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**CFS Retrospective Forecast Daily Climatology**

**in the EMC/NCEP NOMAD public server**

Ǻke Johansson, Catherine Thiaw and Suranjana Saha,

Environmental Modeling Center, NCEP/NWS/NOAA

Email for all correspondence : <cfs@noaa.gov>

July 2006

**1. Introduction**

The new Climate Forecast System (CFS), which is being run operationally
out to 9 months on a daily basis at NCEP since August 2004 (Saha et al.
2006), is regularly used by a large number of users. It has been
recognized that in many applications there is a need of a daily
climatology that is available at all available forecast lead times. This
is because there is a substantial climate drift in the CFS forecasts. A
smooth daily climatology of the annual cycle has therefore been
prepared. The set of 4320 retrospective forecasts -- the CFS hindcast
data set -- which constitute an integral part of the CFS, has been used
for this purpose. Since the CFS hindcast data set spans the 24 years
from 1981 to 2004, the climatology is based on 24-yrs instead of the WMO
standard of 30-yrs.

**2. Variables and spatial and temporal domains**

The climatology is calculated for a subset of all the variables produced
by the CFS.

The number of atmospheric variables considered is 30 and they are listed
in Table 1. The forecast data is of global extent and are given on a
regular latitude-longitude grid of resolution 2.5°×2.5° (somewhat
coarser than the native T62 Gaussian grid), i.e., at 73 · 144 = 10.512
grid points.

The number of oceanic variables considered is 12 and they are listed in
Table 2. The forecast data is available over the oceans between the
latitudes 74°S and 64°N and are given on a regular latitude-longitude
grid of resolution 1°×2° (somewhat coarser than the native ocean model
grid), i.e., at 139 · 180 = 25.020 grid points. Around 2/3 of these grid
points are over the oceans.

**WARNING**. Oceanic variables at high northern and southern latitudes,
poleward of 50ºN and 65ºS, are of lower quality compared to the interior
of the domain. The reason for this is partly because of the proximity to
the imposed climatological lateral boundary conditions at 64°N and 74°S,
and partly because of the restricted atmosphere-ocean coupling at high
latitudes (for further details see Saha et al. 2006). The daily
climatology that has been produced is therefore likewise of uncertain
quality in these regions.

For each atmospheric variable at each grid point the climatologies are
available at either 00 or 12 UTC on all calendar dates in a 366 day year
for all forecast lead times from 24 hours up to 6480 hours (9 months)
with an increment of 12 hours, i.e., at a total of 366 · 539 = 197.274
time pairs (target time , forecast lead time). For forecast lead times
that are a multiple of 24 the daily climatology is valid at 00 UTC while
for the intermediate forecast lead times, 36 hr, 60 hr ..., it is valid
at 12 UTC.

For each oceanic variable at each ocean grid point the climatologies are
available at the same time pairs as for the atmospheric variables, with
the exception that the longest forecast lead time is only 1080 hours
(1.5 months), i.e., the total number of time pairs is limited to 366 ·
89 = 32.574.

In addition to the climatological annual cycle of the mean of the above
42 variables a climatology for the standard deviation around this smooth
climatological annual cycle of the mean is also calculated.

**3. Methodology**

The problem at hand is to extract, for each variable, at each grid point
and at each forecast lead time, an estimate of the true climatological
annual cycle given 24 years of daily data which are only given at
roughly half of the days in a year. At each calendar date where data is
available the straight average value, determined from the available 24
values, is in general composed of the following components:

(i) The true climatological annual cycle

(ii) Meteorological noise

(iii) Climatological noise (Low-frequency meteorological noise, i.e.,
      variability on time scales comparable with, and longer than, the
      24-yrs considered here.)

