**Appendix E
Cryosphere Extrapolations**

EXTRAPOLATION BY GENERALIZED LINEAR MODEL

Extrapolations into the future, based on observational records, were done separately for the three categories of ice sources (glaciers and ice caps, Greenland Ice Sheet, and Antarctic Ice Sheet) using a generalized linear model (GLM) approach. The data were assumed to have a normal distribution, and the parameters of the model were estimated using a weighted least squares approach, which allows data uncertainty to be incorporated (McCullagh and Nelder, 1989). This is especially important because there are multiple data sets for each category with different estimates of data error. Traditionally, the data uncertainty is discarded in fitting the models, thus exaggerating the trends in the face of uncertain information.

For each data set in a category, the weighted least squares method was applied to obtain a robust linear model. The linear model is

*Y = Xβ + ε,*

where *Y* is the vector of response variable (ice mass loss rate per year), *X* is the matrix of the dependent variables (in this case, the intercept represented as a column of 1 and year), *β* is the vector of model coeffcients (in this case *β _{0}* and

*β = (X ^{T} W X)^{-1}(X^{T} W Y)*.

Here *W* is a diagonal matrix containing the uncertainty of each observation. It is populated as 1/*d _{i}*, where

The variance of the error is obtained from standard linear model theory (McCullagh and Nelder, 1989):

where *N* is the number of observations used in the model fitting and *p* is the number of model coeffcients.

With the fitted model, estimate of the response variable for any time *i* is obtained asand the 95 percent prediction interval (or prediction uncertainty) is obtained as

It is apparent from the above equation that the intervals tend to widen as the extrapolation domain extends further from the observations. The term within the curly bracket is the standard error of the estimate *Ŷ _{i}*

**Multi-Data Averaging**

Several independent data sets are available for loss rates from glaciers and ice caps as well as for the Greenland and Antarctic ice sheets (data sources are described below). The following steps were performed to obtain a multi-data averaged estimate and standard error for each category of ice source. First, a weighted least squares based linear model was fit for each data set for a given category of ice source *c* following the

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Appendix E
Cryosphere Extrapolations
EXTRAPOLATION BY GENERALIZED Here W is a diagonal matrix containing the uncer-
LINEAR MODEL tainty of each observation. It is populated as 1/di, where
di is the error associated with each observation point i.
Extrapolations into the future, based on observa- The variance of the error is obtained from standard
tional records, were done separately for the three cat- linear model theory (McCullagh and Nelder, 1989):
egories of ice sources (glaciers and ice caps, Greenland
(Y - Y^ )
2
N
Ice Sheet, and Antarctic Ice Sheet) using a generalized i =1
2
=
linear model (GLM) approach. The data were assumed (N - p)
to have a normal distribution, and the parameters where N is the number of observations used in the
of the model were estimated using a weighted least model fitting and p is the number of model coefficients.
squares approach, which allows data uncertainty to be With the fitted model, estimate of the response
incorporated (McCullagh and Nelder, 1989). This is variable for any time i is obtained as Y^ = X , and the
i i
especially important because there are multiple data sets 95 percent prediction interval (or prediction uncer-
for each category with different estimates of data error. tainty) is obtained as
Traditionally, the data uncertainty is discarded in fitting
the models, thus exaggerating the trends in the face of
uncertain information.
Y i {
^ ± 1.96 1 + X T ( X T X )-1 X .
i i }
For each data set in a category, the weighted least It is apparent from the above equation that the intervals
squares method was applied to obtain a robust linear tend to widen as the extrapolation domain extends fur-
model. The linear model is ther from the observations. The term within the curly
bracket is the standard error of the estimate Y ^.
i
Y = X + ,
Multi-Data Averaging
where Y is the vector of response variable (ice mass loss
rate per year), X is the matrix of the dependent variables Several independent data sets are available for
(in this case, the intercept represented as a column of 1 loss rates from glaciers and ice caps as well as for the
and year), is the vector of model coefficients (in this Greenland and Antarctic ice sheets (data sources are
case 0 and 1), X is the model estimate ( Y ^ ), and is
described below). The following steps were performed
the error assumed to be normally distributed with mean to obtain a multi-data averaged estimate and standard
zero and variance 2. Using weighted least squares, error for each category of ice source. First, a weighted
is estimated as least squares based linear model was fit for each data
set for a given category of ice source c following the
= (XT W X)-1(XT W Y).
191

