10. Ocean State Estimation Packages

This chapter describes packages that have been introduced for ocean state estimation purposes and in relation with automatic differentiation (see Automatic Differentiation). Various examples in this chapter rely on two model configurations that can be setup as explained in Test Cases For Estimation Package Capabilities

10.1. ECCO: model-data comparisons using gridded data sets

Author: Gael Forget

The functionalities implemented in pkg/ecco are: (1) output time-averaged model fields to compare with gridded data sets; (2) compute normalized model-data distances (i.e., cost functions); (3) compute averages and transports (i.e., integrals). The former is achieved as the model runs forwards in time whereas the others occur after time-integration has completed. Following [FCH+15] the total cost function is formulated generically as

(10.1)\[\mathcal{J}(\vec{u}) = \sum_i \alpha_i \left(\vec{d}_i^T R_i^{-1} \vec{d}_i\right) + \sum_j \beta_j \vec{u}^T\vec{u}\]

(10.2)\[\vec{d}_i = \mathcal{P}(\vec{m}_i - \vec{o}_i)\]

(10.3)\[\vec{m}_i = \mathcal{S}\mathcal{D}\mathcal{M}(\vec{v})\]

(10.4)\[\vec{v} = \mathcal{Q}(\vec{u})\]

(10.5)\[\vec{u} = \mathcal{R}(\vec{u}')\]

using symbols defined in Table 10.1. Per Equation (10.3) model counterparts (\(\vec{m}_i\)) to observational data (\(\vec{o}_i\)) derive from adjustable model parameters (\(\vec{v}\)) through model dynamics integration (\(\mathcal{M}\)), diagnostic calculations (\(\mathcal{D}\)), and averaging in space and time (\(\mathcal{S}\)). Alternatively \(\mathcal{S}\) stands for subsampling in space and time in the context of Section 10.2 (PROFILES: model-data comparisons at observed locations). Plain model-data misfits (\(\vec{m}_i-\vec{o}_i\)) can be penalized directly in Eq. (10.1) but penalized misfits (\(\vec{d}_i\)) more generally derive from \(\vec{m}_i-\vec{o}_i\) through the generic \(\mathcal{P}\) post-processor (Eq. (10.2)). Eqs. (10.4)-(10.5) pertain to model control parameter adjustment capabilities described in Section 10.3 (CTRL: Model Parameter Adjustment Capability).

Table 10.1 Symbol used in formulating generic cost functions.

symbol	definition
\(\vec{u}\)	vector of nondimensional control variables
\(\vec{v}\)	vector of dimensional control variables
\(\alpha_i, \beta_j\)	misfit and control cost function multipliers (1 by default)
\(R_i\)	data error covariance matrix (\(R_i^{-1}\) are weights)
\(\vec{d}_i\)	a set of model-data differences
\(\vec{o}_i\)	observational data vector
\(\vec{m}_i\)	model counterpart to \(\vec{o}_i\)
\(\mathcal{P}\)	post-processing operator (e.g., a smoother)
\(\mathcal{M}\)	forward model dynamics operator
\(\mathcal{D}\)	diagnostic computation operator
\(\mathcal{S}\)	averaging/subsampling operator
\(\mathcal{Q}\)	Pre-processing operator
\(\mathcal{R}\)	Pre-conditioning operator

10.1.1. Generic Cost Function

The parameters available for configuring generic cost function terms in data.ecco are given in Table 10.2 and examples of possible specifications are available in:

• MITgcm_contrib/verification_other/global_oce_cs32/input/data.ecco

• MITgcm_contrib/verification_other/global_oce_cs32/input_ad.sens/data.ecco

• MITgcm_contrib/gael/verification/global_oce_llc90/input.ecco_v4/data.ecco

The gridded observation file name is specified by gencost_datafile. Observational time series may be provided as on big file or split into yearly files finishing in ‘_1992’, ‘_1993’, etc. The corresponding \(\vec{m}_i\) physical variable is specified via the gencost_barfile root (see Table 10.3). A file named as specified by gencost_barfile gets created where averaged fields are written progressively as the model steps forward in time. After the final time step this file is re-read by cost_generic.F to compute the corresponding cost function term. If gencost_outputlevel = 1 and gencost_name=‘foo’ then cost_generic.F outputs model-data misfit fields (i.e., \(\vec{d}_i\)) to a file named ‘misfit_foo.data’ for offline analysis and visualization.

In the current implementation, model-data error covariance matrices \(R_i\) omit non-diagonal terms. Specifying \(R_i\) thus boils down to providing uncertainty fields (\(\sigma_i\) such that \(R_i=\sigma_i^2\)) in a file specified via gencost_errfile. By default \(\sigma_i\) is assumed to be time-invariant but a \(\sigma_i\) time series of the same length as the \(\vec{o}_i\) time series can be provided using the variaweight option (Table 10.4). By default cost functions are quadratic but \(\vec{d}_i^T R_i^{-1} \vec{d}_i\) can be replaced with \(R_i^{-1/2} \vec{d}_i\) using the nosumsq option (Table 10.4).

In principle, any averaging frequency should be possible, but only ‘day’, ‘month’, ‘step’, and ‘const’ are implemented for gencost_avgperiod. If two different averaging frequencies are needed for a variable used in multiple cost function terms (e.g., daily and monthly) then an extension starting with ‘_’ should be added to gencost_barfile (such as ‘_day’ and ‘_mon’). 1 If two cost function terms use the same variable and frequency, however, then using a common gencost_barfile saves disk space.

Climatologies of \(\vec{m}_i\) can be formed from the time series of model averages in order to compare with climatologies of \(\vec{o}_i\) by activating the ‘clim’ option via gencost_preproc and setting the corresponding gencost_preproc_i integer parameter to the number of records (i.e., a # of months, days, or time steps) per climatological cycle. The generic post-processor (\(\mathcal{P}\) in Eq. (10.2)) also allows model-data misfits to be, for example, smoothed in space by setting gencost_posproc to ‘smooth’ and specifying the smoother parameters via gencost_posproc_c (name of a smoothing scale file) and gencost_posproc_i (an integer specifying the smoother number of time steps, see Table 10.4). The smoothing scale file can be be based on the large-scale parameter specified in data.smooth or prepared as a factor of the model resolution DXG and DYG. As an example, one can read in offline the model DXG and DYG and create a characteristic length-scale as sqrt(DXG^2 + DYG^2), then multiply by a factor of 3 if one wants the smoothed (large scale) field to be of length-scale 3x that of the model grid spacing. The smoother number of time steps gencost_posproc_i can be the same as that used in data.smooth. Other options associated with the computation of Eq. (10.1) are summarized in Table 10.4 and further discussed below. Multiple gencost_preproc / gencost_posproc options may be specified per cost term.