(iv) Model noise

The extraction of (i) is here done by fitting, through the method of
least squares, the annual cycle of the average values to a truncated
Fourier series with sine and cosine as basis functions. There is no a
priori reason why the climatological annual cycle should necessarily be
best represented by a low-order Fourier series. However, previous
studies like the ones by Trenberth (1985), Epstein (1988) and Schemm et
al. (1998) have shown that this method gives reasonable results. The
method requires a decision on truncation, i.e., how many Fourier
components should be included to give optimum results. Using too few
components -- underfitting -- implies that part of the true
climatological annual cycle is not included in the estimate, while on
the other hand using too many components -- overfitting -- means that
part of the noise is included in the estimate. Epstein (1991) proposed a
method, based on statistical models, to determine the optimum numbers to
use. That work indicated that the optimum number of components is likely
to vary with variable and geographical location. Given the large number
of variables, the global domain and the large number of forecast lead
times in the CFS data the determination of optimum numbers would require
the construction and execution of an objective and automated procedure.
No attempt to device such a procedure has here been made. Instead, in
accordance with the experience and practice at NCEP (Schemm et al.
1998), a truncation at wave number 4 is used for all variables at all
locations and at all forecast lead times.

Note that no smoothing or filtering in the spatial domain is performed.
The rationale being that many variables have quite localized
geographical characteristics.

A special circumstance here is that the 24 year average values are not
available at all days in a year. This is due to the fact that the CFS
hindcasts do not start from all days in a given year. Initial conditions
are only given at roughly half of the days in a given month, namely at 3
separate sets of 5 consecutive days. The first set consists of day
9-10-11-12-13, the second set of day 19-20-21-22-23 and finally the
third set of the second to last day of the month, the last day of the
month and day 1-2-3 of the next month. For later reference such a set of
5 consecutive days is denoted a 5-group.

The fact that data is not available on all calendar dates requires a
slight modification to the most common straightforward method of
calculating Fourier coefficients. The specific technique used here is
described in section 4 and is based on the method of least squares.

**4. Calculation of Fourier coefficients**

Let y be a periodic function with period T=365 days. Values of y are
given at N discrete points in time, $t_{j}$. These time points are on an
equidistant net with a spacing of 1 day, but roughly half of all the
points on the net have no data. N is therefore considerably less than
365. The periodic function y can be approximated by a truncated Fourier
series

$\overset{\sim}{y}(t_{j\ })\  = \ \sum_{k = - K}^{K}{c_{k}\ e^{\text{ikt}_{j^{'}}}}$

where $c_{k}$ are complex Fourier coefficients, K is the maximum wave
number and

$t_{j^{'}}\  = \ {\omega t}_{j}\  = \ \frac{2{\pi t}_{j}}{T}$

is a nondimensional time. From here on the prime is omitted for brevity.
The spectral coefficients are obtained from a minimization of the
quantity

$S\  = \ \sum_{j = 1}^{N}{\ \left\lbrack \ \overset{\sim}{y}(t_{j})\  - \ y(t_{j}) \right\rbrack^{\ 2}} = \ \sum_{j = 1}^{N}{\ \left\lbrack \ \sum_{k = - K}^{K}{c_{k}\ e^{\text{ikt}_{j}}}\  - \ y(t_{j}) \right\rbrack^{\ 2}}$

S is thus the sum of the squared difference between the truncated
Fourier series and the actual data. The minimization is obtained by
differentiating S with respect to each complex coefficient, i.e.,

$\frac{\partial S}{\partial c_{l}}\  = \ 2\ \sum_{j = 1}^{N}{\ \left\lbrack \ \sum_{k = - K}^{K}{c_{k}\ e^{\text{ikt}_{j}}}\  - \ y(t_{j}) \right\rbrack\ }e^{\text{ilt}_{j}}\  = \ 0$

which is equivalent to

$\sum_{k = - K}^{K}{c_{k}\ }\sum_{j = 1}^{N}e^{i(l + k)t_{j}}\  = \ \sum_{j = 1}^{N}{\ y(t_{j})\ e^{\text{ilt}_{j}}}$
(1)