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192 APPENDIX E
method described above. Second, the multiple esti- The second and third steps were repeated for all
mates were combined using a weighted averaging ap- of the projection years. The mass loss rates and the
proach in which each estimate was weighted according interval were subsequently converted to sea-level rates
to its standard error. In the following discussion, the and then cumulatively summed.
subscript ic refers to a variable within category c at
time i. Suppose that, for a category, K data sets are RAPID DYNAMIC RESPONSE
available; thus, for any year i in the future, K estimates
(Y ij )
^ , j = 1, 2, K can be obtained along with their Simple extrapolation of existing trends will not
standard errors ( eij , j = 1, 2, K ) . A normalized set of capture the effect of rapid dynamic response that begins
weights is computed as after the period of observation. The committee calcu-
lated the effects of both acceleration and deceleration
1 1 1
m = + ++ in ice discharge relative to observed present-day rates,
ei 1 ei 2 eiK
as described below. The term "rapid dynamic response"
and is defined here as mass changes in a glacier or ice sheet
that occur at rates faster than accompanying climatic
1
^ j = m -1
w mass balance and which force glacier or ice sheet condi-
eij tions further away from equilibrium with climate.
for j = 1,2...K. The weighted estimate is obtained as
K Increases in Dynamic Discharge
Yic = Y^w
ij
^j
j =1 To simulate the effect of rapid dynamics, supple-
and its standard error as mentary ice fluxes were added to the loss rates deter-
KN
mined by extrapolation. The parameters for the added
eic = e w
^ 2
ij
2
j . rapid dynamic response are summarized in Box E.1.
j =1
The choice of dynamic variations was intended to
The 95 percent confidence interval of this weighted capture the general magnitude of plausible changes.
estimate is provided as Although these particular events may not occur, the cal-
culations provide a means to quantitatively estimate the
Yic = ±1.96 eic . influence of rapid dynamic response on sea-level rise
and to translate ranges of plausible future glaciological
These calculations were applied for each of the changes into equivalent sea-level changes.
three ice categories, yielding a multi-data averaged The range of added rapid dynamic response for
estimate and standard error for each. each ice source for each projection period is given in
Third, the global ice mass loss rate for any year i Table E.1, and the effect of the simulated rapid dynamic
and error were calculated. The global ice mass loss rate response on the projections, summed for all three sources,
was estimated as is shown in Table E.2. The top rows ("base values") of
C
Table E.2 show the integrated cumulative sea-level rise
Yi = Yic from the extrapolation and the low and high values based
c =1 on uncertainties in the extrapolation. The middle rows
and the standard error as of Table E.2 show the effect of additional rapid dynamic
C
response on the projections of sea-level rise. The bottom
ec = e ic
2
. rows ("percentage effect") in Table E.2 show the effect
c =1 of added dynamics expressed as a percentage of total
The 95 percent confidence interval of this multi-data sea-level rise. Rapid dynamic response is not an insig-
averaged global mass loss rate is nificant factor in future sea-level rise, but according to
this simple analysis, it is also not a "wild card" variable
Yic = ±1.96 eic . that will swamp all other sources if it comes into play.

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APPENDIX E 193
BOX E.1
Parameters for Added Rapid Dynamic Response
Added dynamics (additional discharge assigned to each land ice source for simulation of increased rapid dynamic contribution)
· Glaciers and ice caps: 50 percent of 324.8 GT yr-1 = 162.4 GT yr-1
· Greenland Ice Sheet: Increase outlet glacier speed by 2 km yr-1 = 375.1 GT yr-1
· Antarctic Ice Sheet: Double outlet glacier discharge to 264 GT yr-1
Variations (perturbations to base values selected for discharge used in the sensitivity calculation)
· Glaciers and ice caps: Use 30 percent and 70 percent of 324.8 GT yr-1
· Greenland Ice Sheet: Use 80 percent and 120 percent of 375.1 GT yr-1
· Antarctic Ice Sheet: Use 80 percent and 120 percent of 264 GT yr-1
TABLE E.1 Range of Added Rapid Dynamic Response (cm) for the Cryosphere Components of Sea-Level Rise
Term 2030 2050 2100
Glaciers and ice caps 00.5 01.4 03.7
Greenland 01.2 03.3 08.4
Antarctica 00.8 02.3 05.9
Total cryosphere 02.5 07.0 018.0
TABLE E.2 Effect of Rapid Dynamic Response and Uncertainty on Future Cumulative Sea-Level Rise
2030 2050 2100
Base values: Projected sea-level rise Z with uncertainty dZ (cm)
Z 6.6 17.7 57.0
Z - dZ 5.9 14.1 44.2
Z + dZ 7.3 19.0 69.6
Projected sea-level rise with added dynamics Zd (cm)
Zd 2.5 7.0 18.0
Z + Zd 9.1 24.7 75.0
Z - dZ + Zd 8.4 21.1 62.2
Z + dZ + Zd 9.8 26.0 87.6
Percentage effect of added dynamics
Z + Zd 38% 40% 32%
Z - dZ + Zd 42% 50% 41%
Z + dZ + Zd 34% 37% 26%
Sensitivity of the Added Dynamics Analysis from glaciers and ice caps, Greenland, and Antarctica)
were varied by ± 20 percent individually (no two inputs
The choice of fluxes for added rapid dynamic were varied at the same time), and the calculation was
response (Box E.1) was guided by the analogy to the repeated to determine the response in the output (ad-
doubling of the Greenland mass balance deficit in ditional sea-level rise). The result of this sensitivity test
20002006 but was otherwise rather arbitrary. To in- is summarized in Table E.3. For variations of 20 percent
vestigate how sensitive this calculation is to the choice magnitude in the inputs, output magnitudes varied by
of added discharge flux, the input variables (discharge no more than 7 percent.

