January 13, 2018 | Author: Anonymous | Category: Math, Statistics And Probability, Statistics
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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 © Crown copyright Met Office

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

© Crown copyright Met Office

Adaptive grid in 3D-Var: formulation © Crown copyright Met Office

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.

© Crown copyright Met Office

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 © Crown copyright Met Office

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. © Crown copyright Met Office

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


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. © Crown copyright Met Office

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. © Crown copyright Met Office

Adaptive grid in 3D-Var: application © Crown copyright Met Office

Monitor Function and Adaptive Grid

© Crown copyright Met Office

Single Ob tests: Stratocumulus

Single Ob qT inserted above the inversion

Single Ob  inserted above the inversion © Crown copyright Met Office

Nominal physical mesh

Nominal physical mesh

Computational mesh

Computational mesh

Initial implementation: limitations © Crown copyright Met Office

RMS error: Analysis - Observations theta

relative humidity

© Crown copyright Met Office

zonal wind

meridional wind

RMS error: surface observations Improvement of analysis RMS error for temperature at the surface.

Adaptive grid at Camborne


T (K)

RH (%) u (m/ s) v (m/s)
















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.

© Crown copyright Met Office

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. © Crown copyright Met Office



UK index


8/2/2010 - 20/2/2010

Overall: +0.44% T2m: +0.23%


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 © Crown copyright Met Office

Improved implementation © Crown copyright Met Office

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) © Crown copyright Met Office

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!

© Crown copyright Met Office

Control case

Normalisation factor UK4 domain: 3 Jan 2011 00z

Test case

© Crown copyright Met Office

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) © Crown copyright Met Office



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

© Crown copyright Met Office

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



Cloud amount

Cloud base




23 Dec 2010 – 3 Jan 2011








10 Aug 2010 - 20 Aug 2010








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?

© Crown copyright Met Office

RMS error vs sondes: mean relative difference [%] to control case O-B

© Crown copyright Met Office


RMS error vs other obs types: mean relative difference [%] to control case black = O – B; white = O – A

© Crown copyright Met Office

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. © Crown copyright Met Office

Summary and future work © Crown copyright Met Office

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

© Crown copyright Met Office

Thank you for your attention. Any questions? © Crown copyright Met Office

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