chapter 7: Data Assimilation Flashcards
The observed data collected worldwide every ……………. are
6 hours (say 00, 06, …UTC) are blended into forecast values appropriate to the map time by the process known as Data assimilation
The observed data collected worldwide every 6 hours (say 00, 06, …UTC) are blended into forecast values appropriate to the map time by the process known as Data assimilation, and consists of two steps namely
‘objective analysis’ and ‘initialization’
objective analysis
The automatic process of transforming observed data from geographic, but irregularly distributed, locations at various times to numerical values at regularly spaced grid points at fixed times
The step of objective analysis is carried out by
statistical interpolation
The step of objective analysis is carried out by statistical interpolation by
taking into account all of the available observations plus other prior information.
background fields.
Short‐term forecasts from the previous analysis cycle are used as prior information
first guess
also called the background
The forecast used to produce the initial conditions for a new forecast,
observation increment
The difference between the observation and the background field
Data assimilation utilizes these observation increments as
corrections to first guess to create the analysis, which represents current or recent‐past weather
Analysis =
First guess + Correction (weighted average of obs. Increments)
The solutions of primitive equation models correspond to two distinct types of motion.
- One type has low frequency. Its motion is quasi‐geostrophic and meteorologically dominant.
- The other corresponds to high‐frequency gravity‐inertia modes.
One type has low frequency. Its motion is quasi‐geostrophic and meteorologically dominant.
• The other corresponds to high‐frequency gravity‐inertia modes.
The amplitude of the latter type of motion is
small in the atmosphere
The amplitude of the latter type of motion is small in the atmosphere.
Hence, it is important to ensure that
the amplitudes of high‐frequency motions are small initially and remain small during the time integration when the primitive equations are solved using initial conditions
initialization
The process of adjusting the input data to ensure this dynamical balance
Since gravity‐inertial motions are filtered out in
quasi‐geostrophic models
Since gravity‐inertial motions are filtered out in quasi‐geostrophic models, no special procedure was necessary and
the objectively analyzed data could be used immediately as the input data to quasi‐geostrophic models.
In Data Assimilation, observations are used as follows:
First, an automated initial screening of the raw data is performed. During this quality control phase, some observations are rejected because they are unphysical (e.g., negative humidities), or they disagree with most of the surrounding observations.
First, an automated initial screening of the raw data is performed. During this quality control phase, some observations are rejected because they are unphysical (e.g., negative humidities), or they disagree with most of the surrounding observations.
• In locations of the world where the observation network is especially dense,
neighboring observations are averaged together to make a smaller number of statistically‐robust observations.
When incorporating the remaining weather observations into the analysis, the raw data from various sources are not treated equally
Some sources have greater
likelihood of errors, and are weighted less than those observations of higher quality.
When incorporating the remaining weather observations into the analysis, the raw data from various sources are not treated equally.
- Some sources have greater likelihood of errors, and are weighted less than those observations of higher quality.
- Also, observations made
slightly too early or too late, or made at a different altitude, are weighted less.
When incorporating the remaining weather observations into the analysis, the raw data from various sources are not treated equally.
- Some sources have greater likelihood of errors, and are weighted less than those observations of higher quality.
- Also, observations made slightly too early or too late, or made at a different altitude, are weighted less.
- In some locations such as the tropics where Coriolis force and pressure‐ gradients are weak,
more weight can be given to the winds than to the pressures.
Optimum Interpolation
It is one of the objective analysis methods.
Let o-o be
the standard deviation of raw‐observation errors from a sensor such as a rawinsonde. Larger o- indicates larger errors.
let o-f be
the standard deviation of raw‐observation errors associated with the
first guess from a previous forecast. These are also known as background errors.
If the observation has larger errors than the first guess, then
the analysis weights the observation less and the first‐guess more
If the observation has larger errors than the first guess, then the analysis weights the observation less and the first‐guess more
Optimum interpolation is
“local” in the sense that it considers only the observations near a grid point when producing an analysis for that point.
Optimum interpolation is not perfect, leaving
some imbalances that cause atmospheric gravity waves to form in the subsequent forecast.
Initialization
modifies the analysis further by removing the characteristics that might excite gravity waves
Variance – Covariance Matrix
The variance and covariance are the measures of deviation from the mean.
Error Covariance
A correct specification of observation and background error covariances is crucial to the quality of the analysis, because they determine the extent to which the background fields will be corrected to match the observation.
A correct specification of observation and background error covariances is crucial to the quality of the analysis, because they determine the extent to which the background fields will be corrected to match the observation.
