ISAS-Tool

Version 5.3: Method and configuration

L a b o r a t o i r e d e P h y s i q u e d e O c a n s , U M R 6 5 2 3

History
Auteur F. Gaillard R. Charraudeau F. Gaillard F. Gaillard F. Gaillard F. Gaillard F. Gaillard F. Gaillard Mise jour Cration du document V4 beta V4.00 Version franaise V4.01 Version franaise V4.1b - English version Minor corrections V5.1 V5.2b Date 03/02/2007 23/11/2007 11/02/2008 19/03/2008 25/09/2008 18/06/2009 11/01/2010

V5.3: Split method and configuration/program 15/12/2010

Content
1 Introduction ................................................................................................................. 5 2 Estimation method ....................................................................................................... 5 3 Grid and bathymetry .................................................................................................... 6 4 Reference climatology .................................................................................................. 8 4.1 Mean field from NODC climatology .................................................................................................................. 8 4.2 Mean field from ISAS climatology ..................................................................................................................... 8 4.3 Variance ....................................................................................................................................................................... 9 5. Covariance scales ...................................................................................................... 10 5 Areas and masks ........................................................................................................ 11 6 References ................................................................................................................. 12

1 Introduction

ISAS (In Situ Analysis System) is an analysis tool for the temperature and salinity fields. Originally designed for the synthesis of ARGO dataset, it has been tested for the first time on the POMME area in the North-East Atlantic in 2000, it was later extended to the Atlantic and the Global ocean as the Argo array was setting up. It is developed and maintained at LPO (Laboratoire de Physique des Ocans) within the Argo Observing Service (SO-ARGO) where it is used for research purposes on ocean variability. ISAS is made available to the Coriolis datacenter for exploitation in operational mode. It can accommodate a wide range of in situ measurements if they are provided in the standard NetCdf format distributed by the Coriolis datacenter (http://www.coriolis.eu.org/ ) . It is based on optimal interpolation and the estimated quantity is the anomaly on depth levels relative to a reference climatology. ISAS is uni-variate, which means that temperature and salinity variables are estimated independently. This document describes the statistical method used to produce the estimate and the specific choices performed to implement the method. The practical implementation and use of the tool is described in the users manual corresponding to the appropriate ISAS version.

2 Estimation method

ISAS uses estimation theory for mapping a scalar field on a regular grid from sparse and irregular data (Bretherton et al.,1976) the first implementation is described in Gaillard et al. (2008). We use here the unified terminology recommended for data assimilation (Ide et al, 1997). The interpolated field, represented by the state vector x , is constructed as the departure from the value of a reference field at the grid points . This reference is derived from previous knowledge (climatology or forecast). Only the unpredicted part of the observation vector, or departure from the reference field at the data points, called ! innovation, is used:
Eq. 1 d = yo " x f a The analyzed field x is a linear least square estimator, obtained as the linear combination of the innovation that minimizes the statistical error. A covariance matrix of error is associated to this solution ( P a ). ! ! x a = x f + K OI d
a OI T P ! = P " K Cao In the objective analysis formalism, the gain matrix K OI is built from the matrices that express the covariance of the field, from grid point to data point and from data point to data point and the observation noise covariance matrix. ! ! "1 K OI = Cao (Coo + R) Eq. 3

Eq. 2

!
5

The error on the estimation is given by the diagonal of this matrix, usually presented as a percentage of the a priori variance. This solution makes implicit use of an observation or mapping matrix H , such that : y o = Hx + " , which by analogy with the Ide et al. (1997) formalism can be expressed as: HT = P "1Cao .

! !

! It should be noticed that this formalism provides at the same time an estimate of the misfit between observations and analysis, also called analysis residuals:

" = y o # Hx a
#1

" = R(Coo + R) d We make use of these residuals in order to detect erroneous data, either outliers, biases or drifts. The advantage of such method is that the residuals are computed with the correct mapping matrix. Moreover, it is not necessary to perform an analysis at each ! the residuals, they are obtained at once for the whole data set. data point to obtain
The definition of the variables is as follows: -

Eq. 4

! ! ! ! ! ! ! ! ! !

d : Innovation vector y o : Observation vector x f : Climatology or forecast at observation location (vector) x a : Value of the analyzed field at the grid points (vector) " : Observation error (measurement noise + representativity error) " : Data misfit or residual (vector) K OI : Optimal estimation matrix, similar to the Kalman gain matrix P : A priori Covariance matrix of the anomaly field at grid points P a : Covariance matrix of the error on the analyzed field R : Covariance matrix of the error on observations (cumulates measurement error and representation error) Cao : Covariance matrix of the anomaly field between grid points and observation points Coo : Covariance matrix of the anomaly field at observation points.

! !

3 Grid and bathymetry

The horizontal grid is degree Mercator from 77S to 66.5N, where it is thus isotropic with a resolution that increases with latitude, from 66.5N to the North pole, the latitude step is fixed. The vertical resolution increases from 20 m at 2000m to 5 meter in the upper layer. Near surface two levels have been added at 0 and 3 m. The grid properties are summarized Figure 1. The bathymetry is an interpolation over our grid of the file etopo2bedmap.nc produced by MERCATOR from the 2 minutes bathymetry file of the NGDG Bathy_Etopo2.nc. The interpolation is done using the median of the 4 surrounding points.

Levels (m): [0 3] [5:100] [110:800] [820:2000]

Step 3 5 10 20

Figure 1 : Horizontal and vertical resolution of the analysis grid

Figure 2 : Bathymetry interpolated on the analysis grid (global view).