In general the specification of gencost_name is optional, has no impact on the end-result, and only serves to distinguish between cost function terms amongst the model output (STDOUT.0000, STDERR.0000, costfunction000, misfit*.data). Exceptions listed in Table 10.6 however activate alternative cost function codes (in place of cost_generic.F) described in Section 10.1.3. In this section and in Table 10.3 (unlike in other parts of the manual) ‘zonal’ / ‘meridional’ are to be taken literally and these components are centered (i.e., not at the staggered model velocity points). Preparing gridded velocity data sets for use in cost functions thus boils down to interpolating them to XC / YC.

The gencost_kLev_select option allows the user to select the vertical level of a 3D model field, thereby creating a 2D field out of that slice which is used for the cost computation. For example, drifter velocities correspond to the second depth level of the grid used in ECCOv4, so model velocities are selected from this depth level to compare to the drifter observations. The user can specify this in data.ecco with: gencost_kLev_select ( i ) = 2, where i is replaced with the index for that cost function term.

Table 10.2 Run-time parameters used in formulating generic cost functions and defined via ecco_gencost_nml` namelist in data.ecco. All parameters are vectors of length NGENCOST (the # of available cost terms) except for gencost_proc* are arrays of size NGENPPROC\(\times\)NGENCOST (10 \(\times\)20 by default; can be changed in ECCO_SIZE.h at compile time). In addition, the gencost_is3d internal parameter is reset to true on the fly in all 3D cases in Table 10.3.

parameter	type	function
gencost_name	character(*)	Name of cost term
gencost_barfile	character(*)	File to receive model counterpart \(\vec{m}_i\) (See Table 10.3)
gencost_datafile	character(*)	File containing observational data \(\vec{o}_i\)
gencost_avgperiod	character(5)	Averaging period for \(\vec{o}_i\) and \(\vec{m}_i\) (see text)
gencost_outputlevel	integer	Greater than 0 will output misfit fields
gencost_errfile	character(*)	Uncertainty field name (not used in Section 10.1.2)
gencost_mask	character(*)	Mask file name root (used only in Section 10.1.2)
mult_gencost	real	Multiplier \(\alpha_i\) (default: 1)
gencost_preproc	character(*)	Preprocessor names
gencost_preproc_c	character(*)	Preprocessor character arguments
gencost_preproc_i	integer(*)	Preprocessor integer arguments
gencost_preproc_r	real(*)	Preprocessor real arguments
gencost_posproc	character(*)	Post-processor names
gencost_posproc_c	character(*)	Post-processor character arguments
gencost_posproc_i	integer(*)	Post-processor integer arguments
gencost_posproc_r	real(*)	Post-processor real arguments
gencost_spmin	real	Data less than this value will be omitted
gencost_spmax	real	Data greater than this value will be omitted
gencost_spzero	real	Data points equal to this value will be omitted
gencost_startdate1	integer	Start date of observations (YYYMMDD)
gencost_startdate2	integer	Start date of observations (HHMMSS)
gencost_is3d	logical	Needs to be true for 3D fields
gencost_enddate1	integer	Not fully implemented (used only in Section 10.1.3)
gencost_enddate2	integer	Not fully implemented (used only in Section 10.1.3)
gencost_kLev_select	integer	Vertical level of a 3D field to create a 2D field for cost computation
gencost_useDensityMask	logical	Needs to be true if density following feature is used
gencost_sigmaLow	real	Use to define minimum density surface chosen
gencost_sigmaHigh	real	Used to define maximum density surface chosen
gencost_refPressure	real	Defines reference pressure used in density following feature
gencost_tanhScale	real	Used in defining density levels in density following feature

Table 10.3 Implemented gencost_barfile options (as of checkpoint 65z) that can be used via cost_generic.F (Section 10.1.1). An extension starting with ‘_’ can be appended at the end of the variable name to distinguish between separate cost function terms. Note: the ‘m_eta’ formula depends on the ATMOSPHERIC_LOADING and ALLOW_PSBAR_STERIC compile-time options and ‘useRealFreshWaterFlux’ run-time parameter.

variable name	description	remarks
m_eta	sea surface height	free surface + ice + global steric correction
m_sst	sea surface temperature	first level potential temperature
m_sss	sea surface salinity	first level salinity
m_bp	bottom pressure	phiHydLow
m_siarea	sea-ice area	from pkg/seaice
m_siheff	sea-ice effective thickness	from pkg/seaice
m_sihsnow	snow effective thickness	from pkg/seaice
m_theta	potential temperature	three-dimensional
m_salt	salinity	three-dimensional
m_UE	zonal velocity	three-dimensional
m_VN	meridional velocity	three-dimensional
m_ustress	zonal wind stress	from pkg/exf
m_vstress	meridional wind stress	from pkg/exf
m_uwind	zonal wind	from pkg/exf
m_vwind	meridional wind	from pkg/exf
m_atemp	atmospheric temperature	from pkg/exf
m_aqh	atmospheric specific humidity	from pkg/exf
m_precip	precipitation	from pkg/exf
m_swdown	downward shortwave	from pkg/exf
m_lwdown	downward longwave	from pkg/exf
m_wspeed	wind speed	from pkg/exf
m_diffkr	vertical/diapycnal diffusivity	three-dimensional, constant
m_kapgm	GM diffusivity	three-dimensional, constant
m_kapredi	isopycnal diffusivity	three-dimensional, constant
m_geothermalflux	geothermal heat flux	constant
m_bottomdrag	bottom drag	constant

Table 10.4 gencost_preproc and gencost_posproc options implemented as of checkpoint 65z. Note: the distinction between gencost_preproc and gencost_posproc seems unclear and may be revisited in the future.