By defining

$a_{\text{lk}}\  = \ \frac{1}{N}\sum_{j = 1}^{N}e^{i(l + k)t_{j}}\ ,\ b_{l}\  = \ \frac{1}{N}\sum_{j = 1}^{N}{\ y(t_{j})\ e^{\text{ilt}_{j}}}$

equation (1) can be written as a matrix equation

$\sum_{k = - K}^{K}{c_{k}a_{\text{lk}}\ }\  = \ b_{l}\  \Leftrightarrow \ A\ X\  = \ B$
(2)

where

$$A\  = \ \begin{pmatrix}
1 & \frac{1}{N}\sum_{j = 1}^{N}e^{\text{it}_{j}} & \frac{1}{N}\sum_{j = 1}^{N}e^{2\text{it}_{j}} & & \frac{1}{N}\sum_{j = 1}^{N}e^{i2\text{Kt}_{j}} \\
\frac{1}{N}\sum_{j = 1}^{N}e^{- \text{it}_{j}} & 1 & & & \\
\frac{1}{N}\sum_{j = 1}^{N}e^{- 2\text{it}_{j}} & & & & \\
 & & & & \frac{1}{N}\sum_{j = 1}^{N}e^{\text{it}_{j}} \\
\frac{1}{N}\sum_{j = 1}^{N}e^{- i2\text{Kt}_{j}} & & & \frac{1}{N}\sum_{j = 1}^{N}e^{- \text{it}_{j}} & 1
\end{pmatrix}\ X\  = \ \begin{pmatrix}
c_{- K} \\
 \\
 \\
 \\
c_{0} \\
 \\
 \\
 \\
c_{K}
\end{pmatrix}\ $$

$$B\  = \ \begin{pmatrix}
\frac{1}{N}\sum_{j = 1}^{N}{y(t_{j})e^{\text{iKt}_{j}}} \\
 \\
 \\
\frac{1}{N}\sum_{j = 1}^{N}{y(t_{j})} \\
 \\
 \\
\frac{1}{N}\sum_{j = 1}^{N}{y(t_{j})e^{- \text{iKt}_{j}}}
\end{pmatrix}$$

Note that the matrix **A** is Hermitian. If data are available on all
points on the equidistant net then the off-diagonal elements are all
zero. This implies that the coefficients c~k~ are independent of each
other and can be solved without matrix manipulations. However, in the
general case **A** is a full Hermitian matrix, the coefficients c~k~ are
dependent of each other and the different wave numbers are not
orthogonal to each other. The complex matrix equation (2) is solved by
using a dense linear algebraic equation subroutine, in which the matrix
**A** is factored using Gaussian elimination with partial pivoting to
compute the LU decomposition of **A**.

A smooth daily climatological annual cycle, c, is defined by only
considering 4 wave numbers, i.e.,

$c(t_{j\ })\  = \ \sum_{k = - 4}^{4}{c_{k}\ e^{\text{ikt}_{j}}}$ (3)

The climatology is thus determined by 9 real coefficients. Note that the
climatology is defined on each day in a 365 day year. At 00 UTC for
forecast lead times of 24 hr, 48 hr, ... and at 12 UTC for forecast lead
times of 36 hr, 60 hr, ... The climatological value for the leap day, 29
February, is defined as the mean of the values at 28 February and 1
March. Note the inconsistency that even though the annual cycle has been
assigned a period of 365 days it is defined at 366 days. The root of
this inconsistency is of course that the true period of the annual
cycle, the Tropical Year, is 365.2422 days, i.e, an uneven multiple of a
day.

