AMIP II Sea Surface
Temperature and
Sea Ice Concentration Boundary
Conditions
23 June 1997 (revised: 6 April 2001; 8 January 2008)
Karl E. Taylor^{1}, David Williamson^{2}
and Francis Zwiers^{3}
^{1}Program for Climate Model Diagnosis and Intercomparison
^{2}National Center for Atmospheric Research
Climate and Global Dynamics Division
^{3}Canadian Centre for Climate Modelling and Analysis
Atmospheric Environment Service
Table of Contents
Introduction and links to data
Summary
1. Shortcomings
of the "traditional" method
2. Construction
of the AMIP II boundary conditions
3. Comparison of
"traditional" and "new" methods
4. Use of boundary
condition data set in AMIP II simulations
5. Boundary conditions
for suggested spinup procedure
Appendix 1: Mathematical details
Appendix 2: Recipe for obtaining and applying
the AMIP II boundary condition data sets
Appendix 3: Table of midmonth dates and times
Introduction and links to
data
NOTE: this description is somewhat out of date, but provides an accurate
summary of how boundary conditions were prepared until about 2003.
Perhaps the most important boundary conditions for AMIP type experiments
are the prescribed sea surface temperature (SST) and sea ice concentration
(i.e., percentage of area covered by sea ice). Observed monthly mean SSTs
and sea ice concentration have been compiled for the period January 1956
through near present (currently December 2000). This data set will
be updated periodically (at least every 6 months). These observed monthly
means do not constitute the AMIP II boundary conditions, but the boundary
conditions are based on them, as explained in this brief report.
The original AMIP II period extended from January 1979 through February
1996, but simulations need not be strictly limited to this period.
We therefore routinely distribute the observed data for the full period
available.
The AMIP SST and sea ice concentration boundary conditions should be
specified such that the monthly means computed from the model output precisely
agree with the observations. There are several ways to assure agreement,
but if your model relies on linear interpolation between monthly values,
you can obtain from PCMDI the appropriate boundary condition data sets.
There are two ways to proceed:

We strongly recommend that you complete a short
form requesting that PCMDI prepare the boundary condition data sets
specifically for the grid used by your model. On that form you will be
asked to provide some essential information concerning your model grid.
You may also request that PCMDI create an optimal land/sea mask for your
grid. Otherwise, you should provide PCMDI with your own land/sea mask,
which is needed for proper interpolation. Appendix
2 contains a stepbystep procedure for obtaining and applying the
AMIP II boundary conditions.

Alternatively, you may obtain the boundary
condition data sets at the highest resolution available (1x1 degree),
and then interpolate to your own grid. This option is discouraged for several
reasons, as discussed in this report.
If you have already developed a method to generate daily (or higher frequency)
boundary condition data that, when averaged over the month, preserves the
monthly mean, then it may be appropriate to apply your method to the observed
monthly mean SSTs and sea ice concentration data sets available at
PCMDI.
The boundary condition data sets are generated from the observed monthly
means, but to prevent slight modifications to the boundary conditions as
more data become available, the boundary conditions data set will stop
short of the full observational period by 3 months (currently extending
only through September 2001).
The rest of this report contains a more complete description and justification
for the procedure recommended here to prepare and use the AMIP II boundary
condition data sets. A slightly revised version of this report
is part of a series of PCMDI reports and can be cited as follows:
Taylor, K.E., D. Williamson, and F. Zwiers, 2000: The sea surface temperature
and seaice concentration boundary conditions for AMIP II simulations,
PCMDI Report No. 60, Program for Climate Model Diagnosis and Intercomparison,
Lawrence Livermore National Laboratory, Livermore, California, 25 pp .
A printable copy of this report is available as a PDF
file.
Summary
In the initial AMIP simulations (before AMIP II) the observed monthly
mean SSTs were specified for the middle of each month, and in most models
daily SSTs were obtained by linearly interpolating between these monthly
mean values. This is the traditional method for prescribing SSTs in GCM
simulations, and we shall refer to it as such. Under this procedure much
of the SST variance on submonthly timescales is filtered out. Perhaps
less obvious is the fact that the amplitudes of seasonal and interannual
variations in SSTs are also damped under this "traditional" procedure.
In fact the monthly mean SST computed from the linearly interpolated timeseries
will differ in general from the monthly means from which it was derived.
(Consider, for example, the warmest month of the year. In order to recover
the monthly mean value, the maximum temperature in the month must exceed
the mean, but this is clearly impossible if temperatures are linearly interpolated
between monthly mean values.)
In AMIP II the SST and sea ice concentration should be prescribed in
such a way that the correct (observed) monthly means are recovered. Clearly,
a number of different procedures can be devised to satisfy this constraint.
Harzallah and Sadourny (J. Clim., 8, 474495), for example, have
interpolated monthly mean data to daily values by an iterative cubic spline
method that preserves the monthly means. Many models, however, are formulated
such that monthly, not daily, data are accessed, and daily values are then
obtained by linear interpolation. For these models the simplest procedure
is to generate (artificial) midmonth values that, upon interpolation to
daily values, yield the observed monthly means. Here we describe how such
a data set can be constructed and how AMIP II participants can obtain and
use this data set in their simulations. We refer to this procedure of specifying
midmonth values that preserve the observed monthly means as the "new"
method, although variations on it are already in use by some groups. In
particular, Sheng and Zwiers (submitted to Climate Dynamics, 1997)
describe a similar approach to specifying sea surface temperature.
In this report we:

provide evidence that under the "traditional" method followed in earlier
AMIP simulations, the damping of the seasonal and interannual variations
of SSTs is not negligible.