name	description	gencost_preproc_i , _r, or _c
gencost_preproc
clim	Use climatological misfits	integer: no. of records per climatological cycle
mean	Use time mean of misfits	—
anom	Use anomalies from time mean	—
variaweight	Use time-varying weight \(W_i\)	—
nosumsq	Use linear misfits	—
factor	Multiply \(\vec{m}_i\) by a scaling factor	real: the scaling factor
offset	subtract mean misfit	—
mindepth	mask (ignore) misfit above minimum depth	real: minimum water depth (\(< 0\))
gencost_posproc
smooth	Smooth misfits	character: smoothing scale file
		integer: smoother # of time steps

10.1.2. Generic Integral Function

The functionality described in this section is operated by cost_gencost_boxmean.F. It is primarily aimed at obtaining a mechanistic understanding of a chosen physical variable via adjoint sensitivity computations (see Automatic Differentiation) as done for example in [FWL+15, HWP+11, MGZ+99]. Thus the quadratic term in Eq. (10.1) (\(\vec{d}_i^T R_i^{-1} \vec{d}_i\)) is by default replaced with a \(d_i\) scalar 2 that derives from model fields through a generic integral formula (Eq. (10.3)). The specification of gencost_barfile again selects the physical variable type. Current valid options to use cost_gencost_boxmean.F are reported in Table 10.5. A suffix starting with ‘_’ can again be appended to gencost_barfile.

The integral formula is defined by masks provided via binary files which names are specified via gencost_mask. There are two cases: (1) if gencost_mask = ‘foo_mask’ and gencost_barfile is of the ‘m_boxmean*’ type then the model will search for horizontal, vertical, and temporal mask files named foo_maskC, foo_maskK, and foo_maskT; (2) if instead gencost_barfile is of the ‘m_horflux_’ type then the model will search for foo_maskW, foo_maskS, foo_maskK, and foo_maskT.

The ‘C’ mask or the ‘W’ / ‘S’ masks are expected to be two-dimensional fields. The ‘K’ and ‘T’ masks (both optional; all 1 by default) are expected to be one-dimensional vectors. The ‘K’ vector length should match Nr. The ‘T’ vector length should match the # of records that the specification of gencost_avgperiod implies but there is no restriction on its values. In case #1 (‘m_boxmean*’) the ‘C’ and ‘K’ masks should consists of +1 and 0 values and a volume average will be computed accordingly. In case #2 (‘m_horflux*’) the ‘W’, ‘S’, and ‘K’ masks should consists of +1, -1, and 0 values and an integrated horizontal transport (or overturn) will be computed accordingly.

In order to define a control volume using both a depth range and a density range, use a ‘K’ mask and also set gencost_useDensityMask =.TRUE.. When the density range feature is active, the control volume is defined at each timestep by the bounds set in the ‘K’ mask and also by the density range specified by the parameters gencost_sigmaLow (the minimum density to be included in the control volume) and gencost_sigmaHigh (the maximum density to be included in the control volume). As a default gencost_refPressure should be set to 0, but other values can be used (e.g. 1000 dbar, 2000 dbar).

Table 10.5 Implemented gencost_barfile options (as of checkpoint 67x) that can be used via cost_gencost_boxmean.F (Section 10.1.2).

variable name	description	remarks
m_boxmean_theta	mean of theta over box	specify box
m_boxmean_salt	mean of salt over box	specify box
m_boxmean_eta	mean of SSH over box	specify box
m_boxmean_shifwf	total shelfice freshwater flux over box	specify box
m_boxmean_shihf	total shelfice heat flux over box	specify box
m_boxmean_vol	total volume over box	specify box
m_horflux_vol	volume transport through section	specify transect

10.1.3. Custom Cost Functions

This section (very much a work in progress…) pertains to the special cases of cost_gencost_bpv4.F, cost_gencost_seaicev4.F, cost_gencost_sshv4.F, cost_gencost_sstv4.F, cost_gencost_transp.F, and cost_gencost_moc.F. The cost_gencost_transp.F function can be used to compute a transport of volume, heat, or salt through a specified section (non quadratic cost function). To this end one sets gencost_name = ‘transp*’, where * is an optional suffix starting with ‘_’, and set gencost_barfile to one of m_trVol, m_trHeat, and m_trSalt.

The cost_gencost_moc.F function is similar to transport function, but is intended to compute the meridional overturning streamfunction maximum based on the volumetric transport integrated from the floor to surface, as in Smith and Heimbach (2019) [SH19]. Therefore, this function is intended to work with gencost_barfile = m_trVol, and note that the first 3 characters of gencost_name must be moc, as depicted in Table 10.6. Users can specify a latitude band to compute the MOC with appropriately defined West (‘W’) and South (‘S’) masks as described in Section 10.1.2. As an example see parameter group (3) in this data.ecco file .

Note: the functionality in cost_gencost_transp.F is not regularly tested. Users interested in computing volumetric transports through a section are recommended to use the m_horflux_vol capabilities described above as it is regularly tested. Users interested in computing heat and salt transport should note the following about m_trHeat and m_trSalt:

1. The associated advection scheme with transports may be inconsistent with the model unless ENUM_CENTERED_2ND is implemented

2. Bolus velocities are not included

3. Diffusion components are not included

Table 10.6 Pre-defined gencost_name special cases (as of checkpoint 65z; Section 10.1.3).

name	description	remarks
sshv4-mdt	sea surface height	mean dynamic topography (SSH - geod)
sshv4-tp	sea surface height	Along-Track Topex/Jason SLA (level 3)
sshv4-ers	sea surface height	Along-Track ERS/Envisat SLA (level 3)
sshv4-gfo	sea surface height	Along-Track GFO class SLA (level 3)
sshv4-lsc	sea surface height	Large-Scale SLA (from the above)
sshv4-gmsl	sea surface height	Global-Mean SLA (from the above)
bpv4-grace	bottom pressure	GRACE maps (level 4)
sstv4-amsre	sea surface temperature	Along-Swath SST (level 3)
sstv4-amsre-lsc	sea surface temperature	Large-Scale SST (from the above)
si4-cons	sea ice concentration	needs sea-ice adjoint (level 4)
si4-deconc	model sea ice deficiency	proxy penalty (from the above)
si4-exconc	model sea ice excess	proxy penalty (from the above)
transp_trVol	volume transport	specify masks (Section 10.1.2)
transp_trHeat	heat transport	specify masks (Section 10.1.2)
transp_trSalt	salt transport	specify masks (Section 10.1.2)
moc_trVol	meridional ovt. streamfn. maximum	specify masks (Section 10.1.2)

