The optimal observation placement in weather forecast and research (WRF) data assimilation is investigated using a sensitivity analysis method. The method quantifies the sensitivity of observation location to assimilated results as an unobservability index. The empirical observability Gramian matrix composed from a time series of WRF model outputs is used to obtain the unobservability index in the WRF domain. A three-dimensional variational data assimilation (3 D-VAR) method is employed in the WRF model to assimilate the observations of horizontal winds, whose locations are selected based on the unobservability index. The results from the identical-twin experiments show a correlation between improvement in the assimilated wind field and the magnitude of unobservability index. The temporal variation of the vertical component of vorticity is strongly related to the unobservability index, which confirms that an observation location exhibiting a high unobservability index contributes to error reduction in the data assimilation owing to the reduction in the uncertainty caused by the strong vorticity changes.

Wind farms are recognised as one of the most promising reusable energy sources (REN21,

The sensitivity of numerical weather prediction (NWP) to initial conditions originates from the chaotic behaviour of an atmospheric model (Lorenz,

Concerning data assimilation in a limited area, such as a wind farm, it is important to know which observation location is optimal for data assimilation, because there are many methods to be used for observations depending on the locations, such as an anemometer, a wind profiler, a radiosonde, and light detection and ranging (LIDAR). The optimisation of observation locations to improve the NWP is known as targeted observations. Several methods have been proposed to conduct targeted observations. Baker and Daley (

Kang and Xu (

In this study, the method formulated by Kang and Xu (

This paper is organised as follows. The computational methods are presented in

In this study, sensitivity analysis is conducted using the method proposed by Kang and Xu (

Here,

Using

In this study, the observability is evaluated with

The calculation of an observability Gramian

The empirical observability Gramian

The vector

POD, also known as principal component analysis (PCA), or empirical orthogonal function (EOF), can be used to decompose the spatiotemporal variation of a field into orthogonal bases (POD bases or POD modes, hereafter) which optimally represent the variation. In this study, the POD modes obtained from the WRF model run are used as initial disturbance ^{7}) that is too large to solve an eigenvalue problem. The snapshot POD procedure reduces the size of the problem on the order of ^{2}, ∼ the number of snapshots) and calculates the eigenvalue problem for an

These POD modes are expected to be effective as an initial perturbation to calculate the empirical observability Gramian in

The resulting POD modes

In this study, 3 D-VAR in WRFDA Version 3.8 is utilised (Barker et al.,

WRFDA decreases the computational cost of 3 D-VAR using a control-variable transform and an incremental method to avoid the direct calculation of

Five variables compose the control variables

After this transformation, the increment

In WRFDA, it is possible to adjust a background error correlation in

As will be discussed in subsequent sections, we focus on the atmosphere below the planetary boundary layer; therefore, the spatial length scale is adjusted depending on the altitude levels of interest.

The computational conditions, the domain and a wind field at level 12, are presented in

Computational domain, geographical height, and streamline at the assimilation time (2017/2/7 12:00 GMT) for level 12. The strong northwest wind observed in the figure is caused by a typical pressure pattern of winter in Japan. A small turbulence is found in circle A.

Conditions for the observability calculation.

The initial condition for the observability analysis is set to the instantaneous field at 2017/2/7 06:00 GMT, and the energy dominant four POD modes (^{−4} by normalisation during SVD.

Proper orthogonal decomposition (POD) bases for wind components

The effectiveness of the empirical observability Gramian on WRF data assimilation is evaluated in an identical-twin experiment.

Computational conditions for the identical-twin experiment.

^{rd}-order

Data assimilation is performed with one observation every 10 grids horizontally, which means 13 × 13 = 169 assimilations for each level. To evaluate the data assimilation result with/without the targeted observation, the change in the layer-averaged root-mean-square error (RMSE) due to data assimilation is evaluated at each observation point. The RMSE and its change are evaluated by:

These RMSE changes are plotted at 169 points for each level and their spatial distributions are given. We carry out the above evaluation at four layers: levels 2, 7, 12, and 17. In addition, we evaluate the RMSE changes right after assimilation because 3 D-VAR can improve a finite-time forecast. When the RMSE change is negative, the analysis has a higher accuracy than the first guess.