The essential parameters are the
variances, but the correlations are also very important because they specify how the observed information will be used in model space if there is a mismatch between the resolution of the model and the density of observations.
The reported atmospheric observations used in data assimilation are
not perfect, they contain several kinds of errors, including instrumental errors and representativeness errors.
The reported atmospheric observations used in data assimilation are not perfect, they contain several kinds of errors, including instrumental errors and representativeness errors.
• The representativeness errors, i.e., actually
correct observations may reflect the presence of a sub‐grid scale atmospheric phenomenon that cannot be resolved by the model or the analysis.
The representativeness errors, i.e., actually correct observations may reflect the presence of a sub‐grid scale atmospheric phenomenon that cannot be resolved by the model or the analysis.
• The representativeness errors indicate that the
observation is not representative of the area averaged measurement required by the model grid.
They are usually estimates of the error variances in the forecast used to produce xb
They are usually estimates of the error variances in the forecast used to produce xb.
• A crude estimate can be obtained by
taking an arbitrary fraction of climatological variance of the fields themselves.
A crude estimate can be obtained by taking an arbitrary fraction of
climatological variance of the fields themselves.
• If the analysis is of good quality (i.e. if there are a lot of observations) a better
average estimate is provided by the variance of the differences between the forecast and a verifying analysis.
A crude estimate can be obtained by taking an arbitrary fraction of
climatological variance of the fields themselves.
- If the analysis is of good quality (i.e. if there are a lot of observations) a better average estimate is provided by the variance of the differences between the forecast and a verifying analysis.
- If the observations can be assumed to be
uncorrelated, much better averaged background error variances can be obtained by using the observational method.
A crude estimate can be obtained by taking an arbitrary fraction of
climatological variance of the fields themselves.
- If the analysis is of good quality (i.e. if there are a lot of observations) a better average estimate is provided by the variance of the differences between the forecast and a verifying analysis.
- If the observations can be assumed to be uncorrelated, much better averaged background error variances can be obtained by using the observational method.
- However, in a system like the
atmosphere the actual background errors are expected to depend a lot on the weather situation, and ideally the background errors should be flow‐dependent.
A crude estimate can be obtained by taking an arbitrary fraction of
climatological variance of the fields themselves.
- If the analysis is of good quality (i.e. if there are a lot of observations) a better average estimate is provided by the variance of the differences between the forecast and a verifying analysis.
- If the observations can be assumed to be uncorrelated, much better averaged background error variances can be obtained by using the observational method.
- However, in a system like the atmosphere the actual background errors are expected to depend a lot on the weather situation, and ideally the background errors should be flow‐dependent.
This can be achieved by
the Kalman filter.
If background error variances are badly specified, it will lead to
too large or too small analysis increments.
The Kalman filter is a set of
mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error.
The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error.
The filter is very powerful in several aspects:
it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modelled system is unknown
The basic idea of a Kalman filter is
The basic idea of a Kalman filter is:
The direct extension of the OI is the KF. The idea is to do successive OIs between each pair of observation times so as to evolve the model state and the error covariance using the model.
The direct extension of the OI is the KF. The idea is to do successive OIs between each pair of observation times so as to evolve the model state and the error covariance using the model.
The application of KF to assimilation of meteorological and oceanographical observations has produced
convincing results by effectively extracting the information contained in the observations and the model.
Kalman Filter ‐ Limitations
From a practical point of view, the major difficulty in applying the method is the
computational cost.
From a practical point of view, the major difficulty in applying the method is the computational cost.
The main disadvantage of the sequential method is that
a given observation has an impact only on the future evolution of the physical process and is not used to correct the last states. This is due to the main characteristic of the sequential method, i.e., the one‐way pass.
The main disadvantage of the sequential method is that a given observation has an impact only on the future evolution of the physical process and is not used to correct the last states. This is due to the main characteristic of the sequential method, i.e., the one‐way pass.
For accurate atmospheric science applications, state size is
~O (106‐108) so that storage of the covariance matrices is nearly impossible.
Ensemble Kalman Filter (EnKF)
is an approximate KF that obtains its covariance information from a set of parallel forecasts that constitute an ensemble.
The EnKF has several advantages:
- The ensemble mean itself is the minimum‐variance best estimate.
- • The forecast model itself can be fully nonlinear, avoiding the need to construct approximate linearized models.
- • The covariance matrix is sampled from the available ensemble members, avoiding the necessity of expensive matrix operations