10.1.4. Key Routines

TBA … ecco_readparms.F, ecco_check.F, ecco_summary.F, cost_generic.F, cost_gencost_boxmean.F, ecco_toolbox.F, ecco_phys.F, cost_gencost_customize.F, cost_averagesfields.F, …

10.1.5. Compile Options

TBA … ALLOW_GENCOST_CONTRIBUTION, ALLOW_GENCOST3D, ALLOW_PSBAR_STERIC, ALLOW_SHALLOW_ALTIMETRY, ALLOW_HIGHLAT_ALTIMETRY, ALLOW_PROFILES_CONTRIBUTION, ALLOW_ECCO_OLD_FC_PRINT, …

packages required for some functionalities: smooth, profiles, ctrl

10.2. PROFILES: model-data comparisons at observed locations

Author: Gael Forget

The purpose of pkg/profiles is to allow sampling of MITgcm runs according to a chosen pathway (after a ship or a drifter, along altimeter tracks, etc.), typically leading to easy model-data comparisons. Given input files that contain positions and dates, pkg/profiles will interpolate the model trajectory at the observed location. In particular, pkg/profiles can be used to do model-data comparison online and formulate a least-squares problem (ECCO application).

The pkg/profiles namelist is called data.profiles. In the example below, it includes two input netcdf file names (ARGOifremer_r8.nc and XBT_v5.nc) that should be linked to the run directory and cost function multipliers that only matter in the context of automatic differentiation (see Automatic Differentiation). The first index is a file number and the second index (in mult* only) is a variable number. By convention, the variable number is an integer ranging 1 to 6: temperature, salinity, zonal velocity, meridional velocity, sea surface height anomaly, and passive tracer.

The netcdf input file structure is illustrated in the case of XBT_v5.nc To create such files, one can use the MITprof matlab toolbox obtained from https://github.com/gaelforget/MITprof . At run time, each file is scanned to determine which variables are included; these will be interpolated. The (final) output file structure is similar but with interpolated model values in prof_T etc., and it contains model mask variables (e.g. prof_Tmask). The very model output consists of one binary (or netcdf) file per processor. The final netcdf output is to be built from those using netcdf_ecco_recompose.m (offline).

When the k2 option is used (e.g. for cubed sphere runs), the input file is to be completed with interpolation grid points and coefficients computed offline using netcdf_ecco_GenericgridMain.m. Typically, you would first provide the standard namelist and files. After detecting that interpolation information is missing, the model will generate special grid files (profilesXCincl1PointOverlap* etc.) and then stop. You then want to run netcdf_ecco_GenericgridMain.m using the special grid files. This operation could eventually be inlined.

Example: data.profiles

#
# \*****************\*
# PROFILES cost function
# \*****************\*
&PROFILES_NML
#
profilesfiles(1)= ’ARGOifremer_r8’,
mult_profiles(1,1) = 1.,
mult_profiles(1,2) = 1.,
profilesfiles(2)= ’XBT_v5’,
mult_profiles(2,1) = 1.,
#
/

Example: XBT_v5.nc

netcdf XBT_v5 {
dimensions:
iPROF = 278026 ;
iDEPTH = 55 ;
lTXT = 30 ;
variables:
double depth(iDEPTH) ;
depth:units = "meters" ;
double prof_YYYYMMDD(iPROF) ;
prof_YYYYMMDD:missing_value = -9999. ;
prof_YYYYMMDD:long_name = "year (4 digits), month (2 digits), day (2 digits)" ;
double prof_HHMMSS(iPROF) ;
prof_HHMMSS:missing_value = -9999. ;
prof_HHMMSS:long_name = "hour (2 digits), minute (2 digits), second (2 digits)" ;
double prof_lon(iPROF) ;
prof_lon:units = "(degree E)" ;
prof_lon:missing_value = -9999. ;
double prof_lat(iPROF) ;
prof_lat:units = "(degree N)" ;
prof_lat:missing_value = -9999. ;
char prof_descr(iPROF, lTXT) ;
prof_descr:long_name = "profile description" ;
double prof_T(iPROF, iDEPTH) ;
prof_T:long_name = "potential temperature" ;
prof_T:units = "degree Celsius" ;
prof_T:missing_value = -9999. ;
double prof_Tweight(iPROF, iDEPTH) ;
prof_Tweight:long_name = "weights" ;
prof_Tweight:units = "(degree Celsius)-2" ;
prof_Tweight:missing_value = -9999. ;
}

10.3. CTRL: Model Parameter Adjustment Capability

Author: Gael Forget

Package ctrl provides an interface to defining the control variables for an optimization. After defining CPP-flags ALLOW_GENTIM2D_CONTROL, ALLOW_GENARR2D_CONTROL, ALLOW_GENARR3D_CONTROL in CTRL_OPTIONS.h <pkg/ctrl/CTRL_OPTIONS.h, the parameters available for configuring generic cost terms in data.ctrl are given in Table 10.7. The control variables are stored as fields on the model grid in files $ctrlvar.$iternumber.data/meta, and corresponding gradients in ad$ctrlvar.$iternumber.data/meta, where $ctrl is defined in data.ctrl (see Table 10.8 for possible options) and $iternumber is the 10-digit iteration number of the optimization. Further, ctrl maps the gradient fields to a vector that can be handed over to an optimization routine (see Section 10.5) and maps the resulting new control vector to the model grid unless CPP-flag EXCLUDE_CTRL_PACK is defined in CTRL_OPTIONS.h.