describe a "new" method for generating midmonth values for use in AMIP
(i.e., the AMIP II boundary conditions).

show that compared with the "traditional" method, the time series of daily
values interpolated from the AMIP II boundary condition data is in better
agreement with observations.

provide instructions on how to obtain and apply this procedure for specifying
AMIP II SST and sea ice boundary condition data.

recommend a spinup procedure that minimizes initial transients and describe
the SST and sea ice data that can be used in the spinup period.
Return to Table of Contents
1.
Shortcomings of the "Traditional" Method
As asserted above, linear interpolation of monthly mean values (under
the "traditional" procedure) will damp seasonal and interannual variations.
In Figure 1 we show, for example, that at a location
in the North Pacific, the maximum temperature observed in 1983 was 20.7°C,
but the monthly mean temperature for the warmest month was only 19.6°C.
In this case if the temperatures were linearly interpolated between monthly
mean values in a GCM simulation, then the peak temperatures would be missed
by more than a degree. The particular grid cell and year shown in Figure
1 are used to illustrate the problem, but may not, of course, be typical.
We have, however, analyzed the monthly mean SSTs globally for a 14 year
period from 1982 through 1995. We find that under the traditional procedure
the climatological mean seasonal cycle for that period over much of the
tropics is damped by a few percent (i.e., typically, no more than a few
tenths of a degree), but in some regions of the midlatitudes, it is damped
by a half degree or more. See Sheng and Zwiers (submitted to Climate
Dynamics, 1997) for further evidence of the reduction in amplitude
of the seasonal cycle.
Perhaps more troubling than the fact that temperature extremes are missed
under the traditional procedure, is the fact that the monthly means are
not preserved. It can be shown analytically that the monthly mean temperature
for a given month, T*_{i }is related to the monthly boundary
condition data (T_{i}) by the following formula (assuming
here for simplicity that all months comprise the same number of days):
T*_{i} = (T_{i1} + 6 T_{i} + T_{i+1})
/ 8.
Thus for month i, the monthly mean resulting from this procedure is
a weighted average of the temperature for three months centered on month
i. The monthly mean, T*_{i}, will therefore generally differ
from T_{i}. The process of linear interpolation to daily
values in effect filters the data in such a way as to decrease the variance
of the monthly means on all timescales, but especially at the highest
frequencies. Under the traditional procedure generally followed in earlier
AMIP simulations, T_{i} was taken as the observed monthly
mean, so these monthly means were clearly not preserved.
How large are the differences between T*_{i} and T_{i}?
Figure
2a shows the maximum positive difference between T_{i}
and T*_{i} (when T_{i} is set to the observed
monthly mean values) for icefree regions of the oceans, considering all
months of the AMIP II period (January 1979 through February 1996). In most
places this maximum difference is less than a half degree, but in the North
Pacific and North Atlantic it reaches a degree or more. If the observed
monthly mean values were specified in a GCM simulation, the resulting monthly
mean temperature (after daily interpolation) would therefore differ from
the observed by a few tenths of a degree or more almost everywhere during
at least one month of the AMIP period. The difference shown almost invariably
occurs in the warmest month of the year, so in fact the warmest month under
the traditional method of prescribing SSTs is always too cool. Similarly,
the coolest month of the year is always too warm, but typically the discrepancy
is smaller in this case, as shown in Figure 2b.
The size of monthly temperature anomalies (i.e., the variations in temperature
once the climatological mean seasonal cycle has been removed) is also reduced
under the traditional procedure of interpolating to daily values from monthly
mean SSTs. The fractional reduction in the variance of monthly anomalies
can be calculated:
where T ' is the SST anomaly, the overbar indicates an average
over all months, and R_{1} and R_{2} are
the 1month and 2month lag correlations, respectively, for temperature.
An analysis of the SST anomalies during the AMIP II period gives the
following global mean values for the lag correlations: R_{1 }=
0.67 and R_{2 }= 0.47. According to the above formula,
this implies that under the traditional method of specifying monthly mean
temperatures in GCM simulations, the variance of the monthly mean temperature
anomalies is typically reduced below the observed variance of monthly means
by 14%. Clearly, under this procedure the true strength of the SST anomalies
will not be preserved in the actual forcing of the GCM. If the GCM response
to this forcing scales linearly, then obviously the response will also
be reduced typically by 14%. (For sea ice concentration, the corresponding
number is 18%.)
Return to Table of Contents
2.
Construction of the AMIP II Boundary Conditions
Because of the shortcomings described above concerning the traditional
method of specifying boundary conditions, in AMIP II a new approach to
specifying the boundary condition data sets has been devised. The boundary
conditions prepared by PCMDI were originally based on the monthlymean
observationallybased
dataset maintained by Mike Fiorino
(Fiorino, 1996),
but starting in 2007
these observations were superceded by a dataset described by
Hurrell et al. (2008),
which is maintained by the National Center for Atmospheric Research
(NCAR). The boundary conditions comprise a set of artificially constructed
midmonth temperatures and sea ice concentrations
that on average correspond more closely to the actual midmonth temperatures
than the observed monthly means. More importantly, the boundary conditions
are more realistic in that when they are used in GCM simulations, they
lead to monthly means that are identical to the observed. In contrast,
as indicated in Section 1, the monthly means are not maintained under the
traditional procedure of linearly interpolating between monthly mean values.
The importance of maintaining the observed monthly mean in GCM simulations
has been recognized by several researchers. Notably, Harzallah and Sadourny
(J. Clim., 8, 474495) have interpolated monthly mean data to daily
values by an iterative cubic spline method that preserves the monthly means.
This method may yield a smoother representation of the data than the method
describe here, but may be more difficult to adapt to the architecture of
some climate models. Both methods satisfy the AMIP II boundary condition
requirement that monthly means be preserved when the data are interpolated
to daily values. Many groups may find it easier, however, to continue their
practice of linearly interpolating between midmonth values, and those
groups may therefore prefer our method. Our procedure differs only slightly
from that described by Sheng and Zwiers (submitted to Climate Dynamics,
1997).
To create the AMIP II boundary conditions satisfying the constraints
discussed above, we have:

Calculated a climatological monthly mean for each month of the year, based
on observations Hurrell et al. (2008)
for all the months of the AMIP II time period (i.e., the eighteen January's
and February's and the seventeen March's, April's, May's, etc.). The climatological
monthly means can be used to create a climatological boundary condition
data set for use during model spinup and for use in studies of the causes
of interannual variability. The climatology (along with the temporal autocorrelation
structure of the monthly mean anomalies) can also be used to generate reasonably
realistic monthly data for the months prior to the AMIP II period when
the monthly mean observations are not globally available.

Generated twelve midmonth values that (when interpolated to daily values)
exactly reproduce the observed climatological monthly mean values. These
values can be used during most of the spinup period prior to the AMIP
II simulation period.

Generated midmonth values (for the 206 AMIP II months plus December of
1978 and March of 1996) that (when interpolated to daily values) exactly
reproduce the observed monthly means. (In order to do this, observed monthly
means were required for several months before and for several months beyond
the AMIP II period.)

Generated a data set of midmonth values for 1978 that can be used in the
last year of the model spinup, prior to the AMIP II simulation period.

Written computer code to generate midmonth values appropriate for any
model, given its grid structure and an appropriate landsea mask. (An optimal
landsea mask, based on a U.S. Navy landfraction data set with onesixth
degree latitudelongitude resolution, can also be generated on any model
grid.)
Appendix 1 contains the mathematical details
concerning steps 2, 3, and 4. For sea ice concentration (C), some of the
midmonth values in the boundary condition data set are negative or greater
than 100%. These values are clearly unphysical, but have been deliberately
defined this way so that the correct monthly means will be preserved if,
after a daily value is calculated based on linear interpolation, it is
"clipped", if necessary, by applying the following filter (see a FORTRAN
manual for the function definitions):
C = amax1(amin1(C, 100.0), 0.0)
Similarly for sea surface temperature (T), the following filter will
ensure that the observed monthly mean is preserved and the temperature
is held at or above the freezing point (taken to be 271.38 K).
T = amax1(T, 271.38)
Accurate preservation of the monthly means requires use of a realistic
(Julian or Gregorian) calendar and requires that daily (or higher frequency)
data be generated through linear interpolation between the appropriate
midmonth values, assuming the midmonth values apply precisely at the
middle of each month (i.e., 12Z January 16, 0Z February 15 (for nonleap
years), 12Z February 15 (for leap years), etc.). We note, however, that
for models with unrealistic calendars (e.g., twelve 30day months), shifting
these times by a day or so probably makes little practical difference.
(See Appendix 3 for a complete table of midmonth
dates and times.)
Temperature anomaly data for the period preceding and following the
AMIP II period have been used to minimize the influence of a somewhat arbitrary
mathematical boundary condition that must be applied to solve for midmonth
values. (See Appendix 1.)
Return to Table of Contents
3.
Comparison of "Traditional" and "New" Methods
In Figure 3 we show the observed weekly temperature
timeseries for a threeyear period (19821984) at a location in the North
Pacific. Also plotted are timeseries generated by linear interpolation
from observed monthly mean values (identified as "traditional") and by
linear interpolation from the AMIP II boundary condition data set (identified
as "new"). For 1983 the observed maximum temperature occurs near September
1, whereas both the traditional method and new method can produce maxima
only at the center of a month. Still the new method produces a maximum
value closer to the observed than the traditional method, and, more importantly,
it yields the correct monthly mean.
In order to determine whether in general the new method, compared to
the traditional method, yields daily values in closer agreement with observations,
we have considered the daily time series at each grid cell for the period
January 1982 through December 1995 (during which weekly observed SSTs are
available for computing an approximate daily time series). At each grid
cell we have computed two rootmeansquare (RMS) differences: the RMS difference
between the observed daily values and the daily values computed from the
"traditional" method (E_{t}), and, similarly, the RMS difference
between the observations and the time series resulting from the "new" method
(E_{n}). The RMS difference represents the average error
in the daily temperatures that are used to force GCMs under each of the
methods. Next we computed the fractional difference in these errors between
the "traditional" and "new" methods:
(E_{t}  E_{n}) / E_{n}
This difference, expressed as a percent, indicates by what percent the
error in daily temperatures is reduced by the "new" method. Figure
4a shows that the reduction in error is positive everywhere, indicating
that the "new" method is everywhere better than the "traditional" method.
In the tropics the "new" method reduces the error typically by less than
10%, but in middle latitudes the error is reduced in some areas by more
than 20%. Since the RMS errors are of the order of a degree or so, the
actual reduction in error is quite small, as shown in Figure
4b. Still it is remarkable that the "new" method, which gives perfect
agreement with monthly mean observations, also yields daily temperature
series that are everywhere in better agreement with observations than the
traditional method.
As noted earlier, the variance of monthly mean temperature anomalies
is reduced by 14% under the "traditional" method, whereas the new method
again forces exact agreement with observations. Thus, the new method is
also superior in this regard.
Return to Table of Contents
4.
Use of Boundary Condition Data Sets in AMIP II Simulations
We have described how midmonth values of SST and seaice concentration
can be generated from observed monthly mean data. Here we describe how
to prepare the boundary condition data for use in climate model simulations.