**5. Archetypal examples**

To demonstrate the characteristics of the daily climatological annual
cycle for an archetypal, well behaved, variable in the Northern
Hemisphere extratropics Fig. 1 is presented. With well behaved is meant
a variable that is continuous, always defined and with a probability
density function (PDF) that is quasi-Gaussian. The variable shown is 2 m
temperature at a forecast lead time of 660 hours (27.5 days) at a grid
point close to Washington, DC. The mean climatological annual cycle
(left panel, red curve) possess a distinct sinusoidal behavior with an
amplitude of around 25 ºC. The shape of the curve clearly indicates the
dominance of the first harmonic and the suitability of using a fit to
sine/cosine functions. Compared to the time series of the raw 24-yr
average values (blue discontinuous curve) the smooth climatological
annual cycle (red curve) intuitively looks more like a true
climatological annual cycle. The standard deviation around the
calculated mean climatological annual cycle is shown in the right panel.
The red curve displays a form that indicates the influence from more
than just the first harmonic in establishing the climatology. As
expected, the variability is out of phase with the mean annual cycle
with a maximum in winter that is almost 8 times as large as in summer
(in terms of variance). Figure 2 demonstrates the undesirable climate
drift in the CFS forecasts. The left panel shows that the climatology
after 8.1 months of integration has undergone a warming during
wintertime, of the order of 2 ºC, relative to 0.9 months of integration.
The right panel shows that the wintertime variability during the same
time has decreased by around 15% (in terms of variance).

An example of a corresponding archetypal behavior in the tropics is
shown in Fig. 3 which is for the same variable at a grid point at the
equator in Indonesia. In contrast to the extratropics the mean
climatological annual cycle has considerably less amplitude -- of the
order of 1 ºC - and is furthermore dominated by the second harmonic. The
use of a Fourier series seems to be reasonable even in this case. The
standard deviation displays a more complex behavior indicating that all
four wave numbers contribute to the final climatology. The precision of
0.5 ºC in the gribbed hindcast data is here clearly evident and may well
have a detrimental effect on the determination of the climatology.
However, the visual impression is that it looks reasonable.

The examples discussed above indicate that the methodology adopted works
satisfactorily for well behaved variables. However, there are also other
variables with different properties that make it difficult to fit their
climatology to a low-order Fourier series. The next sections are devoted
to a discussion of how the procedure described in section 4 is modified
to accommodate for the difficulties that these properties present.

**6. Variables that can not be negative**

Some variables can not have a mean climatological annual cycle with
negative values. Furthermore, for all variables the standard deviation,
by definition, can not be negative. In most cases this circumstance does
not pose any problem. However, for variables that are zero, or close to
zero, during part of the year, a straightforward application of the
methodology of section 4 will create unphysical climatologies. This
unwanted behavior is due to the Gibbs phenomenon which can create
unphysical

(i) negative values in parts of the year, which is impossible.

(ii) positive values in times of the year when it is obvious that the
     climatology is zero.

The methodology of section 4 has therefore been modified to ensure that
a physically reasonable climatology is calculated. The modification to
the smooth climatology given by the solution of equation (2) is
determined by the following procedure:

1.  Denote the 5 consecutive days which have raw mean values defined as
    a 5-group.

2.  If there are no raw mean values in a 5-group that is greater than a
    small value, $\varepsilon\  = \ \text{10}^{- 6}$, then all days in
    the 5-group is designated the final climatological value of zero,
    instead of the smooth climatology given by equation (2).

3.  If a day outside of a 5-group is surrounded by 5-groups on either
    side that have been designated the final climatological value of
    zero, then that day is assigned the climatological value of zero as
    well, instead of the smooth climatology given by equation (2).

4.  If there remains days with a negative climatological value given by
    equation (2), then those days are instead assigned the
    climatological value of zero.

A set of variables with this kind of behavior are short wave radiation
fluxes which are zero at night time. These variables are relatively easy
to correct since there are no instances of sudden departure from zero
during the time of the year when they are zero. An example is shown in
Fig. 4 for the downward short wave radiation flux at the earth's
surface. A slight discontinuity in the curvature of the smooth curve is
seen at the boundary to the zero line. However, this only occurs for
values less than the resolution (1 W/m^2^) of the input gribbed forecast
data (the discontinuous blue curve).

A variable which is considerably more difficult to account for is
precipitation. In arid regions like in the Saharan desert it can have a
mean value of zero during part of the year. However, due to the episodic
nature of precipitation, there may be an occasional day with non-zero
precipitation in the middle of such a dry spell. It has been found to be
very difficult to design an algorithm that will always create a smooth
looking climatology for precipitation. As seen in Fig. 5 there are
instances when the final climatology

-   suddenly departs from a smooth behavior to become zero.