Table 10.7 Parameters in ctrl_nml_genarr namelist in data.ctrl. The * can be replaced by arr2d, arr3d, or tim2d for time-invariant two and three dimensional controls and time-varying 2D controls, respectively. Parameters for genarr2d, genarr3d, and gentime2d are arrays of length maxCtrlArr2D, maxCtrlArr3D, and maxCtrlTim2D, respectively, with one entry per term in the cost function.

parameter	type	function
xx_gen*_file	character(*)	Control Name: prefix from Table 10.8 + suffix.
xx_gen*_weight	character(*)	Weights in the form of \(\sigma_{\vec{u }_j}^{-2}\)
xx_gen*_bounds	real(5)	Apply bounds
xx_gen*_preproc	character(*)	Control preprocessor(s) (see Table 10.9 )
xx_gen*_preproc_c	character(*)	Preprocessor character arguments (see Table 10.10)
xx_gen*_preproc_i	integer(*)	Preprocessor integer arguments
xx_gen*_preproc_r	real(*)	Preprocessor real arguments
gen*Precond	real	Preconditioning factor (\(=1\) by default)
mult_gen*	real	Cost function multiplier \(\beta_j\) (\(= 1\) by default)
xx_gentim2d_period	real	Frequency of adjustments (in seconds)
xx_gentim2d_startda te1	integer	Adjustment start date
xx_gentim2d_startda te2	integer	Default: model start date
xx_gentim2d_cumsum	logical	Accumulate control adjustments
xx_gentim2d_glosum	logical	Global sum of adjustment (output is still 2D)

Table 10.8 Generic control prefixes implemented as of checkpoint 67x.

	name	description
2D, time-invariant controls	genarr2d
	xx_etan	initial sea surface height
	xx_bottomdrag	bottom drag
	xx_geothermal	geothermal heat flux
	xx_shicoefft	shelfice thermal transfer coefficient (see Section 10.3.1)
	xx_shicoeffs	shelfice salinity transfer coefficient (see Section 10.3.1)
	xx_shicdrag	shelfice drag coefficient (see Section 10.3.1)
	xx_depth	bottom topography requires to define ALLOW_DEPTH_CONTROL
	xx_siheff	initial sea ice thickness
	xx_siarea	initial sea ice area
3D, time-invariant controls	genarr3d
	xx_theta	initial potential temperature
	xx_salt	initial salinity
	xx_uvel	initial zonal velocity
	xx_vvel	initial meridional velocity
	xx_kapgm	GM coefficient
	xx_kapredi	isopycnal diffusivity
	xx_diffkr	diapycnal diffusivity
2D, time-varying controls	gentim2D
	xx_atemp	atmospheric temperature
	xx_aqh	atmospheric specific humidity
	xx_swdown	downward shortwave
	xx_lwdown	downward longwave
	xx_precip	precipitation
	xx_runoff	river runoff
	xx_uwind	zonal wind
	xx_vwind	meridional wind
	xx_tauu	zonal wind stress
	xx_tauv	meridional wind stres
	xx_gen_precip	globally averaged precipitation?
	xx_hflux	net heat flux
	xx_sflux	net salt (EmPmR) flux
	xx_shifwflx	shelfice melt rate

Table 10.9 xx_gen????d_preproc options implemented as of checkpoint 67x. Notes: \(^a\): If noscaling is false, the control adjustment is scaled by one on the square root of the weight before being added to the base control variable; if noscaling is true, the control is multiplied by the weight in the cost function itself.

name	description	arguments
WC01	Correlation modeling	integer: operator type (default: 1)
smooth	Smoothing without normalization	integer: operator type (default: 1)
docycle	Average period replication	integer: cycle length
replicate	Alias for docycle	(units of xx_gentim2d_period)
rmcycle	Periodic average subtraction	integer: cycle length
variaweight	Use time-varying weight	—
noscaling \(^{a}\)	Do not scale with xx_gen*_weight	—
documul	Sets xx_gentim2d_cumsum	—
doglomean	Sets xx_gentim2d_glosum	—

Table 10.10 xx_gen????d_preproc_c options implemented as of checkpoint 67x.

name	description	arguments
log10ctrl	Control adjustments to base 10 logarithm of 2D or 3D array (not available for xx_gentim2d).	See Section 10.3.2

The control problem is non-dimensional by default, as reflected in the omission of weights in control penalties [(\(\vec{u}_j^T\vec{u}_j\) in (10.1)]. Non-dimensional controls (\(\vec{u}_j\)) are scaled to physical units (\(\vec{v}_j\)) through multiplication by the respective uncertainty fields (\(\sigma_{\vec{u}_j}\)), as part of the generic preprocessor \(\mathcal{Q}\) in (10.4). Besides the scaling of \(\vec{u}_j\) to physical units, the preprocessor \(\mathcal{Q}\) can include, for example, spatial correlation modeling (using an implementation of Weaver and Coutier, 2001) by setting xx_gen*_preproc = ’WC01’. Alternatively, setting xx_gen*_preproc = ’smooth’ activates the smoothing part of WC01, but omits the normalization. Additionally, bounds for the controls can be specified by setting xx_gen*_bounds. In forward mode, adjustments to the \(i^\text{th}\) control are clipped so that they remain between xx_gen*_bounds(i,1) and xx_gen*_bounds(i,4). If xx_gen*_bounds(i,1) \(<\) xx_gen*_bounds(i+1,1) for \(i = 1, 2, 3\), then the bounds will “emulate a local minimum;” otherwise, the bounds have no effect in adjoint mode.

For the case of time-varying controls, the frequency is specified by xx_gentim2d_period. The generic control package interprets special values of xx_gentim2d_period in the same way as the exf package: a value of \(-12\) implies cycling monthly fields while a value of \(0\) means that the field is steady. Time varying weights can be provided by specifying the preprocessor variaweight, in which case the xx_gentim2d_weight file must contain as many records as the control parameter time series itself (approximately the run length divided by xx_gentim2d_period).

The parameter mult_gen* sets the multiplier for the corresponding cost function penalty [\(\beta_j\) in (10.1); \(\beta_j = 1\) by default). The preconditioner, \(\cal{R}\), does not directly appear in the estimation problem, but only serves to push the optimization process in a certain direction in control space; this operator is specified by gen*Precond (\(=1\) by default).

Note that control parameters exist for each individual near surface atmospheric state variable, as well as the net heat and salt (EmPmR) fluxes. The user must be mindful of control parameter combinations that make sense according to their specific setup, e.g., with the EXF package.

10.3.1. Shelfice Control Parameters

The available iceshelf control parameters depend on the form of transfer coefficient used in the simulation.

The adjustments xx_shicoefft and xx_shicoeffs are available when the velocity independent form of transfer coefficients is used, by setting #undef SHI_ALLOW_GAMMAFRICT in SHELFICE_OPTIONS.h at compile time (see Table 8.22) and SHELFICEuseGammaFrict =.FALSE. in data.shelfice (see Table 8.23). These parameters provide adjustments to \(\gamma_T\) and/or \(\gamma_S\) directly. If only one of either is used, the value of the other is set based on the control adjustments used together with SHELFICEsaltToHeatRatio, which can be set in data.shelfice. See Run-time parameters and default values; all parameters are in namelist group SHELFICE_PARM01 for the default.