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194 APPENDIX E
TABLE E.3 Sensitivity of Rapid Dynamic Response Estimate to the Choice of Parameters
Percentage Change mm Sea-Level Rise
Glaciers and ice caps
30 percent of 324.8 GT yr-1 0.99 179
70 percent of 324.8 GT yr-1 1.01 182
Greenland Ice Sheet
80 percent of 375.1 GT yr-1 0.9 165
120 percent of 375.1 GT yr-1 1.09 197
Antarctic Ice Sheet
80 percent of 264 GT yr-1 0.93 169
120 percent of 264 GT yr-1 1.07 192
NOTE: Input parameters were varied by ± 20 percent individually (only one parameter was varied at a time).
TABLE E.4 Effect of Reduced Greenland Dynamic Discharge on Sea-Level Rise Projections
Year Total Sea-Level Rise (mm) Cryosphere Component (mm)
Base-Rate Projection (From Table 5.2)
2030 135 81
2050 280 180
2100 827 584
50 Percent Slowdown in Greenland Dynamic Discharge
2030 128 76
2050 273 168
2100 774 535
Percent Change
2030 -5% -6%
2050 -3% -7%
2100 -6% -8%
Decreases in Dynamic Discharge EFFECT OF SEA-LEVEL FINGERPRINT
To test the effect of decreased dynamic discharge, The influence of melting from Alaska, Greenland,
the projected output of the Greenland Ice Sheet was and Antarctica on regional sea level was described in
reduced by 25 percent from its projected base value and "Sea-Level Fingerprints of Modern Land Ice Change"
all other cryosphere terms were left unchanged. The in Chapter 4. To estimate this effect on projected future
results are summarized in Table E.4. The table shows sea-level rise, land ice loss rates were subdivided into
the cumulative sea-level rise (central value only) for Alaska, Greenland, Antarctic, and all other glacier and
2030, 2050, and 2100 for both the base rate projection ice cap losses other than Alaska. The fingerprint scale
(Table 5.2) and for a 50 percent reduction in Greenland factors for Alaska, Greenland, and Antarctica (specified
calving discharge (equivalent to a 25 percent reduc- in Table 4.1) were then applied to losses from those re-
tion in overall Greenland discharge). The cryosphere gions, on a year-by-year basis, and the losses from other
component totals are for the Greenland and Antarctica glacier and ice cap regions were carried forward without
ice sheets, glaciers, and ice caps, and the total sea-level adjustment. The projected sea level DZ at destination
rise includes the steric component. The percent change region p and time t is then, using the notation defined
rows show that reducing the Greenland discharge af- in Appendix C:
fects sea-level projections by a maximum of 8 percent.

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APPENDIX E 195
3
Z p ( t ) = s k , p Rk ( )d + RGIC - AK ( t ),
t t
0 0
k =1
where R is rate of ice loss (e.g., in GT yr-1), s is the
fingerprint scale factor, k =1,2,3 is the set of source
locations (Alaska, Greenland, and Antarctica), p = 1,2,3
is the set of destination locations (north coast, central
coast, south coast), and t is time. The term RGIC-AK
designates the loss rate from all glacier and ice cap
regions with the exception of Alaska.
REFERENCE
McCullagh, P., and J.A. Nelder, 1989, Generalized Linear Models,
2nd edition, Chapman and Hall, London, 532 pp.

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