> (See end of August and December in Fig. 5)

-   take no account of a single precipitation event in an otherwise dry
    spell of the year.

> (See mid-July in Fig. 5)

The discontinuities that are seen in Fig. 5 are however occurring for
precipitation amounts that are less than the lowest amount measured in
the USA, i.e., 0.01 inches = 0.254 mm, which corresponds to a
precipitation rate of 3 · 10 ^-6^ kg/m^2^/s.

The difficulties seen reflect the non-trivial nature of constructing a
climatology for a variable like precipitation. It is not at all clear
that a low-order Fourier series is the preferred approach to use in this
case. Note also the occasional large positive departures that the raw
mean values can possess relative to the final smooth climatology. This
is a consequence of the non-Gaussian, skewed nature of the PDF of
precipitation.

Another problem associated with the choice of fitting the climatology to
a low-order Fourier series is demonstrated in Fig. 6. In order to avoid
problem (i) of the Gibbs phenomenon as discussed above, both the mean
and the standard deviation have been set to zero during an extended
period in summer when it is clear that it ought to be slightly positive.
However, once again, this only occurs at precipitation amounts that are
of the order of the lowest amount measured.

It should be emphasized that the problems discussed here only occurs for
grid points with very little precipitation. For the vast majority of
grid points there are no problems of this kind and the calculated
climatology looks reasonable. Figure 7 is provided to illustrate a
typical example to this effect. In contrast to 2 m temperature (Fig. 1)
it is clear that for precipitation all four harmonics contribute to the
final climatology. The variability of the raw mean values around the
smooth climatology is appreciable, indicating that the sample size may
not be sufficient to establish a stable estimate of the true climate and
hence of all nine Fourier coefficients of the smooth climatology.

**7. Variables that are not defined during part of the year**

There are a number of variables, mostly oceanographic variables, which
at certain geographical locations are not defined during part of the
year. An example is the variable "depth to the 2.5 ºC isotherm", whose
climatology at a sub-polar location is shown in Fig. 8. The algorithm
used to solve equation (2) does not always produce a correct answer when
the number of days in a year when the variable is defined is small. This
problem is to a large extent resolved by using double precision in the
calculations (as is the case for the smooth climatology displayed in
Fig. 8). However, there are still a number of instances when the problem
persists. A smooth climatology is therefore only calculated by solving
equation (2) when the number of days in a year when the variable is
defined is larger than 2/3 of the maximum available (i.e., 120 \~ 2/3 ·
365 · 0.5). Otherwise the final climatology is defined as follows:

-   For days when raw mean values are available, they are used as the
    final climatology.

-   If a day outside of a 5-group is surrounded by 5-groups on either
    side that have raw mean values available, then a linear
    interpolation to the nearest available raw mean values defines the
    final climatology at that date.

This crude remedy was adopted since it was not deemed meaningful to
elaborate on this issue because of the fact that the grid points
affected are normally at high northern and southern latitudes where the
quality of the modeled oceanographic fields are any way not very
reliable. As pointed out in section 2 the reason for the poor quality in
the neighborhood of the polar lateral boundaries is partly because of
the proximity to the imposed climatological lateral boundary conditions,
and partly because of the restricted atmosphere-ocean coupling at high
latitudes (for further information see Saha et al. 2006).

**REFERENCES**

Epstein, E. S., 1988: A spectral climatology. *J. Climate*, **1**,
88--107.

Epstein, E. S., 1991: Determining the optimum number of harmonics to
represent normals based on multiyear data. *J. Climate*, **4**,
1047--1051.

Saha S., S. Nadiga, C. Thiaw, J. Wang, W. Wang, Q. Zhang, H. M. van den
Dool, H.-L. Pan, S. Moorthi, D. Behringer, D. Stokes, M. Peña, S. Lord,
G. White, W. Ebisuzaki, P. Peng, P. Xie, 2006: The NCEP Climate Forecast
System. *J. Climate*. **19**, 3483-3517.