The adjustment xx_shicdrag is available in the velocity dependent form of the ice-ocean transfer coefficients, which is specified by #define SHI_ALLOW_GAMMAFRICT and SHELFICEuseGammaFrict =.TRUE. at compile time and run time respectively. This parameter provides adjustments to the drag coefficient at the ice ocean boundary, but by default only adjusts the drag coefficient used to compute the thermal and freshwater fluxes, neglecting the momentum contributions. To allow the contribution directly to momentum fluxes, specify xx_genarr2d_preproc_c(*,iarr) = 'mom' in data.ctrl.

10.3.2. Logarithmic Control Parameters

As indicated in Table 10.10, the base-10 logarithm of a control field can be adjusted by specifying the character option genarr*d_preproc_c(k2,iarr) = 'log10ctrl', with k2 and iarr as appropriate, and *d denoting that 2d or 3d are available. As a concrete example, if the control parameter is updating fld2d, then the field will be set as follows:

fld2d(i,j,bi,bj) = 10**( log10InitVal + xx_genarr2d(i,j,bi,bj,iarr) )

where log10InitVal is a scalar with a default value of 0, but can be changed by setting gencost_preproc_r(k2,iarr). This is useful in the case where doInitXX=.TRUE.. Concretely, if we had an initial guess for fld2d = 10^-4 then one could set the following in data.ctrl:

xx_genarr2d_file(1) = 'xx_fld2d'
xx_genarr2d_weight(1) = 'nonzero_weights.data'
xx_genarr2d_preproc_c(1,1) = 'log10ctrl'
xx_genarr2d_preproc_r(1,1) = -4. ,

Note that the log10ctrl option can only be used when a weight file is provided, and finally that this log-option cannot be used with xx_gen*_preproc(k2,iarr) = 'noscaling',.

10.4. SMOOTH: Smoothing And Covariance Model

Author: Gael Forget

TO BE CONTINUED…

10.5. The line search optimisation algorithm

Author: Patrick Heimbach

10.5.1. General features

The line search algorithm is based on a quasi-Newton variable storage method which was implemented by [GLemarechal89].

TO BE CONTINUED…

10.5.2. The online vs. offline version

• Online version

  Every call to simul refers to an execution of the forward and adjoint model. Several iterations of optimization may thus be performed within a single run of the main program (lsopt_top). The following cases may occur:

  • cold start only (no optimization)

  • cold start, followed by one or several iterations of optimization

  • warm start from previous cold start with one or several iterations

  • warm start from previous warm start with one or several iterations

• Offline version

  Every call to simul refers to a read procedure which reads the result of a forward and adjoint run Therefore, only one call to simul is allowed, itmax = 0, for cold start itmax = 1, for warm start Also, at the end, x(i+1) needs to be computed and saved to be available for the offline model and adjoint run

In order to achieve minimum difference between the online and offline code xdiff(i+1) is stored to file at the end of an (offline) iteration, but recomputed identically at the beginning of the next iteration.

10.5.3. Number of iterations vs. number of simulations

- itmax: controls the max. number of iterations

- nfunc: controls the max. number of simulations within one iteration

10.5.3.1. Summary

From one iteration to the next the descent direction changes. Within one iteration more than one forward and adjoint run may be performed. The updated control used as input for these simulations uses the same descent direction, but different step sizes.

10.5.3.2. Description

From one iteration to the next the descent direction dd changes using the result for the adjoint vector gg of the previous iteration. In lsline the updated control

\[\tt xdiff(i,1) = xx(i-1) + tact(i-1,1)*dd(i-1)\]

serves as input for a forward and adjoint model run yielding a new gg(i,1). In general, the new solution passes the 1st and 2nd Wolfe tests so xdiff(i,1) represents the solution sought:

\[{\tt xx(i) = xdiff(i,1)}\]

If one of the two tests fails, an inter- or extrapolation is invoked to determine a new step size tact(i-1,2). If more than one function call is permitted, the new step size is used together with the “old” descent direction dd(i-1) (i.e. dd is not updated using the new gg(i)), to compute a new

\[{\tt xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1)}\]

that serves as input in a new forward and adjoint run, yielding gg(i,2). If now, both Wolfe tests are successful, the updated solution is given by

\[\tt xx(i) = xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1)\]

In order to save memory both the fields dd and xdiff have a double usage.

• 
  

  - in lsopt_top: used as x(i) - x(i-1) for Hessian update

  - in lsline: intermediate result for control update x = x + tact*dd
• 
  

  - in lsopt_top, lsline: descent vector, dd = -gg and hessupd

  - in dgscale: intermediate result to compute new preconditioner

10.5.3.3. The parameter file lsopt.par

• NUPDATE max. no. of update pairs (gg(i)-gg(i-1), xx(i)-xx(i-1)) to be stored in OPWARMD to estimate Hessian [pair of current iter. is stored in (2*jmax+2, 2*jmax+3) jmax must be > 0 to access these entries] Presently NUPDATE must be > 0 (i.e. iteration without reference to previous iterations through OPWARMD has not been tested)

• EPSX relative precision on xx bellow which xx should not be improved

• EPSG relative precision on gg below which optimization is considered successful

• IPRINT controls verbose (>=1) or non-verbose output

• NUMITER max. number of iterations of optimisation; NUMTER = 0: cold start only, no optimization

• ITER_NUM index of new restart file to be created (not necessarily = NUMITER!)