Most of the work can be avoided by simply asking PCMDI to prepare the data
on the grid appropriate for a particular model. Appendix
2 provides a stepbystep procedure for requesting the data and using
it in AMIP II simulations. Here we summarize the PCMDI method for generating
boundary condition data on any model's grid.
The monthly mean AMIP II observed data sets are available on a 1 degree
by 1 degree latitudelongitude grid as described by
Hurrell et al. (2008)
Before this data can be used in models it must be mapped to
a model's (generally coarser) grid and it must be processed, so that interpolation
to daily values preserves the monthly means. There are two ways to proceed:
map the monthly mean data to the model grid and then produce the midmonth
values, or vice versa. The boundary condition data sets resulting from
these alternative procedures will not be identical in regions of seaice,
where, because of the maximum and minimum limits imposed (100% and 0%,
respectively), the problem is nonlinear. Under our procedure, we mapped
the data to the model's grid first. This ensures that when the boundary
condition data are generated on the model's grid, temporal interpolation
will preserve these observed mean values, as required under AMIP II.
It is interesting to see how the monthly mean temperatures calculated
from the boundary condition data depend on the order of interpolation/calculation.
Figure
5a shows for sea ice concentration the rootmeansquare (RMS) difference
between monthly means resulting from interpolating the observed monthly
means first, calculating the midmonth boundary condition data second (i.e.,
the recommended procedure) and vice versa. The differences are small (less
than 4%), but of course the RMS difference is an average over all months.
Figure
5b, on the other hand, shows that for an individual month during the
AMIP period the order of interpolation/calculation can change the monthly
mean by 10% and more (up to 34% at one grid cell for one month).
For SST the monthly mean dependence on order of interpolation/calculation
appears to be less important. Figure 6a shows that
the RMS difference is almost everywhere less than 0.1 K (and never exceeds
0.3 K). Figure 6b shows that at a few grid cells
the maximum difference exceeds 1 K for at least one month of the AMIP period.
Nevertheless, if the objective is to run all AMIP models under the same
boundary conditions (to the extent that this is possible at different resolutions),
we recommend that all groups follow the recommended procedure and obtain
boundary condition data from PCMDI at the resolution needed for their models.
In interpolating data to a model's grid, we use a mapping algorithm
(areaweighted averaging) that preserves area means. Near ocean boundaries
the artificial SST and sea ice data for land grid cells found in the 1x1
degree monthly mean data are generally ignored. (Note that in the observational
data set, the values over land may be reasonable, but in fact they are
extrapolated from nearby ocean regions using a "weaving" algorithm developed
at NCEP.) With our method, if a model's ocean grid cell overlaps a region
that according to the 1x1 degree data set is partly land and partly ocean,
then only values from the 1x1 degree ocean grid cells contribute
to the mean value assigned to the model's grid cell. In this way only real
ocean data are used to produce a data set on the model's grid. If for some
grid cell a model's land/sea "mask" is completely incompatible with the
1x1 degree data (i.e., a model's ocean grid cell contains only land grid
cells as designated on the 1x1 degree grid), then we are forced to use
the artificial land data to estimate the SST and sea ice concentration
for the model. The 1x1 degree land/sea mask used in creating the monthly
mean data is unrealistic in the sense that if only a small fraction of
a cell is ocean, the cell is designated ocean. This tends to exaggerate
the ocean area, but it means that if a model's land/sea mask is reasonably
realistic, then there should be no need to use the artificial land data.
This is also true of model's with fractional ocean area in individual grid
cells.
If a model's land/sea mask is unrealistic, an optimized land/sea mask,
based on a 1/6 by 1/6 degree U.S. Navy data set, can be requested from
PCMDI as explained in Appendix 2. For models
with fractional land area in individual grid cells, the land fraction will
be preserved; for models with traditional landonly or oceanonly grid
cells, the method optimizes the agreement between the model's mask and
the original high resolution data.
Because of the considerable care and effort required to correctly prepare
the AMIP II boundary conditions (and the potential for subtle errors),
we recommend that each modeling group provide information to PCMDI concerning
their model's grid structure and landsea distribution and rely on PCMDI
to produce the boundary condition data sets appropriate to their model.
If PCMDI is asked to generate boundary condition data, the mapping from
the original resolution to the model resolution will preserve areaaverage
temperatures and sea ice concentration. On request, PCMDI will also produce
an optimal land/sea mask. See Appendix 2 for
a stepbystep procedure for obtaining the boundary condition data sets.
If a modeling group elects to create the boundary condition data sets
itself, then it is suggested that rather than proceeding as described above
(i.e., interpolating the monthly mean data to the model grid and then generating
the midmonth values), it is better to obtain the 1 degree by 1 degree
midmonth boundary condition data set available from PCMDI and interpolate
this data to the model's grid. Although in regions of partial sea ice coverage
this procedure will not exactly recover the observed monthly means (obtained
by interpolating the 1x1 degree monthly mean data to the model grid), it
is considerably easier than solving the (nonlinear) problem of creating
a midmonth data set from an observed monthly mean data set.
Once the midmonth boundary condition data set has been obtained, it
will not be difficult to use it in AMIP II simulations. There are, however,
certain details that need to be taken into consideration:

SST and sea ice concentration should be updated at least once each simulated
day by linearly interpolating between the midmonth values.

