Alternative data-driven methods to estimate wind from waves by inverse modeling

Abstract

An attempt is made to derive wind speed from wave measurements by carrying out an inverse modeling. This requirement arises out of difficulties occasionally encountered in collecting wave and wind data simultaneously. The wind speed at every 3-h interval is worked out from corresponding simultaneous measurements of significant wave height and average wave periods with the help of alternative data-driven methods such as program-based genetic programming, model trees, and locally weighted projection regression. Five different wave buoy locations in Arabian Sea, representing nearshore and offshore as well as shallow and deep water conditions, are considered. The duration of observations ranged from 15 months to 29 months for different sites. The testing performance of calibrated models has been evaluated with the help of eight alternative error statistics, and the best model for all locations is determined by averaging out the error measures into a single evaluation index. All the three methods satisfactorily estimated the wind speed from known wave parameters through inverse modeling. The genetic programming is found to be the most suitable tool in majority of the cases.

Appendices

Appendices

1.1 Appendix 1: Linear model in LWPR

The four-step algorithm to fit a linear model locally is as below:

Let x _q be query point and p be the training points {x _i , y _i } in memory. Then the prediction is made as follows:

• 1. Compute diagonal weight matrix W with elements as per Eq. 2

• 2. Construct matrix X and vector y such that

$$ {\mathbf{X}} = \left( {\tilde{x}_{1} ,\tilde{x}_{2} , \ldots ,\tilde{x}_{p} } \right)^{T} $$

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where,

$$ \tilde{x}_{i} = \left[ {\left( {x_{i} - x_{q} } \right)^{T} 1} \right]^{T} $$

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$$ {\mathbf{y}} = \left( {y_{1} ,y_{2} , \ldots ,y_{p} } \right)^{T} $$

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• 3. Compute regression parameter for locally linear model

$$ \beta = \left( {{\mathbf{X}}^{{\mathbf{T}}} {\mathbf{WX}}} \right)^{{ - {\mathbf{1}}}} {\mathbf{X}}^{{{\mathbf{T}} }} {\mathbf{Wy}} $$

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• 4. The prediction for given x _q is thus

$$ \hat{y}_{q} = \beta_{n + 1} $$

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where β _n+1 denotes the (n + 1)th element of the regression vector β.

1.2 Appendix 2: The error measures

1.2.1 Correlation coefficient

It measures the linear association of two variables. The measure, however, is very sensitive to deviations at larger observations and does not reflect nonlinear associations.

$$ {\text{Correlation coefficient}},\;R = \frac{{\sum_{i = 1}^{n} {x \cdot y} }}{{\sqrt {\sum_{i = 1}^{n} {x^{2} } \cdot \sum_{i = 1}^{n} {y^{2} } } }} $$

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where, $x = X_{i} - \bar{X}$, and $y = Y_{i} - \bar{Y}$; X _i  = Observed values at time i, $\bar{X} = {\text{Mean of}}\; X$, Y _i  = Modeled value at time i, $\bar{Y} = {\text{Mean of}}\;Y$, n = number of input–output pairs.

1.2.2 Root mean square error

This measure indicates an overall agreement (without any upper bound) between the observed and modeled datasets. This measure is good for predictions arrived at iteratively but gives only an overall picture of errors.

$$ {\text{Root mean square error, RMSE}} = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {(X_{i} - Y_{i} )^{2} } }}{n}} $$

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1.2.3 Mean absolute error

This statistic also gives an overall agreement between the observed and modeled datasets and is useful for practical interpretations. It is non-negative and has no upper bound but provides no information about underestimation or overestimation. It is not weighted towards high or low magnitude events, but instead evaluates all deviations from the observed values, in an equal manner and regardless of sign. MAE is comparable to the total sum of absolute residuals.

$$ {\text{Mean absolute error, MAE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {|X_{i} - Y_{i} |} $$

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1.2.4 Relative absolute error

This measure comprises the total absolute error made relative to what the total absolute error would have been if the forecast had simply been the mean of the observed values. It is a non-negative metric that has no upper bound. It is dimensionless.

$$ {\text{Relative absolute error, RAE}} = \frac{{\sum_{i = 1}^{n} {|X_{i} - Y_{i} |} }}{{\sum_{i = 1}^{n} {|X_{i} - \bar{X}|} }} $$

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1.2.5 Mean absolute relative error

This metric comprises the mean of the absolute error made relative to the observed record and unlike MAE it is dimensionless and hence suitable to compare across many parameters. It is a non-negative metric that has no upper bound.

$$ {\text{Mean absolute relative error, MARE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\frac{{|X_{i} - Y_{i} |}}{{X_{i} }}} $$

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1.2.6 Mean square logarithmic error

This is similar to RMSE but compares the logged values of observed and modeled data. It is more suitable than RMSE for low values as it is based on logarithmic transformations.

$$ {\text{Mean square logarithmic error, MSLE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {{ \ln }\;X_{i} - { \ln }\;Y_{i} } \right)}^{2} $$

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1.2.7 Inertial root mean square error

This is obtained by dividing the RMSE by the standard deviation of the differential series. The use of differential series enables understanding of the model to predict changes in observation.

$$ {\text{Inertial root mean square error, IRMSE}} = \frac{\text{RMSE}}{{\sigma_{\Updelta } }} $$

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where,

$$ \sigma_{\Updelta } = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {\left( {\Updelta_{i} - \bar{\Updelta }} \right)^{2} } }}{n - 1}} $$

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$$ \bar{\Updelta } = \frac{1}{n}\sum\limits_{i = 1}^{n} {\Updelta_{i} } $$

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$$ \Updelta_{i} = X_{i} - X_{i - 1} $$

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1.2.8 Error ratio of forecasted peak

This metric comprises the difference between the highest value in the modeled dataset and the highest value in the observed dataset, made relative to the magnitude of the highest value in the observed dataset. It can be either positive or negative. It is unbounded.

$$ {\text{ERFP}} = \frac{{Y_{ \max } - X_{ \max } }}{{X_{ \max } }} $$

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