• NFUNC max. no. of simulations per iteration (must be > 0); is used if step size tact is inter-/extrapolated; in this case, if NFUNC > 1, a new simulation is performed with same gradient but “improved” step size

• FMIN first guess cost function value (only used as long as first iteration not completed, i.e. for jmax <= 0)

10.5.3.4. OPWARMI, OPWARMD files

Two files retain values of previous iterations which are used in latest iteration to update Hessian:

• OPWARMI: contains index settings and scalar variables

  n = nn	no. of control variables
  fc = ff	cost value of last iteration
  isize	no. of bytes per record in OPWARMD
  m = nupdate	max. no. of updates for Hessian
  jmin, jmax	pointer indices for OPWARMD file (cf. below)
  gnorm0	norm of first (cold start) gradient gg
  iabsiter	total number of iterations with respect to cold start

• OPWARMD: contains vectors (control and gradient)

  entry	name	description
  1	xx(i)	control vector of latest iteration
  2	gg(i)	gradient of latest iteration
  3	xdiff(i),diag	preconditioning vector; (1,…,1) for cold start
  2*jmax+2	gold=g(i)-g(i-1)	for last update (jmax)
  2*jmax+3	xdiff=tact*d=xx(i)-xx (i-1)	for last update (jmax)

Example 1: jmin = 1, jmax = 3, mupd = 5

  1   2   3   |   4   5     6   7     8   9     empty     empty
|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___|
      0       |     1         2         3

Example 2: jmin = 3, jmax = 7, mupd = 5   ---> jmax = 2

  1   2   3   |
|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___|
              |     6         7         3         4         5

10.5.3.5. Error handling

lsopt_top
    |
    |---- check arguments
    |---- CALL INSTORE
    |       |
    |       |---- determine whether OPWARMI available:
    |                * if no:  cold start: create OPWARMI
    |                * if yes: warm start: read from OPWARMI
    |             create or open OPWARMD
    |
    |---- check consistency between OPWARMI and model parameters
    |
    |---- >>> if COLD start: <<<
    |      |  first simulation with f.g. xx_0; output: first ff_0, gg_0
    |      |  set first preconditioner value xdiff_0 to 1
    |      |  store xx(0), gg(0), xdiff(0) to OPWARMD (first 3 entries)
    |      |
    |     >>> else: WARM start: <<<
    |         read xx(i), gg(i) from OPWARMD (first 2 entries)
    |         for first warm start after cold start, i=0
    |
    |
    |
    |---- /// if ITMAX > 0: perform optimization (increment loop index i)
    |      (
    |      )---- save current values of gg(i-1) -> gold(i-1), ff -> fold(i-1)
    |      (---- CALL LSUPDXX
    |      )       |
    |      (       |---- >>> if jmax=0 <<<
    |      )       |      |  first optimization after cold start:
    |      (       |      |  preconditioner estimated via ff_0 - ff_(first guess)
    |      )       |      |  dd(i-1) = -gg(i-1)*preco
    |      (       |      |
    |      )       |     >>> if jmax > 0 <<<
    |      (       |         dd(i-1) = -gg(i-1)
    |      )       |         CALL HESSUPD
    |      (       |           |
    |      )       |           |---- dd(i-1) modified via Hessian approx.
    |      (       |
    |      )       |---- >>> if <dd,gg> >= 0 <<<
    |      (       |         ifail = 4
    |      )       |
    |      (       |---- compute step size: tact(i-1)
    |      )       |---- compute update: xdiff(i) = xx(i-1) + tact(i-1)*dd(i-1)
    |      (
    |      )---- >>> if ifail = 4 <<<
    |      (         goto 1000
    |      )
    |      (---- CALL OPTLINE / LSLINE
    |      )       |
   ...    ...     ...

...    ...
 |      )
 |      (---- CALL OPTLINE / LSLINE
 |      )       |
 |      (       |---- /// loop over simulations
 |      )              (
 |      (              )---- CALL SIMUL
 |      )              (       |
 |      (              )       |----  input: xdiff(i)
 |      )              (       |---- output: ff(i), gg(i)
 |      (              )       |---- >>> if ONLINE <<<
 |      )              (                 runs model and adjoint
 |      (              )             >>> if OFFLINE <<<
 |      )              (                 reads those values from file
 |      (              )
 |      )              (---- 1st Wolfe test:
 |      (              )     ff(i) <= tact*xpara1*<gg(i-1),dd(i-1)>
 |      )              (
 |      (              )---- 2nd Wolfe test:
 |      )              (     <gg(i),dd(i-1)> >= xpara2*<gg(i-1),dd(i-1)>
 |      (              )
 |      )              (---- >>> if 1st and 2nd Wolfe tests ok <<<
 |      (              )      |  320: update xx: xx(i) = xdiff(i)
 |      )              (      |
 |      (              )     >>> else if 1st Wolfe test not ok <<<
 |      )              (      |  500: INTERpolate new tact:
 |      (              )      |  barr*tact < tact < (1-barr)*tact
 |      )              (      |  CALL CUBIC
 |      (              )      |
 |      )              (     >>> else if 2nd Wolfe test not ok <<<
 |      (              )         350: EXTRApolate new tact:
 |      )              (         (1+barmin)*tact < tact < 10*tact
 |      (              )         CALL CUBIC
 |      )              (
 |      (              )---- >>> if new tact > tmax <<<
 |      )              (      |  ifail = 7
 |      (              )      |
 |      )              (---- >>> if new tact < tmin OR tact*dd < machine precision <<<
 |      (              )      |  ifail = 8
 |      )              (      |
 |      (              )---- >>> else <<<
 |      )              (         update xdiff for new simulation
 |      (              )
 |      )             \\\ if nfunc > 1: use inter-/extrapolated tact and xdiff
 |      (                               for new simulation
 |      )                               N.B.: new xx is thus not based on new gg, but
 |      (                                     rather on new step size tact
 |      )
 |      (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries)
 |      )---- >>> if ifail = 7,8,9 <<<
 |      (         goto 1000
 |      )
...    ...

