ROMS/TOMS Developers

Algorithms Update Web Log

arango - August 18, 2006 @ 10:50
Updated Algorithms, Version 3.0- Comments (0)

I updated few routines to correct few things:

• Fixed a problem with incremental 4DVar (IS4DVAR) that forced the inner loop to converge after two iterations of the inner loop regardless of the values used for the convergence of the total cost function, CostFunFac. This parameter is set in s4dvar.in and represents the percentage of total cost function change between successive iterations in the inner loop. The comparison between old and new values was wrong. Many thanks to Javier and Brian for noticing this.
• Corrected the fifth argument call for routine get_state for several drivers. Now a dummy variable is used instead of a numerical value. This variable has the intent(inout) declaration in get_state. This problem was giving us segmentation violation in a cluster using MPI. Many thanks to Brian for finding this one.
For the current updated file list .

arango - August 16, 2006 @ 18:02

The minimization in the descent algorithm for inner loop iteration n is given by

```
```

where αn is the step size and dn are the conjugate direction vectors:

```

```

or

```
```

The steepest descent occurs when β = 0. This happens either every NiterSD iterations or when dot2 > CGtol * dot1, where

```

```

The refined step size is computed in the second inner loop pass, m = 2, as

```
```

arango - August 16, 2006 @ 17:59
ROMS/TOMS Incremental 4DVAR Diagram- Comments (0)

The ROMS/TOMS incremental, strong contraint 4DVAR (IS4DVAR) driver, is4dvar_ocean.h, is as follows:

arango - August 16, 2006 @ 17:51
Vertical and Horizontal Convolutions- Comments (0)

Following Weaver and Courtier (2001), the vertical and horizontal correlations, C, for any state variable Φ are modeled using separated 1D and 2D diffusion equations:

```

```

where Kv and KH are the vertical and horizontal diffusion coefficients (m2/s), assumed to be of order one. They are set to unity (isotropic correlations), if these coefficients are not found in ROMS/TOMS background/model standard deviation NetCDF file (STDname, variables Kv and Kh). The user can set Kv and Kh to the desired patterns to get anisotropic correlations. If we discretize the above diffusion operators using an explicit FTCS scheme, the stability limit is governed by:

```

```

In practice, we set γ < 1 to be below the stability threshold, (γ = 0.5). This stability parameter is set in 4DVAR input script. There are two variables: Vgamma and Hgamma for vertical and horizontal diffusion operators. In realistic applications, the vertical difussion needs to be solved implicitly (IMPLICIT_VCONV). Otherwise, the it will be very expensive since the vertical scales are much smaller. The implicit algorithm is unconditionally stable for any time-step, so we use the maximum thickness in the above equation instead of the minimum.

The parameters controlling the vertical and horizontal length-scales of the Gaussian correlation function are:

```

```

over a time interval 0 ≤ t ≤ T and

```

```

where Mv and MH are the total number of vertical and horizontal integration steps. Notice that both values are forced to an even number since the squared-root diffusion operators are used to insure symmetry. The squared-root operator iterates only for half of the steps. For any given decorrelation length-scale, the number of integration steps are given by:

```

```

The vertical and horizontal decorrelation scales (m) are set in 4DVar input script variables Vdecay(:) and Hdecay(:) for each state variable.

ROMS/TOMS reports to standard output all the parameters associated with these diffusion operators. For example,

``` 5.0000E-01  Hgamma          Horizontal diffusion stability/accuracy factor.
5.0000E-05  Vgamma          Vertical diffusion stability/accuracy factor.
20000.000  Hdecay(isFsur)  Horizontal decorrelation scale (m), free-sruface.
20000.000  Hdecay(isUbar)  Horizontal decorrelation scale (m), 2D U-momentum.
20000.000  Hdecay(isVbar)  Horizontal decorrelation scale (m), 2D V-momentum.
20000.000  Hdecay(isUvel)  Horizontal decorrelation scale (m), 3D U-momentum.
20000.000  Hdecay(isVvel)  Horizontal decorrelation scale (m), 3D V-momentum.
20000.000  Hdecay(idTvar)  Horizontal decorrelation scale (m), temp.
20000.000  Hdecay(idTvar)  Horizontal decorrelation scale (m), salt.
5.000  Vdecay(isUvel)  Vertical decorrelation scale (m), 3D U-momentum.
5.000  Vdecay(isVvel)  Vertical decorrelation scale (m), 3D V-momentum.
5.000  Vdecay(idTvar)  Vertical decorrelation scale (m), temp.
5.000  Vdecay(idTvar)  Vertical decorrelation scale (m), salt.

Minimum X-grid spacing, DXmin =  5.26835784E+00 km
Maximum X-grid spacing, DXmax =  5.58390264E+00 km
Minimum Y-grid spacing, DYmin =  5.05224079E+00 km
Maximum Y-grid spacing, DYmax =  5.35460591E+00 km
Minimum Z-grid spacing, DZmin =  1.33333333E-01 m
Maximum Z-grid spacing, DZmax =  2.86550363E+02 m

Minimum barotropic Courant Number =  2.99123135E-02
Maximum barotropic Courant Number =  8.17259140E-01
Maximum Coriolis   Courant Number =  3.43845829E-02

Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  zeta
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  ubar
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  vbar
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  u
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  v
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  temp
Horizontal convolution, NHsteps, DTsizeH =    64  3.19064213E+06 s  salt

Vertical   convolution, NVsteps, DTsizeV =     6  2.05277776E+00 s  u
Vertical   convolution, NVsteps, DTsizeV =     6  2.05277776E+00 s  v
Vertical   convolution, NVsteps, DTsizeV =     6  2.05277776E+00 s  temp
Vertical   convolution, NVsteps, DTsizeV =     6  2.05277776E+00 s  salt
```

It is recommended to reduce the Vgamma parameter until a value of Mv is around six for vertical diffusion accuracy.

arango - August 16, 2006 @ 17:38
Incremental, Strong Constraint 4DVAR- Comments (0)

The incremental, strong contraint 4DVAR (IS4DVAR) approximated approach was proposed by Courtier et al. (1994) to reduce the computational cost and facilitate better preconditioning. We follow the algorithm proposed by Weaver et al (2002) that defines the increment to the difference from previous reference state.

Let xk denote the state vector on the k-th outer loop such that δxk is the increment difference from the previous state

```
```

where xk -1 is the current estimate of the ocean state. It is equal to the background state for the first minimization (x0 = xb, δx1 = 0). The cost function on the k-th loop can be written as

```

```

where dk – 1 = xoHxk – 1 is the innovation vector, xo is the observation vector, xb is the background state, H is a linear approximation of the observation operator H, and B and O are the background and observation error covariance matrices. Let’s introduce a new minimization variable δv, such that:

```

```

It is more convenient to work in terms of δv (minimization space: v-space). The conditioning of the Jb in v-space is optimal since it’s Hessian, 2Jb/∂v2, yields the identity matrix. The gradient of J in v-space, denoted vJ, is given by:

```
```

where B = BT/2B1/2 and BT/2 denotes (B1/2)T, and xJo is the gradient of Jo in model space (x-space) which is computed using the adjoint model and given by:

```
```

Following Weaver and Courtier (2001), the background-error covariance matrix can be factored as:

```
```

where S is a diagonal matrix of background-error standard deviations, C is a symetric matrix of background-error correlations which can be factorized as C = C1/2CT/2, G is the normalization matrix which ensures that the diagonal elements of C are equal to unity, L is a 3D self-adjoint filtering operator, and W is the grid cell area (2D fields) or volume (3D fields).