Hello I am Arthur, postdoc at the University of Western Australia, where I work in a group seeking to improve numerical wave forecast using data assimilation or machine learning. In both cases we use data retrieved by directional wave buoy hence the following question:
What are wave buoys actually measuring and how confidently ?
Motivation
The question is motivated by the assimilation of wave buoy measurements into spectral model like WW3. As a brief reminder, data assimilation aims at estimating an initial condition combining observation and model information, to later produce a forecast.
The link between model and observation space is classically described by the following observation equation:
Y = H(X) + Îľ
H: Observation operator
X: state (model space)
Y: observation
Îľ: observational noise
In the case of interest, X=E(f, θ) is a directional wave spectrum and Y is an in-situ wave buoy measurement. It is often considered in the literature that the first-5 directional Fourier coefficients [a0, a1, b1, a2, b2](f) of E(f, θ) can be observed so that:
H(X) = int_θ{ E(f, θ) [1, cos(θ), sin(θ), cos(2θ), sin(2θ)] dθ } (un-normalized convention)
But [a0, a1, b1, a2, b2]=Y are actually estimated from measurements. The estimation involves a chain of various processes and assumptions. So the observational noise Îľ could arise from different sources and errors may accumulate and grow along the chain, which makes Îľ difficult to describe. Refining the question I am interested in:
What is the likelihood of Y knowing X?
i.e. what are the statistics of Îľ, which characterize the confidence in the observation and so act as weighting in the assimilation process?
Identifying error sources
The series of operations to go from raw sea state measurements to estimated coefficients can be quite complex as illustrated in [1] (fig. 2). From my understanding, there are three majors operations at which errors might be introduced:
1) Sensing
Raw data are produced by calibrated instruments. Calibration and intrinsic instrumental sensitivity introduce an error.
2) Response Amplitude Operator inversion
The Response Amplitude Operator expresses how a specific buoy-sensor system turns the true sea-surface particle motion into the buoyâs measured quantities. In the process, the RAO is inverted to infer the equivalent surface-particle. The operator is not perfectly known so it introduces an error.
To the best of my knowledge, commercial pitch-roll buoys assume an approximately flat RAO in the 0.03â0.5 Hz band and therefore omit an explicit inversion step. I keep the RAO error source as (i) it reminds that the assumption can break in different see condition, and (ii) the same concept re-appears whenever we assimilate data from sensors with non-flat response.
3) Fourier analysis
The corrected surface-particle time series are windowed, detrended, correlated and Fourier-transformed to estimate auto- and cross-spectra. From such spectra, under linear wave and stationarity assumption, directional coefficients [a0,a1,b1,a2,b2](f) can be deduced. The spectral density estimation process, by definition, introduces an error.
Quantifying observation uncertainty i.e. covariance(Îľ)
Unfortunately most of these errors seem untraceable. However what has been done in the article [2] is to use the covariance of the spectral density estimation. Calculation of such covariance can be found in [3] or [4], it is relatively heavy, even working with un-nornmalized directional Fourier coefficients.
I agree thatâs it is better than nothing to use the spectral density estimation uncertainty. For instance if you consider a0(f) alone, the variance is proportional to a0**2(f), which translates into âthe more energy in a frequency bin, the less certain the energy estimationâ. In the classical Gaussian Data Assimilation framework, my understanding is that it means more flexibility for the model to fit high energy bins but less for low energy ones, which seems reasonable.
Finally, the same question becomes:
Is there a way to characterise Îľ more accurately or usefully than by relying solely on the covariance of spectral-density estimates ?
[1] Steele et al. (1992) Wave direction measurements using pitch and roll buoys. Ocean Engineering
[2] Crosby et al. (2017) Assimilating global wave model predictions and deep-water wave observations in nearshore swell predictions. Journal of Atmospheric and Oceanic Technology
[3] Borgman et al. (1982) Statistical precision of directional spectrum estimation with data from a tilt-and-roll buoy. Topics in Ocean Physics
[4] Long et al. (1980) The statistical evaluation of directional spectrum estimates derived from pitch/roll buoy data. Journal of Physical Oceanography