Figure 3 : Bathymetry interpolated on the analysis grid (polar view).

4 Reference climatology

The climatology is an important component of the a priori statistical information used by ISAS to compute the estimate. The mean ocean state defines the background or reference state ( x f ) and the variance is necessary to compute the elements of the covariance matrices that appear in equations 2-4. Two types of climatologies can be used with ISAS: a climatology derived from the NODC ! or a climatology constructed from a previous ISAS analysis. In each case WOA atlas gridded fields of the following variables must be provided on the ISAS grid: Monthly mean for temperature and salinity Variance for temperature and salinity relative to the monthly mean. At the moment this variance is assumed constant over the annual cycle.

4.1 Mean field from NODC climatology

In the case of NODC climatology, provided on a 1 degree grid and low vertical resolution, the data must be interpolated on the ISAS higher resolution grid but it is also necessary to extrapolate the NODC atlas from the ocean to the land. This interpolation is thus is done in three steps: 1. 2. 3. 4. Extend NODC atlas at constant latitude over land at NODC levels Perform 2D horizontal bilinear interpolation onto ISAS grid at NODC levels Apply land mask Perform vertical linear interpolation on ISAS vertical levels

Monthly fields are provided by NODC only from 0-1500m. They are extended to 2000m using the annual mean. The transition is smoothed with a linear filter (1/4, 1/2, 1/4).

4.2 Mean field from ISAS climatology

An ISAS monthly mean climatology can be easily computed by averaging over several years the gridded fields from a previous analysis. For example, the ARV09 climatology results from the averaging of an analysis of years 2002-2009.

4.3 Variance
The variance fields are computed from a data set prepared for a previous analysis, that have been interpolated on ISAS levels. In the case of ARV09 all data over the period 2002-2009 were used. The variance is computed at each grid point as the mean square anomaly relative to the monthly climatology for all data within a square of size 10 grid points. To take into account that some areas remain under-sampled (the southern ocean in particular), a lower bound is imposed to the variance. First, the new computed variance must be higher than 0.05 (0.02 PSS) for temperature (salinity) and when no data are available, the NODC variance is used.

Figure 4 : Standard deviation of temperature and salinity at 300 m in ARV09 configuration.

5. Covariance scales
Statistical information on the field and data noise are introduced through the covariance matrices that appear in equation 1. We assume that the covariance of the analyzed field can be specified by a structure function modeled as the sum of two Gaussian function, each function associated with two horizontal space scales and one time scale:
! !", !", !" =
! ! !! ! ! !

!"#

!"! !!! !"

!"! !!! !"

!"! !!! !"

Eq. 5

where !" , !", !", are the space and time separations, !!" , !!" , !!! the corresponding e- folding scales. The weight given to each ocean scale is controlled by the variances "i2 . The total variance is computed as the variance of the anomaly relative to the monthly reference field as explained in the previous section. It is considered as the sum of four terms: !
! ! !! ! and !!! are the two terms appearing in eq. 5, their sum is the total field variance. ! The remaining sum is the total error variance: !!" corresponds to the measurement ! errors and !!" represents small scales unresolved by the analysis and considered as
! ! ! !! = !! !" + !!" + !!" + !!"

Eq. 6

noise, also called representativity error.

! A unique !!" profile has been computed from the measurement errors of the standard database and substracted from the total variance to obtain the ocean variance (first three terms of the sum). We express the variances associated to each scales as a function of the ocean variance by introducing normalized weights: ! ! ! ! ! ! !! ! ; !!! = !!"#$% !! ; !!" = !!"#$% !!" ! = !!"#$% !

!! +!! +!!" = 1

Figure 5 : Covariances scales L2 based on the Rossby Radius.

10

1 2

The free parameters of the system are the weights !! that define the distribution of variance over the different scales. The error matrix combines the measurement error and the representativity error due to unresolved scales, it is assumed diagonal, although this is only a crude approximation since both errors are likely to be correlated for measurements obtained with the same instrument, or within the same area and time period. The first scale length !! is fixed over the ocean, separate values can be used in x and y. Most of the time it is set to 300km, the target Argo resolution. The second length !! is proportional to the Rossby Radius computed from the annual climatology. This value is bounded by 300 km for the highest limit and twice the grid size for the lowest limit. This correlation is isotropic.

5 Areas and masks

Figure 6 : Areas defined for the analysis

For the practical implementation of the method, the global ocean has been divided in areas. Each area defines the group of points that are processed at once (the yellow area of Figure 7) . A mask associated to each area indicates which data are taken into account ( the yellow and green area of Figure 7). Generally the useful area include all points surrounding the analyzed area, but some points can be masked to avoid mixing data from different basins.
Figure 7 : Mask for area 105 : In yellow the analyzed area, in green the useful area.

11

6 References
Bretherton, F., R. Davis, and C. Fandry, (1976), A technique for objective analysis and design of oceanic experiments applied to Mode-73. Deep Sea Research, 23, 1B, 559--582. Gaillard, F., E. Autret, V.Thierry, P. Galaup, C. Coatanoan, and T. Loubrieu , 2009 : Quality control of large Argo data sets. JOAT, Vol. 26, No. 2. 337351 Ide, K., P. Courtier, M. Ghil and A. C. Lorenc (1997), Unified Notation for Data Assimilation: Operationnal, Sequential and Variational. Journal of the Meteorological Society of Japan, 75, no 1B, 181-189.

12