...    ...
 |      )
 |      (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries)
 |      )---- >>> if ifail = 7,8,9 <<<
 |      (         goto 1000
 |      )
 |      (---- compute new pointers jmin, jmax to include latest values
 |      )     gg(i)-gg(i-1), xx(i)-xx(i-1) to Hessian matrix estimate
 |      (---- store gg(i)-gg(i-1), xx(i)-xx(i-1) to OPWARMD
 |      )     (entries 2*jmax+2, 2*jmax+3)
 |      (
 |      )---- CALL DGSCALE
 |      (       |
 |      )       |---- call dostore
 |      (       |       |
 |      )       |       |---- read preconditioner of previous iteration diag(i-1)
 |      (       |             from OPWARMD (3rd entry)
 |      )       |
 |      (       |---- compute new preconditioner diag(i), based upon diag(i-1),
 |      )       |     gg(i)-gg(i-1), xx(i)-xx(i-1)
 |      (       |
 |      )       |---- call dostore
 |      (               |
 |      )               |---- write new preconditioner diag(i) to OPWARMD (3rd entry)
 |      (
 |---- \\\ end of optimization iteration loop
 |
 |
 |
 |---- CALL OUTSTORE
 |       |
 |       |---- store gnorm0, ff(i), current pointers jmin, jmax, iterabs to OPWARMI
 |
 |---- >>> if OFFLINE version <<<
 |         xx(i+1) needs to be computed as input for offline optimization
 |          |
 |          |---- CALL LSUPDXX
 |          |       |
 |          |       |---- compute dd(i), tact(i) -> xdiff(i+1) = x(i) + tact(i)*dd(i)
 |          |
 |          |---- CALL WRITE_CONTROL
 |          |       |
 |          |       |---- write xdiff(i+1) to special file for offline optim.
 |
 |---- print final information
 |
 O

1

ecco_check may be missing a test for conflicting names…

2

The quadratic option in fact does not yet exist in cost_gencost_boxmean.F…

10.5.4. Alternative code to optim and lsopt

The non-MITgcm package optim_m1qn3 is based on the same quasi-Newton variable storage method (BFGS) [GLemarechal89] as the package in subdirectory lsopt, but it uses a reverse communication version of the latest (and probably last) release of the subroutine m1qn3. This avoids having to define a dummy subroutine simul and also simplifies the code structure. As a consequence this package is simple(r) to compile and use, because m1qn3.f contains all necessary subroutines and only one extra routine (ddot, which was copied from BLAS) is required.

The principle of reverse communication is outlined in this example:

external simul_rc
...
reverse = .true.
do while (.true.)
  call m1qn3 (simul_rc,...,x,f,g,...,reverse,indic,...)
  if (reverse) break
  call simul (indic,n,x,f,g)
end while

simul_rc is an empty ‘’model simulator’’, and simul generates a new state based on the value of indic.

The original m1qn3 has been modified to work “offline”, i.e. the simulator and the driver of m1qn3_offline are separate programs that are called alternatingly from a (shell-)script. This requires that the “state” of m1qn3 is saved before this program terminates. This state is saved in a single file OPWARM.optXXX per simulation, where XXX is the simulation number. Communication with the routine, writing and restoring the state of m1qn3 is achieved via three new common-blocks that are contained in three header files. simul is replaced by reading and storing the model state and gradient vectors. Schematically the driver routine optim_sub does the following:

external simul_rc
...

call optim_readdata( nn, ctrlname, ...,   xx ) ! read control vector
call optim_readdata( nn, costname, ..., adxx ) ! read gradient vector
call optim_store_m1qn3( ..., .false. )         ! read state of m1qn3
reverse = .true.
call m1qn3 (simul_rc,...,xx,objf,adxx,...,reverse,indic,...)
call optim_store_m1qn3( ..., .true. )          ! write state of m1qn3
call optim_writedata( nn, ctrlname, ..., xx )  ! write control vector

The optimization loop is executed outside of this program within a script.

The code can be obtained at https://github.com/mjlosch/optim_m1qn3. The README contains short instructions how to build and use the code in combination with the verification/tutorial_global_oce_optim experiment. The usage is very similar to the optim package.

10.6. Test Cases For Estimation Package Capabilities

First, if you have not done so already, download the model as explained in Getting Started with MITgcm via the MITgcm git repository:

% git clone https://github.com/MITgcm/MITgcm.git

Then, download the setup from the MITgcm verification_other git repository:

% cd MITgcm
% git clone https://github.com/MITgcm/verification_other

and follow the additional directions provided for global_oce_cs32 or for global_oce_llc90. These model configurations are used for daily regression tests to ensure continued availability of the tested estimation package features discussed in Ocean State Estimation Packages. Daily results of these tests, which currently run on the glacier cluster, are reported on https://mitgcm.org/testing-summary. To this end, one sets a crontab job that typically executes the script reported below. The various commands can also be used to run these examples outside of crontab, directly at the command line via the testreport capability.

Note

Users are advised against running global_oce_llc90 tests with fewer than 12 cores (96 for adjoint tests) to avoid potential memory overloads. global_oce_llc90 (595M) uses the same LLC90 grid as the production ECCO version 4 setup [FCH+15]. The much coarser resolution global_oce_cs32 (614M) uses the CS32 grid and can run on any modern laptop.

% #!/bin/csh -f
% setenv PATH ~/bin:$PATH
% setenv MODULESHOME /usr/share/Modules
% source /usr/share/Modules/init/csh
% module use /usr/share/Modules
% module load openmpi-x86_64
% setenv MPI_INC_DIR $MPI_INCLUDE
%
% cd ~/MITgcm
% #mkdir gitpull.log
% set D=`date +%Y-%m-%d`
% git pull -v > gitpull.log/gitpull.$D.log
%
% cd verification
%
% #ieee case:
% ./testreport -clean -t 'global_oce_*'
% ./testreport -of=../tools/build_options/linux_amd64_gfortran -MPI 24 -t 'global_oce_*' -addr username@something.whatever
% ../tools/do_tst_2+2 -t 'global_oce_*' -mpi -exe 'mpirun -np 24 ./mitgcmuv' -a username@something.whatever
%
% #devel case:
% ./testreport -clean -t 'global_oce_*'
% ./testreport -of=../tools/build_options/linux_amd64_gfortran -MPI 24 -devel -t 'global_oce_*' -addr username@something.whatever
% ../tools/do_tst_2+2 -t 'global_oce_*' -mpi -exe 'mpirun -np 24 ./mitgcmuv' -a username@something.whatever
%
% #fast case:
% ./testreport -clean -t 'global_oce_*'
% ./testreport -of=../tools/build_options/linux_amd64_gfortran -MPI 24 -t 'global_oce_*' -fast -addr username@something.whatever
% ../tools/do_tst_2+2 -t 'global_oce_*' -mpi -exe 'mpirun -np 24 ./mitgcmuv' -a username@something.whatever
%
% #adjoint case:
% ./testreport -clean -t 'global_oce_*'
% ./testreport -of=../tools/build_options/linux_amd64_gfortran -MPI 24 -ad -t 'global_oce_*' -addr username@something.whatever