Figure 1. Schematic wave number spectrum of ocean variability. (from Figure 2 in White et al., 1982)
Ocean variability has multi-scale temporal and spatial structures. Ocean modeling is faced with sampling problems in verification of the model using observed data. Discrete grid and simple forcing of the model inevitably restrict range of temporal and spatial scale in the simulation. The autocorrelation analysis (Gandin, 1963; White et al, 1982) extracts a component within a range of specific temporal and spatial scale (signal) from the data. Residual component of the data is considered as noise. 'Kmin' and 'kgrid' shown in Figure 1 denote minimum wavenumber and maximum wave number in the model simulation, respectively. Variance of the signal component V2s corresponds to stippled area in Figure 1. Variance of the noise component (V2n) consists of the part excluded in the model and white noise, which exists in whole range of the spectrum. We must compare the model result with the signal component in the same range of temporal and spatial scale as in the model.
The optimum interpolation (Gandin, 1963) provides a spatial map of the data with the signal scale using the above signal/noise information. Moreover, this method provides the interpolation error of the map. If the model can perfectly simulate the signal scale events, the map of the error gives us the 'error bar' of the model. Although the model always suffers from the imperfectness of the formulation or missing physics (representation error) for even simulating the target scale events, this error bar is considered as the ultimate goal for the modeling.
In order to evaluate the model result comparing with the signal component in the observation data, we statistically estimate the signal/noise characteristics of available data. Using this information, we create the map of the signal component of the data. Sub surface temperature data, sea surface temperature (SST) data and the sea surface height anomaly (SSHA) data are processed. Because our model resolves the meso-scale eddies and the Kuroshio path variation south of Japan (Miyazawa et al, 2002), we determine that the target scales are longer 10 days and larger 50km.
We can easily obtain the necessary data through the internet as a result of the international cooperation. Sub surface temperature data (January 1990 to June 2002) are obtained from the Global Temperature and Salinity Profile Program (GTSPP) . Sea surface temperature is created using the Multi Channel Sea Surface Temperature (MCSST) algorithm from the satellite brightness temperature (August 2001 to July 2002) provided by NAVOCEANO-JPL . The satellite altimetry data of the TOPEX/POSEIDON(T/P) and ERS-2 (January 1998 to July 2002) are obtained from the Colorado Center for Astrodynamics Research (CCAR). The each data is interpolated onto a quarter degree grid (117E-180E, 12N-56N). Examples of the data observed during a period from 2 May 2002 to 22 May 2002 are shown in Figure 2. The satellites (SST and sea surface height anomaly ) provides us a large amount of data.
Number of the sub surface data increased owing to development of the ARGO floats since 2000. In fact, number of data of observed temperature at 200m depth increased by 290 (2-22 May 1992, left panel of Figure 3) to 580 (2-22 May 2002, right panel of Figure3).
Lag correlation for anomaly from the temporal mean with interval of 10 days and 50km is calculated from the above data. It is averaged in 5 degree and 5 degree mesh to remove a bias due to low sampling. Examples of the lag correlation map for SSHA calculated in 140E-145E and 20N -25N are shown in Figure 4. Westward propagation of the signal is clearly found in the x-t correlation map (Left panel). On the other hand, the signal shows no propagation for north-south direction (Right panel).
Therefore we approximates the correlation by the following exponential type function according to Kuragano and Kamachi (1999).
where X,Y,T are east-west lag, north-south lag, and time lag, respectively, C0 is zero-lag, Cx is phase velocity for east-west direction, Lx and Ly are east-west and north-south spatial scale, respectively, Lt is temporal scale. The correlation parameters (Cx, Lx, Ly, Lt) are determined by the least square method. In 145E-150E and 20N-25N for SSHA, (Cx, Lx, Ly, Lt) = (-0.07m/s, 152km, 108km, 53days ) (Figure 5). The value of zero-lag C0, 0.626 is taken from the x-t correlation map (Left panel fo Figure 4). The Variance of the signal V2sand the noise V2n are specified by V2m C0 and V2m (1-C0) , respectively (V2m is variance of the data). Therefore noise/signal ratio is (1-C0)/C0.
The correlation parameters in (1) for SSHA are estimated in each 5 degree mesh in the North West Pacific (125E-180E, 12N-55N). The estimated values are smoothed using the Gaussian function and specified onto the quarter degree grid (Figure 6). The signal scales are small in the marginal seas and the subarctic region. Large magnitude of noise/signal ratio in these regions suggests that physical processes with smaller scales than 10-days and 50km dominate there. Westward propagation is significant in the subtropical region south of 30N. The variance of signal is large due to the meso-scale eddy activities in the Kuroshio-Kuroshio extension region, the mixed water region and the subtropical front region.
Correlation parameters for sub surface temperature data from the GTSPP are shown in Figure 7. No parameter is estimated in some regions owing to low sampling. As the depth increases, the signal scales decrease. Large scale variation related to seasonal thermocline dominates upper 100m. On the other hand, at the depth of lower 100m, signal scales show complex structures. In general, large scale variation exists in the subtropical region. Large noise/signal ratios in the coastal area also suggest the unresolved small-scale physical processes there.
For MCSST, Large scale variation is also extracted from the autocorrelation analysis. However, it is meaningful to extract the smaller scale signal such as the meso-scale eddy activities because the density of data sampling is dense as shown in left panel of Figure 2. Therefore we calculate lag correlation for deviation from zonal mean anomaly since the zonal mean anomaly almost corresponds to the large scale variation. The result shown in Figure 8 indicates that the meso-scale variation with 100km-300km scale is detected in the mid-latitude region from 30N to 45N by this method.
Using the estimated parameters, the observed data are interpolated according to following formula (Gandin, 1963).
where VAij is interpolated anomaly on a target point (i,j), VAobs (m,n) is observed anomaly data on point(m,n) around point (i,j), p(m,n) is weight. The weights p(m,n) are determined so that interpolation error is minimized. These p(m,n) are specified from the following formula.
where C(obs<->obs) is a correlation matrix among the data points (m,n), Rns is a diagonal matrix which consists of noise/signal variance ratio for points (m,n), and c(target<->obs) is a correlation vector between the target point (i,j) and the data points (m,n). Interpolation error variance e2ij is
Maps of SSHA, MCSST, and temperature at 200m depth on 12 MAY 2002 are created(Figure 9,10,11). The SSHA data with interpolation error shown in lower panel of Figure 9 is assimilated into the model. The MCSST and GTSPP data (not assimilated) will be compared with the model result. Lower panels of Figure 10 and 11 provides error bars for comparison. The ARGO float effectively reduce the interpolation error (Figure 11 and 12).
Gandin, L., S., 1963: Objective analysis of meteorological field. Gidrometeorologicheskoe Izdate'stvo., Leningrad, U.S.S.R., 286pp.
Kuragano, T., and M. Kamachi, 1999: Global statistical space-time scales of oceanic variability estimated from the TOPEX/POSEIDON altimeter data. J. Geophys. Res., 955-974.
Miyazawa, Y., X. Guo and T. Yamagata, 2002: Roles of meso-scale ediies on the Kuroshio meandering, submitted to J. Phys. Oceanogr.
White, W., B., Meyers, G., and K. Hasunuma, 1982: Space/time statistics of short-term climatic variability in the Western North Pacific. J. Phys. Oceanogr., 1979-1989.
Point of contact: miyazawa@jamstec.go.jp. Last modified: July 17, 2002