As noted above, the midmonth boundary condition SST values at high latitudes
may in some months be less than the freezing point, but after interpolating
in time, the filter, T = amax1(T, 271.38), should be applied, preventing
the temperature specified in the model from falling below freezing (even
if the linearly interpolated value does). (n.b.: We have generated the
midmonth values such that if this algorithm is followed, the correct monthly
means will result). For this purpose the freezing point of sea water is
taken to be 271.38 K.

For sea ice a similar approach is taken. For models with fractional sea
ice concentration, the midmonth values will again be used with an algorithm
for linear temporal interpolation and limits placed at 0% and 100% coverage.
(Here the midmonth values may in a few cases exceed 100% or be negative,
but if after interpolation the sea ice concentration (C) is "clipped" by
filtering through the FORTRAN statement, C = amax1(amin1(C, 100.0), 0.0),
then the correct monthly mean concentration will be recovered.)

For models with sea ice that either completely covers a grid cell or is
completely absent (i.e., models without fractional sea ice coverage), the
best procedure is to again read in the midmonth sea ice concentrations,
and then proceed as above, but with a final step added, whereby a threshold
value of 50% is used to determine whether or not each grid cell is actually
covered by sea ice. This test should be carried out at least once each
day after interpolating linearly between midmonth values. It is not recommended
that a flag in the SST data set (such as the value 1.8°C) be used
to determine whether or not sea ice is present (as has been historically
the practice in some models). This may therefore require some groups to
modify their computer codes to read in the monthly sea ice concentration
data set in addition to the SSTs.
Return to Table of Contents
5.
Boundary conditions for suggested spinup procedure
Before beginning the simulation of the AMIP II time period, the models
should be spunup in such a way as to minimize initial transients. PCMDI
has created a climatological SST and sea ice data set (the midmonth values
created in step 2 of Section 2) that can be used as a boundary condition
during most of the spinup period. In order to avoid an initial shock when
the observed SST anomalies are first imposed at the beginning of the AMIP
pe riod, a special transition boundary condition data set has been prepared
(see step 4 of Section 2) for use during the last year of the spinup period
(i.e., a representation of the year 1978). This data set of midmonth values
has been constructed based on climatology and on artificially generated
monthly SST anomalies that are initially 0, but gradually approach the
monthly anomalies observed in December 1978 (the first month where adequate
global data are available). The artificially generated anomalies have an
autocorrelation structure that agrees approximately with the observed (i.e.,
for SST on a 3 degree grid, a 1month lag correlation of 0.69, 2month
lag correlation of 0.47, and similarly for 3, 4, 5, ... 12 month lags,
correlations of 0.34, 0.27, 0.21, 0.17, 0.14, 0.12, 0.09, 0.06, 0.03, 0.00,
respectively, and for sea ice concentrations, lag correlations of 0.58,
0.27, 0.15, 0.09, 0.06, 0.03, 0.0, 0.0, 0.0, 0.0, 0.0, and 0.0). The temporal
autocorrelation structure depends somewhat on location and on spatial resolution,
but not enough to significantly affect the boundary conditions in the months
needed for AMIP II.
This method of constructing anomalies prior to December of 1978 is theoretically
more realistic, but probably differs little in practice from simply linearly
ramping up to the December anomalies over a few month period. In any case
the 1978 data set is artificial and only based on climatology, the December
1978 temperature anomalies, and the temporal autocorrelation structure
of the temperature anomalies.
Return to Table of Contents
Appendix 1: Mathematical Details
Here we outline the mathematical procedure for calculating midmonth
SSTs that, when linearly interpolated to generate a continuous timeseries,
will yield monthly mean temperatures that are in exact agreement with the
observed monthly mean SSTs. For simplicity we describe the procedure under
the approximation that all months are of equal length (say 30 days), but
in the actual creation of the AMIP II boundary condition data set, months
comprised the correct number of days, and the equations given here were
appropriately modified. In addition, the complications of physical limits
on the SSTs (i.e. the freezing point) or sea ice concentration (i.e. nonnegative
values no greater than 100%) are not included in this summary, but again
were correctly accounted for in constructing the AMIP II boundary conditions.
(Note that placing lower and/or upper limits on the data introduces a nonlinearity
that requires an iterative method for solution.)
As will be shown below, solution for the AMIP boundary condition data
requires specification of monthly mean data for several months preceding
and several months following the AMIP period (January 1979  February 1996).
Since good data are not available prior to December 1978, artificial data
were generated for those months, based on the monthly mean climatological
data (as described further below). In addition, the climatological data
are needed for model spinup. The first step is then to generate a climatological
monthly mean boundary condition data set. This data set, comprising twelve
midmonth values for each grid cell, is generated from the observed climatological
monthly mean data as follows (described here with specific reference to
sea surface temperature, but applicable also to sea ice):
Let S_{i} be the observed mean SST for month i
(i=1,I, where I=206 is the number of months in the AMIP II
period: January 1979 through February 1996).
Let T_{i} be the midmonth SST for month i, specified
such that the monthly mean temperatures calculated from the timeseries
resulting from linear interpolation of these midmonth values are identical
to the observed monthly mean SSTs. T_{i} will constitute
the AMIP II SST boundary condition data set.
Let an overbar represent a climatological monthly mean value and a prime
an anomaly (relative to the climatological monthly mean value) such that
and similarly for T_{i}.
The climatological monthly mean is defined as
and similarly for ,
where K is the number of years in the AMIP II period. For January
and February (j = 1 and j = 245), there are 18 years in the
AMIP period (K=18), and for all other months there are 17 years
(K=17). Note that if a grid cell is completely covered by sea ice,
the SST is interpreted to be the temperature of the ocean just below the
ice and is set (following
Hurrell et al., 2008)
to 271.38 K. This temperature will contribute to the climatological monthly
mean temperature.
Setting the observed climatological monthly mean temperatures to
the monthly means obtained after linearly interpolating from the climatological
midmonth ,
we obtain the following set of equations:
and a cyclic boundary condition:
Given the observed climatological mean SSTs, these equations can be
solved for the midmonth temperatures that constitute the climatological
boundary conditions:
The above climatological boundary condition data should be used in the
initial spinup period, as described in Section
5. For AMIP II simulations, however, SSTs should be specified that
include the interannual variability and are consistent with the observed
monthly mean data for the period January 1979 through February 1996. The
midmonth temperatures for this period are generated similarly to the climatological
boundary condition data, but simple application of a cyclic boundary condition
is no longer appropriate. The set of equations to be solved is:
(1/8) T _{i 1} + (3/4) T _{i} + (1/8) T _{i
+1} = S_{i} i=1, I
where I = 206. We have I equations and I + 2 unknowns
(noting that T_{0 }and T_{I+1} appear in
the equation for i = 1 and i = I, respectively). Two additional
constraints are needed to close the problem mathematically. We note that
however T_{0} and T_{I+1} are determined,
they primarily influence the temperatures generated for months 1
and I, respectively. They have less influence on the temperatures