Schemm, J-K. E., H. M. van den Dool, J. Huang, and S. Saha, 1997:
Construction of daily climatology based on the 17-year NCEP-NCAR
reanalysis. *Proceedings of the First WCRP International Conference on
Reanalyses*. Silver Spring, Maryland, USA. 290-293.

Trenberth, K. E., 1985: Persistence of daily geopotential heights over
the Southern Hemisphere. *Mon. Wea. Rev.*, **113**, 38-53.

[Table 1]{.smallcaps}. List of atmospheric variables for which daily
climatologies has been calculated.

═════════════════════════════════════════════════════════

  ------------------------------- --------------- ----------- ------------------
  **VARIABLE**                    **LEVEL**       **UNIT**    **ABBREVIATION**

  Temperature                     2 m             K           T2M

  Pressure                        Surface         Pa          PS

  Pressure                        Mean sea level  Pa          PMSL

  Zonal component of momentum     Surface         N/m^2^      UWSTR
  flux                                                        

  Meridional component of         Surface         N/m^2^      VWSTR
  momentum flux                                               

  Precipitation rate              Surface         kg/m^2^/s   PRATE

  Precipitable water              Whole           kg/m^2^     PWAT
                                  atmosphere                  

  Sensibel heat flux              Surface         W/m^2^      SHTFL

  Latent heat flux                Surface         W/m^2^      LHTFL

  Upward long wave radiation flux Top of          W/m^2^      OLR
                                  atmosphere                  

  Upward short wave radiation     Top of          W/m^2^      USWTOP
  flux                            atmosphere                  

  Upward long wave radiation flux Surface         W/m^2^      ULWSFC

  Downward long wave radiation    Surface         W/m^2^      DLWSFC
  flux                                                        

  Upward short wave radiation     Surface         W/m^2^      USWSFC
  flux                                                        

  Downward short wave radiation   Surface         W/m^2^      DSWSFC
  flux                                                        

  Geopotential height             200 hPa         m           Z200

  Geopotential height             500 hPa         m           Z500

  Geopotential height             700 hPa         m           Z700

  Geopotential height             850 hPa         m           Z850

  Geopotential height             1000 hPa        m           Z1000

  Streamfunction                  200 hPa         m^2^/s      PSI200

  Streamfunction                  850 hPa         m^2^/s      PSI850

  Velocity potential              200 hPa         m^2^/s      CHI200

  Velocity potential              850 hPa         m^2^/s      CHI850

  Zonal velocity                  200 hPa         m/s         U200

  Zonal velocity                  850 hPa         m/s         U850

  Meridional velocity             200 hPa         m/s         V200

  Meridional velocity             850 hPa         m/s         V850

  Zonal velocity                  10 m            m/s         U10M

  Meridional velocity             10 m            m/s         V10M
  ------------------------------- --------------- ----------- ------------------

[Table 2]{.smallcaps}. List of oceanic variables for which daily
climatologies has been calculated.

═════════════════════════════════════════════════════════

  ------------------------------- --------------- ---------- ------------------
  **VARIABLE**                    **LEVEL**       **UNIT**   **ABBREVIATION**

  Depth to the 2.5 °C isotherm    \-              m          DT2.5C

  Depth to the 5 °C isotherm      \-              m          DT5C

  Depth to the 10 °C isotherm     \-              m          DT10C

  Depth to the 15 °C isotherm     \-              m          DT15C

  Depth to the 20 °C isotherm     \-              m          DT20C

  Depth to the 25 °C isotherm     \-              m          DT25C

  Depth to the 28 °C isotherm     \-              m          DT28C

  Ocean heat content[^1]          0-300 m         J/m^2^     OHC

  Ocean mixed layer depth         \-              m          OMLD

  Ocean surface isothermal layer  \-              m          OSILD
  depth                                                      

  Ocean temperature               Surface         K          SST

  Ocean tropical cyclone heat     \-              J/m^2^     TCHP
  potential                                                  
  ------------------------------- --------------- ---------- ------------------

