Adaptive mesh method in the Met Office
variational data assimilation system
Chiara Piccolo and Mike Cullen
Adaptive Multiscale Methods for the Atmosphere and Ocean
23 August 2012
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Contents
This presentation covers the following areas
• Adaptive grid in 3D-Var: formulation
• Adaptive grid in 3D-Var: application
• Initial implementation and limitations
• Improved implementation
• Summary and future work
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Adaptive grid in 3D-Var: formulation
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Adaptive Mesh Transform
in 3D-Var
• Motivation: improving accuracy in areas of large spatial variations
of the 3D-Var solution, e.g. strong temperature inversions due to the
presence of stratocumulus clouds.
• Static Adaptive Mesh Methods relocate grid points in a mesh so that
they concentrate the grid points in areas where there is a rapid
variation of the atmospheric field. For this kind of method,
interpolation is required to pass the analysis increments on the old
mesh to the new mesh.
• The transformation from the physical grid to the computational grid
is guided by a monitor function, which controls the mesh
distributions.
• The choice of the monitor function is problem- and user-dependent
and can be determined a priori from consideration of the geometry
or the physics of the atmospheric feature we want to diagnose.
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3D-Var Transforms
 The adaptive mesh transformation aims at changing the vertical
background-error correlations by moving the vertical levels to
concentrate mesh points around temperature inversions.
The movement of the levels is guided by a scalar monitor function
which here we choose to be a function of the static stability which
strongly controls the vertical mixing of the atmosphere and thus
probably the vertical correlation structure of model variables.
 We introduce the adaptive method within the Met Office Var
system as an another transformation in the sequence of variable
transformations aimed to simplify the background term of the cost
function:
x  Uχ  U p Ua Uv Uh χ
T
B

UU
and
where Ua is the "adaptive mesh transform" which is placed
between the parameter transform Up and the vertical transform Uv
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Adaptive Grid Formulation
The first step of the Ua transform is to calculate a monitor function
M ( >0 ) in physical space z  [0,1]:
1
 M ( z' )dz'  1
0
The second step is to generate the adaptive mesh in physical
space by defining a computational coordinate z  [0,1]:
z
 ( z)   M ( z' )dz'
0
The map from computational domain to physical domain is thus
defined by the a unique one-dimensional map which connects
intervals of a prescribed length.
Finally, the control variables c which will be generated at points z
by the vertical transform are then interpolated to the true levels z.
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Choice of the Monitor Function
Lorenc (2007) showed that a significant part of the observed
background-error correlation structure could be explained by
regarding it as a function of static stability (  / z ).
Most choices of monitor function need regularisation to perform
effectively and ensure that a good mesh resolution is maintained
everywhere, an example is given by:
M  1  c ( z )
2
2
M is always positive and can be modulated by a scaling factor c.
If the scaling factor c is set to zero, the computational grid and the
physical grid are the same.
Since mesh points will be clustered where the monitor function is
large, this choice of M will cluster mesh points in regions of large
static stability.
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Horizontal Smoothing
The adaptive mesh transform is a 1D transformation in the vertical only.
The transformation depends on horizontal position.
The monitor function is calculated for every horizontal grid point.
In order to avoid a loss of horizontal coherence in the mesh a horizontally
smoother mesh can be generated by smoothing the regularised monitor
function prior to the mesh calculation.
The smoothing at point i, j (longitude/latitude) can be expressed as:
The degree of smoothing applied can be increased by iterating this
smoothing procedure N times.
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Adaptive grid in 3D-Var: application
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Monitor Function and Adaptive Grid
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Single Ob tests: Stratocumulus
Single Ob
qT inserted
above the
inversion
Single Ob
 inserted
above the
inversion
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Nominal physical mesh
Nominal physical mesh
Computational mesh
Computational mesh
Initial implementation: limitations
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RMS error: Analysis - Observations
theta
relative humidity
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zonal wind
meridional wind
RMS error: surface observations
Improvement of analysis RMS
error for temperature at the
surface.
Adaptive grid at Camborne
RMS
T (K)
RH (%) u (m/ s) v (m/s)
Control
0.76
0.045
1.32
1.16
Test
0.64
0.045
1.29
1.16
Nobs
1011
901
819
819
Reduction of the background vertical
correlation distance for the potential
temperature and an increase of its
variance
Variance increase induced by increase
of pressure vertical gradient resulting
from the reduced vertical correlation
distance.
Near the surface less weight is given to
the background state and more weight
is given to the observations.
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Forecast RMS error for T2m
Results from the full coupled
analysis/forecast system:
 positive impact up to
T+15 h for 2m temperature
forecasts
 relatively neutral impact
for the other variables.