generated for months 2 and I1, and their influence decreases
exponentially for months further away from the beginning and end of the
period. If observed monthly mean data were available for many months prior
to the AMIP period and for many months following the AMIP period, then
one could simply extend the period for which the above equation applied,
and one could then prescribe the temperature prior to the beginning and
following the end of this extended period to be equal to the climatological
monthly mean, knowing this would not significantly affect the midmonth
boundary condition data for the AMIP period. The problem is that good global
monthly mean data are not available prior to December 1978. We have therefore
created an artificial set of monthly mean data for the months from January
through December 1978, and have also extended the data set beyond the end
of the AMIP II period.
Prior to December 1978, artificial "observed" monthly mean values were
prescribed. The artificial data were generated such that temperature anomalies
smoothly approach the actual anomalies at the beginning of the AMIP period,
but approach zero (i.e., the monthly means approaches climatology) within
1 year preceding the AMIP period. Thus for January 1978 through November
1978, the monthly mean data were specified as:
where S_{i} is the monthly mean temperature for month
i
preceding December 1978,
is the climatological mean temperature for month i, S'_{0}
is the observed temperature anomaly (with climatological mean removed)
for December 1978, and r_{i} is the global mean of the auto
correlation coefficient (with a lag of i months) calculated from
the time series for the AMIP period at each grid cell. (Values are given
in Section 5.)
Following the end of the AMIP period, observed monthly mean data were
available through August of 1996, and these were supplemented by a full
year of data generated artificially using a similar procedure to that described
above. Finally, four more months of data (corresponding to September 1997
through December 1997) were prescribed to be equal to the climatological
monthly mean values.
Through this procedure a 20year hybrid data set of observed and artificial
monthly mean data was obtained, covering the period January 1978 through
December 1997. This data, along with a cyclic boundary condition (in effect
setting the month prior to January 1978 equal to the value for December
1997 and setting the month following December 1997 equal to January 1978),
were used to generate the AMIP II boundary condition data sets.
There is one final detail concerning the calculation of the artificial
monthly mean for a month in which the climatological value is near one
of its limits. In this case the equation given above involving the lag
correlation coefficient is applied, but the resulting value is "clipped,"
if necessary, to avoid exceeding the limits.
We reiterate that the artificial data used to generate the AMIP II boundary
conditions have only a small effect on the AMIP period. Analysis indicates
that in general an error in the monthly mean data has a large effect on
the midmonth value for the month where the error occurs, but the effect
of this error on the adjacent months is diminished by a factor of about
6. The effect on the next nearest neighbors is diminished even further,
by another factor of about 6, so the error does not propagate significantly
to later or earlier times. We might therefore expect that the use of artificial
data for months preceding the AMIP period will only affect the first few
months of the simulation, and the effect on those months will be small.
Following the end of the AMIP II period, we used observed data through
August of 1996 (six months after the last AMIP month), so we do not expect
the artificial data following these months to have any noticeable effect
on any of the AMIP months.
In the above derivations, we assumed that all months were of equal length.
In creating the actual AMIP II boundary condition data sets, we did not
make this assumption. Months were defined according to the Gregorian Calendar.
This requires minor changes in the coefficients in the equations being
solved (i.e., changes in the coefficients, 1/8 and 3/4, which depend on
the month).
Also in the above derivation, the values for temperature (or sea ice)
were not constrained to lie within certain limits. In fact the ocean temperature
should never drop below freezing and sea ice concentration should never
be negative or greater than 100%. If these constraints are imposed, then
the linear equations given above no longer apply everywhere. In the case
of temperature, if linear interpolation from midmonth values yields a
value less than 271.38 K, the value is set to 271.38 K, so that when a
monthly mean is computed, the simple 1/8, 3/4, 1/8 weighting of midmonth
temperatures is no longer correct. In general the coefficients depend on
the midmonth temperatures and the equations become nonlinear. This leads
to a number of complications, but the equations can be solved using an
iterative technique as long as the following additional constraint is imposed
(which eliminates the possibility of more than one solution for a given
set of observed monthly mean data): If for some month the observed monthly
mean is very near its upper limit (e.g., a grid cell is completely icecovered
for an entire month), then the midmonth value is set to the minimum value
that (when interpolated in time and "clipped" as described in Section 4
and Appendix 2) will yield the correct (i.e., observed) monthly means.
This midmonth value will always be no less than the maximum allowed monthly
mean. A similar constraint is imposed when the observed monthly mean is
equal to its lower limit.
In the case of sea ice, an additional complication arises when a grid
cell is completely icecovered in one month but becomes completely icefree
the next month. In this case the midmonth value would have to approach
negative infinity in the first month and positive infinity in the next
month in order for linear interpolation followed by the "clipping" filter
to yield the correct monthly mean. In this case unacceptably large "ringing"
can also occur in neighboring months. It is of course highly unusual for
sea ice to completely cover a grid cell through an entire month and then
completely melt at the very end of that month and become completely ice
free for the entire next month. Nevertheless, the observed monthly mean
data set does contain occasional inland grid cells (which have been assigned
artificial data through an extrapolation technique described in
Fiorino, (1996)
with nearly complete ice cover in one month followed by nearly
complete melting at or just following the end of the month. To treat this
case the data are temporally filtered in such a way as to preserve the
timemean over the two months, but preventing abrupt jumps from one limit
to the other. Specifically, if the change in the sea ice concentration
from one month to the next exceeds 96%, then a small amount of sea ice
is moved from one month to the other to reduce the difference to less than
96%. This is done in such a way as to preserve the mean over the two months.
These smoothed monthly means are then used in calculating the midmonth
AMIP boundary condition values.
Return to Table of Contents
Appendix 2: Recipe for Obtaining
and Applying the AMIP II Boundary Condition Data Sets
The AMIP II SST and sea ice concentration boundary conditions should
be specified such that the monthly means computed from the model output
precisely agree with the observations as given by Fiorino
(1996). There are several ways to assure agreement, but if your model
relies on linear interpolation between monthly values, we strongly recommend
that you obtain from PCMDI the data sets AT THE RESOLUTION OF YOUR MODEL,
not the original 1x1 degree resolution. The following procedure should
be followed (but if you elect not to follow this recommendation, the next
best alternative is to obtain the boundary condition midmonth data sets
on the 1x1 degree grid and proceed as explained
in Section 4):
Fill out the form (or send an email message
to ktaylor@pcmdi.llnl.gov) requesting the midmonth boundary condition
data sets and answering the following questions:

What is your model's grid structure? (sample answer: T42 spherical harmonic
gaussian grid with a longitude/latitude resolution of 128x64 and with the
the first longitude grid point, i.e., the center of the first grid box
for evenly spaced grids) located at 0 degrees East, and with the first
latitude grid point at 87.8638 degrees.)

Does your model accommodate fractional land coverage in individual grid
cells, or does it use a "binary" land/sea mask?

Do you want PCMDI to create an optimal land/sea mask for your grid or do
you prefer to use your own land/sea mask? If you plan to rely on your own
land/sea mask, please send either an ascii file containing the land fraction
(expressed as a percentage) for each grid cell or a file containing the
same information written by LATS.
If you send an ascii file with the land percentage data, please put it
in the following format:
write(*,'(a80)') name
write(*,'(2i5)') ii, jj
write(*,'(10f8.3)') ((sftl(i,j), i=1,ii), j=1,jj)
where "name" should be a string containing information that identifies
the modeling group and model (e.g., "NCAR ccm3"), "ii" and "jj" are the
longitude and latitude dimensions, respectively, and "sftl" is an array
containing the percent land in each grid cell (0.000 or 100.000 for models
without fractional coverage). The first longitude in the array,
sftl, should coincide with the Greenwich meridian (or the first grid point
to the east of the Greenwich meridian if there if no grid point coincides
with 0 degrees), and longitudes should be stored from west to east. Latitudes
should be stored from south to north. Please send a note to PCMDI (ktaylor@pcmdi.llnl.gov)
indicating how your land/sea mask data can be retrieved (e.g., provide
an anonymous ftp address).

Which of the following file types do you prefer for the data sets that
you will receive from PCMDI: netcdf (COARDS compatible), grib (also readable
by grads), drs, or ascii? If netcdf or grib files are requested, they will
be written with the LATS
utility. In all SST and sea ice concentration files, grid cells that are
entirely land will have a sea ice concentration of 0.0 and an SST value
of 0.0.
You will be notified when the data sets appropriate for your model's grid
have been prepared and placed on anonymous ftp. (This should take only
a few days.) The following files will be found:

SST and sea ice concentration midmonth boundary condition data (for the
AMIP period and the preceding year: January 1978  March 1996).

SST and sea ice concentration midmonth boundary condition climatology
based on the period January 1979  February 1996.

SST and sea ice concentration observed monthlymean data (for model verification
from January 1979  February 1996).

SST and sea ice concentration observed climatological monthlymean data
based on the period January 1979  February 1996.