Levels 2, 7, 12, and 17 of the WRF model outputs are referred for the following discussion. Levels 2 and 7 are lower than the planetary boundary layer, where the temporal and spatial scales of the wind field become smaller. Therefore, we set the spatial length scale

Schematic of identical-twin experiment and the calculation of observability.

With the conditions mentioned in

Examples of the spatial distributions of the minimum eigenvalue are shown in

Spatial distribution of minimum eigenvalues at each level, where (a), (b), (c), and (d) correspond to the distribution at level 2 (150 m), 7 (850 m), 12 (3200 m) and 17 (8000 m), respectively.

We now discuss the relationship between the eigenvalues and corresponding wind fields. A meteorological field can include several flow configurations, such as convection caused by pressure and temperature gradients, and the unsteady wake of terrain roughness. The eigenvalue from the empirical observability Gramian appears to have a relationship with the vertical components of the vorticity vector (

The magnitude of the time-averaged vorticity:

The time-averaged magnitude of the perturbation component of vorticity:

The time-averaged magnitude of the temporal variation of vorticity:

The calculations of abovementioned three quantities are performed with the same data as the observability computation, with the setting of

Correlations of the minimum eigenvalues and the actual distribution at levels 2, 7, 12, and 17. The vertical and horizontal axes correspond to the correlation coefficient and vertical height, respectively. The correlations with the time-averaged vorticity (red line), time-averaged perturbation of vorticity (blue line), and time-averaged temporal variation of vorticity (green line) are plotted. Each label on the left plot corresponds to the vorticity distributions on the right.

The decay of the

From the above discussion, it is confirmed that the eigenvalues from this observability analysis have a strong relationship with

In this section, we verify the effect of the observability analysis on the result of WRF data assimilation.

The distribution of eigenvalue, RMSE changes of wind magnitude, and time-averaged temporal variation of vorticity for levels 2, 7, 12, and 17, respectively. Area A shows the region where the correlation with the eigenvalue distribution is observed. Area B shows the region where RMSE reduction is observed regardless of almost zero eigenvalue. The first 6 h of the wind data (2017/2/7 6:00 GMT – 2017/2/7 12:00 GMT) are utilised to compose the Gramian.

The relationship between the distribution of the Gramian eigenvalues and the RMSE changes after the assimilation is observed, because the eigenvalues are strongly related to temporal changes in vertical component of vorticity as discussed in

The exceptions at levels 2 and 7 (the area B in

The comparison of the RMSE and

In this study, the effectiveness of the observability Gramian for the targeted observation was investigated in the identical-twin experiment with the WRF model and 3 D-VAR, assuming a meso-scale wind prediction. There have not been prior attempts to use the observability for weather data assimilation in targeted observations.

We discussed the relationship between the spatial eigenvalue distribution of the empirical observability Gramian and corresponding wind fields. Their correlation coefficient are calculated and evaluated for levels 2, 7, 12 and 17. The correlation coefficients showed strong relationships between the eigenvalue and time-averaged magnitude of the temporal variation of vorticity

Identical-twin experiments were then conducted to investigate the effect of the observability analysis by comparing assimilation results and the eigenvalue distribution of the observability Gramian. As a result, the forecast error was effectively reduced by choosing the area with high observability as an observation location.

As for future study, we will focus more on small-scale flows in an effort to produce accurate wind prediction around a wind farm. To realise this, observability analysis will be carried out in smaller domains with large eddy simulations. Moreover, the observability of other variables (e.g. atmospheric pressure, wind speed, temperature, humidity) will be assessed. The empirical observability Gramian has the capability to evaluate the observabilities for different variables simultaneously. The eigenvalue decomposition of this Gramian considers the correlations among those variables and is expected to produce information about which variable has a large impact on a forecast.

The data that support the findings of this study are available from the corresponding author, Ryoichi Yoshimura, upon reasonable request.

The authors declare no conflicts of interest.