[Fig. 1.]{.smallcaps} The climatological annual cycle of 2 m temperature
at a forecast lead time of 660 hours (27.5 days) at a grid point close
to Washington, DC, is shown as a smooth red curve. Shown are both the
mean (left panel) and the standard deviation (right panel). The 24-yr
mean values as calculated directly from the forecast data is shown by
the blue discontinuous curve. The reason for the discontinuities is that
these values are only available during roughly half of the days in a
year. The time on the abscissa is the initial time of the forecast, not
the actual time when the forecast is valid. In this case this implies an
offset by roughly one month. Unit is K.

![](media/image4.png){width="3.999212598425197in"
height="5.195669291338582in"}![](media/image5.png){width="4.0in"
height="5.203543307086615in"}

[Fig. 1.]{.smallcaps} The climatological annual cycle of 2 m temperature
at a forecast lead time of 660 hours (27.5 days) at a grid point close
to Washington, DC, is shown as a smooth red curve. Shown are both the
mean (left panel) and the standard deviation (right panel). The 24-yr
mean values as calculated directly from the forecast data is shown by
the blue discontinuous curve. The reason for the discontinuities is that
these values are only available during roughly half of the days in a
year. The time on the abscissa is the initial time of the forecast, not
the actual time when the forecast is valid. In this case this implies an
offset by roughly one month. Unit is K.

![](media/image6.png){width="4.040157480314961in"
height="5.194488188976378in"}![](media/image7.png){width="4.040157480314961in"
height="5.194488188976378in"}

[Fig. 2.]{.smallcaps} As in [Fig.]{.smallcaps} 1 but for a forecast lead
time of 5844 hours (8.1 months).

![](media/image8.png){width="4.0in"
height="5.199212598425197in"}![](media/image9.png){width="3.999212598425197in"
height="5.199212598425197in"}

[Fig. 3.]{.smallcaps} As in [Fig.]{.smallcaps} 1 but for a grid point at
the equator in Indonesia.

![](media/image10.png){width="4.040157480314961in"
height="5.195275590551181in"}![](media/image11.png){width="4.040157480314961in"
height="5.195275590551181in"}

[Fig. 4.]{.smallcaps} As in [Fig.]{.smallcaps} 1 but for downward short
wave radiation flux at the earth's surface. Unit is W/m^2^.

![](media/image12.png){width="4.0in"
height="5.2in"}![](media/image13.png){width="4.0in" height="5.2in"}

[Fig. 5.]{.smallcaps} As in [Fig.]{.smallcaps} 1 but for precipitation
rate at a grid point in the eastern part of the Sahara desert. Unit is
kg/m^2^/s.

![](media/image14.png){width="4.040157480314961in"
height="5.195275590551181in"}![](media/image15.png){width="4.040157480314961in"
height="5.195275590551181in"}

[Fig. 6.]{.smallcaps} As in [Fig.]{.smallcaps} 5 but for a grid point in
the central part of the Sahara desert.

![](media/image16.png){width="4.040157480314961in"
height="5.195275590551181in"}![](media/image17.png){width="4.040157480314961in"
height="5.195275590551181in"}

[Fig. 7.]{.smallcaps} As in [Fig.]{.smallcaps} 5 but for a grid point
close to Washington, DC.

![](media/image18.png){width="4.040157480314961in"
height="5.195275590551181in"}![](media/image19.png){width="4.040157480314961in"
height="5.195275590551181in"}

[Fig. 8.]{.smallcaps} As in [Fig]{.smallcaps}. 1 but for the oceanic
variable "depth to the 2.5 ºC isotherm" at a grid point close to
Antarctica at a forecast lead time of 1080 hours (1.5 months). Unit is
m.

[^1]: Only defined when the ocean depth is larger than 300 m.