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**Estimation of Tropical Sea Level Anomaly by an Improved Kalman
Filter**

**Ngai Hang Chan, Joseph B. Kadane, Robert N. Miller and
Wilfredo Palma**

### Abstract:

*Kalman filter theory and autoregressive time series are used to map sea
level height anomalies in the tropical Pacific. Our Kalman filters are
implemented with a linear state space model consisting of evolution equations
for the amplitudes of baroclinic Kelvin and Rossby waves and data from the
Pacific tide gauge network. In this study, three versions of the Kalman filter
are evaluated through examination of the innovation sequences, i.e., the time
series of differences between the observations and the model predictions before
updating. In a properly tuned Kalman filter, one expects the innovation
sequence to be white (uncorrelated, with zero mean).
A white innovation sequence can thus be taken as an indication that
there is no further information to be extracted from the sequence of
observations. This is the basis for the frequent use of whiteness, i.e., lack
of autocorrelation, in the innovation sequence as a performance diagnostic for
the Kalman filter.
*
Our long wave model embodies the conceptual basis of current understanding of the large-scale behavior of the tropical ocean. When the Kalman filter was used to assimilate sea level anomaly data, we found the resulting innovation sequence to be temporally correlated, i. e., non-white, and well fitted by an autoregressive process with a lag of one month. A simple modification of the way in which sea level height anomaly is represented in terms of the state vector for comparison to observation results in a slight reduction in the temporal correlation of the innovation sequences and closer fits of the model to the observations, but significant autoregressive structure remains in the innovation sequence. This autoregressive structure represents either a deficiency in the model or some source of inconsistency in the data.

When an explicit first order autoregressive model of the innovation sequence is
incorporated into the filter, the new innovation sequence is white. In an
experiment with the modified filter in which some data were held back from the
assimilation process, the sequences of residuals at the withheld stations were
also white. To our knowledge, this has not been achieved before in an ocean
data assimilation scheme with real data. Implications of our results
for improved estimates of model error statistics and evaluation of adequacy
of models are discussed in detail.

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