The forecast error is reduced
most at the beginning of the
forecast. This reflects the
movement of the grid and the
local interaction between
atmospheric variables near
the surface.
The impact greatly decreases
as the lead time increases.
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Case
Period
UK index
Winter
8/2/2010 - 20/2/2010
Overall: +0.44%
T2m: +0.23%
Spring
25/4/2010 - 20/5/2010
Overall -0.08%
T2m: +0.07%
Time series of the forecast RMS error
verified against surface observations
for the winter trial
Limitations
The monitor function is based on the background-state static stability.
The monitor function is calculated every analysis cycle and depends
uniquely on the previous cycle forecast. This limits its effectiveness
because it relies on the presence and position of low clouds or
temperature inversions in the background-state.
Background
Adaptive Grid
Standard Grid
Observations
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Improved implementation
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Updates to Adaptive Grid
transform in 3D-Var
 More appropriate normalisation factor
 Recalculation of the computational grid based on the
updated background-state in 3 steps:
1. standard assimilation for 10 iterations in order to
get closer to the observations and improve the
background-state (called AG0)
2. calculation of the monitor function from the
preliminary analysis coming from step 1 and
application of the adaptive grid transform in the
assimilation for 10 iterations only (AG1)
3. second calculation of the monitor function and grid
based on the improved analysis coming from step 2
and application of the adaptive grid transform in the
assimilation until convergence (AG2)
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UK4: 3 Jan 2011 00z
MSG image at 3.9 mm over UK
for 3 Jan 2011 at midnight:
emitted radiation from the
surface and cloud tops (3km)
White areas: cold temperatures,
i.e. clouds
Dark areas: warm temperatures,
i.e. sea and land surfaces.
High clouds over North Atlantic
Ocean and west of Ireland
Patches of clear sky over the
sea and France
Low clouds over all UK!
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Control case
Normalisation factor
UK4 domain: 3 Jan 2011 00z
Test case
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Recalculation of the grid
UK4 domain: 3 Jan 2011 00z
LS
LS = background-state (3h forecast)
AG0: standard 3D-Var (10 iterations)
AG1: LS1=LS+PF-AG0 (10 iterations)
AG2: LS2=LS+PF-AG1 (convergence)
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LS1
LS2
RMS error: Analysis - Observations
UK4 domain: 3 Jan 2011 00z
Single Var analysis cycle:
monitor function calculation
started from the same
background-state
 [K] RMS Error
Control
Theta
AG2
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Var analysis cycle from the
full coupled analysis/forecast
system: improvement of the
background-state
 [K] RMS Error
Control
AG2
AG2 cycled
Analysis/forecast system performance
when recalculating the grid twice
All options tested provide a small positive impact to the UK index in winter.
After extensive tuning, the best option in the full analysis/forecast system
was given by using static stability with the recalculation of the grid:
consistent signal for surface temperature and cloud base height.
Period
Vis
Precip
Cloud
amount
Cloud
base
Temp
Wind
Overall
23 Dec 2010 – 3 Jan 2011
-2.56%
5.48%
-1.05%
3.03%
0.22%
-0.04%
+0.25%
10 Aug 2010 - 20 Aug 2010
12.20%
0.00%
0.00%
4.17%
0.23%
0.10%
+0.55%
These results are reinforced by a general improvement of the initial fit
to the observations in all analysis cycles of few % for both seasons.
Which observation types contribute the most?
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RMS error vs sondes:
mean relative difference [%] to control case
O-B
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O-A
RMS error vs other obs types:
mean relative difference [%] to control case
black = O – B; white = O – A
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3 Jan 2011 00z:
Camborne
The monitor function is based on the
background state: if the inversion is
not present, the vertical grid does not
change.
Background
Analysis
Observations
When the monitor function is based on
an update background-state using the
observation’s information in the
minimisation process the analysis has
a clearer inversion.
Limitation: long horizontal and mode
independent length scales in the
background error covariance matrix.
QC: RH sonde not assimilated around
the inversion because observations are
too far from background.
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Summary and future work
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Summary and future work
Initial operational implementation of the vertical adaptive grid
transformation in 3D-Var in November 2010 had an impact mainly
in 2m temperature.
In July 2011 the recalculation of the adaptive grid within the
minimisation process became operational.
The main limitation is due to the long horizontal length scales
which are fixed for each vertical mode in the UK models.
This washes out the local effects of the adaptive grid transform.
More realistic horizontal length scales will allow to detect more
local phenomena.
In the future we will extend the transformation to full 3D:
 see C. Budd’s talk tomorrow on: Monge ampere based moving
mesh methods with applications to numerical weather prediction
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Thank you for your attention.
Any questions?
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