about the actual melting.

Vergent

[Bob Wallace]

After the thick/thin under/over discovery was made I wonder if the model was adjusted?  It seems this would be standard practice with modeling - make it better when possible.

[Vergent]

After the thick/thin under/over discovery was made I wonder if the model was adjusted?  It seems this would be standard practice with modeling - make it better when possible.

That depends on what your definition of "better" is. From the change they did do, it seems to me that "better" for them meant that the model did not go alarmingly to zero. They reduced the trend from -3.6 to -2.8 kkm^3/decade. They did so in the face of a current decade decline of -6 kkm^3.

Vergent

[Pmt111500]

Jim Pettit: "But then I realized that it's the overall percentage of the ice melting each year that's making a difference and that truly matters, not the absolute amount that's melting:"

Nice graph! Though this is a volume thread, a short note on the area. The albedo difference between water and ice makes it so that to predict sea ice area, one would have to know the percentages of the area loss but these in turn can't be predicted from the area data alone, since albedo changes on ice will effect the area loss. Just a complicated way of saying that all the radiation doesn't stop on the surface of the ice.

Jim Pettit: "As was noted last year when I posted that graph, it's true that the annual melt hasn't been increasing much each year; in fact, 1981 saw a total volume loss of 18,259 km³, while 2012 saw a loss of a nearly-identical 18,662 km³. "

Thanks for the info, (joke) on to overtly simplistically check the CO2 levels in 1981 (340,1ppm) and 2012 (393,82ppm) and conclude that excluding physics, positive feedbacks and all sorts of tipping arctic systems, 1 extra ppm of CO2 equals extra 7,50186km³ of ice loss , so doubling the CO₂ from 280 would increase the yearly melt by ~2100km³ ice (Modified 24.5.2013) This simplistic approach (calculating through 1979-2012) results in 18686km³ of ice loss in 2012  (/joke) (insert proper physics to get a better value.).

The natural energy exchange from tropics to arctic to space is so large, that the anthropogenic effect looks small in comparison, but it's still measurable and is having an effect in the Arctic (soon-not-so-)cooler of the Earth. Thanks to all in this thread for your efforts to approach the feedbacks and forcings through physics, statistics alone isn't enough.

[crandles]

It is time to enter May polls if you haven't already.

We don't have PIOMAS volume beyond day 120:
2011 day 120 21.336
2012 day 120 21.568
2013 day 120 21.273 

So .295 K Km^3 below last year at day 120, but what has it been doing since:



Well it has been colder. Sometime this can be misleading: rather than colder meaning less melting sometimes more melting is absorbing heat and making it colder. However we are only in May so there is little surface melting when the absorption of heat can make it cold.

Looking at the pressure, the low is more centrally located this year allowing it to be cold. 2012 the low was displaced drawing in warm air from Russia.

May is more about bottom melt so cold surface temperatures might suggest of little bottom melt, but it probably depends much more on the upward heat flux. The rate of volume loss at the end of April was looking high so the heat flux could be high and this might not have enough effect to raise temperatures with the centrally located low.

There is also the Ekman pumping effect. If the pack is more broken up, perhaps this has more effect than usual.

2012 day 151 volume was 18.186. Where does this leave an estimate for day 151 volume this year?

What has most effect? Cold weather? Lack of heat being brought in by wind from south? Upward heat flux? Ekman pumping?

I don't have much confidence in guessing 17.85. Do other people have guesses (preferably very soon to help me with my extent and area poll guesses later today) ?

[Wipneus]

There was't a poll for volume, was there?

I don't have much confidence in guessing 17.85. Do other people have guesses (preferably very soon to help me with my extent and area poll guesses later today) ?

My guess would be 17.5 --very low, 17.75 -- best guess and 18.0 -- very high.

After you showed min volume = may volume -15 or so, makes the coming PIOMAS update extremely interesting. Hopefully it is in before June 7th.