Optimal land/sea mask or percentage land in each grid cell (if requested).
A "readme" file will accompany the data with a full explanation of
how to read the data, along with a sample FORTRAN code segment for reading
the data.
In applying the boundary condition data in AMIP II simulations, you should:

Read in monthly SST and sea ice concentration midmonth boundary condition
data as needed.

Interpolate linearly in time every model timestep (or at least once each
day) between the appropriate midmonth values. (Note that we have found
that interpolating once each day, rather than continuously, introduces
errors in the monthly mean that are quite small: for SST, maximum errors
of less than 0.01 K, and for sea ice concentration, maximum errors of less
than 0.1%).

For SSTs "clip" the data if necessary, applying the filter T = amax1(T,
271.38), to prevent temperatures below freezing. Do this test after interpolating
in time.

For sea ice concentration "clip" the data if necessary, applying the filter
C = amax1(amin1(C, 100.0), 0.0) to prevent concentrations less than 0%
or greater than 100%. Do this test after interpolating in time.

For models with grid cells that are either completely sea ice covered or
completely ice free (i.e., models without fractional sea ice coverage)
use a threshold value of 50% to determine whether or not sea ice is present.
Do this test after interpolating in time.
A suggested spinup procedure is as follows:

Simulate several years under the climatological boundary condition
data set (supplied by PCMDI), until all transients die out.

Simulate one year (1978) under an artificially generated boundary condition
data set (supplied by PCMDI), with anomalies that are initially zero, but
gradually approach the monthly anomalies observed in December 1978 (the
first month with adequate global data).
Return to Table of Contents
Appendix 3: Table of MidMonth
Dates and Times
Month 
NonLeap Year
Time/Date Day*

Leap Year
Time/Date Day*

January
February
March
April
May
June
July
August
September
October
November
December 
12Z/16
0Z/15
12Z/16
0Z/16
12Z/16
0Z/16
12Z/16
12Z/16
0Z/16
12Z/16
0Z/16
12Z/16

15.5
45.0
74.5
105.0
135.5
166.0
196.5
227.5
258.0
288.5
319.0
349.5

12Z/16
12Z/15
12Z/16
0Z/16
12Z/16
0Z/16
12Z/16
12Z/16
0Z/16
12Z/16
0Z/16
12Z/16

15.5
45.5
75.5
106.0
136.5
167.0
197.5
228.5
259.0
289.5
320.0
350.5

*Number of days following 0Z January 1.
Return to Table of Contents
Fig. 1: Observed weekly temperature near 180 W 40 N for the years 1982,
1983 and 1984. Also shown is a temperature timeseries generated by interpolating
linearly between monthly mean values.
Fig. 2a: Maximum difference found between T and T* (i.e.
the difference between the observed monthly mean and the monthly mean calculated
from daily values obtained by linearly interpolating between observed monthly
means) for the AMIP II period (January 1979 through February 1996). Units:
K.
Fig. 2b: As in Figure 2a, but the maximum negative difference found.
Fig. 3: Observed daily temperature near 180 W 40 N for the years 1982,
1983 and 1984. Also shown is a temperature timeseries generated by interpolating
linearly between monthly mean values (the "traditional" method) and a temperature
time series generated by interpolating between midmonth values constructed
according to the "new" method.
Fig. 4a: The percent decrease in RMS error achieved using the "new"
method in comparison to the "traditional" method. All values are positive,
which demonstrates improvement everywhere. See text for further explanation.
Fig. 4b: The actual reduction in the RMS error achieved using the "new"
method in comparison to the "traditional" method. See text for further
explanation. (Units: K)
Fig. 5a: The RMS difference in monthly mean sea ice concentration resulting
from midmonth boundary condition data that have been generated in two
different ways: 1) areaweighted mapping of 1x1 degree observed monthly
mean data to a 4x5 degree grid followed by calculation of midmonth values,
and 2) calculation of midmonth values from the 1x1 degree observed monthly
mean data followed by areaweighted mapping to a 4x5 degree grid. The first
procedure is the one recommended in this report. (Units: K)
Fig. 5b: The maximum difference in monthly mean sea ice concentration
resulting from midmonth boundary condition data that have been generated
in two different ways: 1) areaweighted mapping of 1x1 degree observed
monthly mean data to a 4x5 degree grid followed by calculation of midmonth
values, and 2) calculation of midmonth values from the 1x1 degree observed
monthly mean data followed by areaweighted mapping to a 4x5 degree grid.
The difference shown is (1)  (2), for the month with the largest difference.
The first procedure is the one recommended in this report. (Units: K)
Fig. 6a: The RMS difference in monthly mean sea surface temperature
(SST) resulting from midmonth boundary condition data that have been generated
in two different ways: 1) areaweighted mapping of 1x1 degree observed
monthly mean data to a 4x5 degree grid followed by calculation of midmonth
values, and 2) calculation of midmonth values from the 1x1 degree observed
monthly mean data followed by areaweighted mapping to a 4x5 degree grid.
The first procedure is the one recommended in this report. (Units: K)
Fig. 6b: The maximum difference in monthly mean sea surface temperature
(SST) resulting from midmonth boundary condition data that have been generated
in two different ways: 1) areaweighted mapping of 1x1 degree observed
monthly mean data to a 4x5 degree grid followed by calculation of midmonth
values, and 2) calculation of midmonth values from the 1x1 degree observed
monthly mean data followed by areaweighted mapping to a 4x5 degree grid.
The difference shown is (1)  (2), for the month with the largest difference.
The first procedure is the one recommended in this report. (Units: K)
Return to AMIP home page
Last updated April 6, 2001. For further information, contact Karl
Taylor.
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