Satellite SAR is an active microwave radar that provides information about sea surface roughness independent of daylight and weather conditions. Using the Doppler information obtained from the returned signals, SAR systems on board European satellites such as Sentinel-1A and Sentinel-1B achieve a high spatial resolution of the order of 10 m in the along and across flight directions. For C-band SAR systems, the surface roughness length scale relevant for the radar cross section is of the order of a few centimetres and depends on the incidence angle. As this part of the surface spectrum is highly influenced by the surface wind, SAR has become an established tool for the derivation of 2D wind speed maps over the ocean . SAR data have also been used in the context of offshore wind farming in a large number of studies . In this study, Sentinel-1A/Sentinel-B data obtained in interferometric wide-swath (IW) mode with VV polarisation are used. Sentinel-1A was launched on 3 April 2014 and is still active, while Sentinel-1B was launched on 25 April 2016 and has become inoperative since December 2021. The two satellites together have an exact repeat cycle of 6 d, acquiring data with the same imaging geometry. Each acquired scene covers a swath approximately 250 km wide. For this study, SAR scenes with different imaging geometries, obtained during both ascending and descending passes, were used. Sentinel-1A/Sentinel-B satellites operate in a sun-synchronous orbit, with overflights of the German Bight at around 06:00 UTC during descending passes and around 17:00 UTC during ascending passes, respectively.

The SAR data were radiometrically calibrated to obtain the normalised radar cross section (NRCS) using SNAP software made available by the European Space Agency (ESA). This radiometric calibration ensures that the pixel values represent physically meaningful backscatter coefficients independent of acquisition geometry. After calibration, the images were terrain corrected. To reduce speckle noise , the SAR images were smoothed down to 200 m grid resolution. The NRCS is a dimensionless quantity, which describes the intensity of the radar return, and it is often expressed in decibel (dB) values. For assimilation into the inversion framework, we used linear units of NRCS, while dB values are shown only for better visualisation in selected figures.

A couple of simple quality checks were performed to exclude image points, which are likely related to perturbations due to ships, wind turbines or shallow-water current features :

Points with NRCS >1 are excluded.

Information about the 30 SAR scenes used in this study is summarised in Table . An overview of the corresponding environmental conditions in terms of the average 10 m wind speeds and the air–sea temperature differences in the OWF areas for the SAR acquisition times is shown in Fig. a. We emphasise that we did not use the Sentinel-1 Level-2 Ocean (OCN) products, even though they provide wind speed, NRCS, and quality flags. Our assimilation framework requires radiometrically calibrated NRCS values as direct input, not preprocessed wind vectors. Moreover, the OCN products are delivered at coarser resolution and do not allow detailed control of calibration and filtering (e.g. removal of ship signatures, turbine returns and shallow-water artefacts), which is essential for reliable wake inversion.

Data from the Icosahedral Non-hydrostatic (ICON) atmosphere model run at the German Weather Service (DWD) were used as first-guess information . This model has a grid with approximately 7 km resolution and is routinely used for numerical weather prediction (NWP) at DWD. The model data contain hourly wind vectors at 10 m height and information about sea surface temperatures (SSTs) and temperatures at 2 m height (T2m).

Figure b shows a 2D histogram of wind speeds and air–sea temperature differences ΔT=T2m-SST computed inside the German Bight OWF areas for the years 2020–2022. One can see that there is a dominance of unstable situations and that the selected SAR data provide a reasonable representation of typical conditions.

As the wind deficits caused by OWFs have an order of magnitude, which can be comparable to errors in the first-guess model wind field, respective corrections have to be applied in addition to the adjustments to the wake model parameters. Technically, this was done using a 2D spline basis, with two example basis functions shown in Fig. . The splines used are quadratic in both dimensions and are defined on a grid with 40 km resolution. One can see that the scale of the splines is larger than the OWF dimensions, and thus it can be expected that the spline corrections are not able to replicate OWF wakes and therefore do not interfere with the wake model.

The quadratic B splines overlap and sum up to unity . Denoting the spline basis as Bj,j=1,…,Nsp, the correction of the u and v components of the first-guess wind field for SAR image number k is computed as 16uBGk(x,y)=uDWDk(x,y)+∑j=1Nspβj,kuBj(x,y),17vBGk(x,y)=vDWDk(x,y)+∑j=1Nspβj,kvBj(x,y), with coefficients β1,ku,…,βNsp,ku,β1,kv,…,βNsp,kv. The total control vector α for the optimisation problem is then defined as 18α=(αwake,β:,1u,…,β:,Nimagesu,β:,1v,…,β:,Nimagesv)∈IR8+2NspNimages. The control vector dimension is 5728 and thus smaller than 8+2NspNimages because B splines completely outside of the area covered by the SAR scene were not considered in the inversion. The number of B splines used for each of the SAR images is given in the last column of Table . We denote the wind field obtained by feeding the corrected wind field UBG=(uBG,vBG) into the empirical wake model by Usim(α). The structure of the complete forward model, including the dependencies on the control vector, is depicted in Fig. .

The model inversion is then equivalent to the minimisation of the following cost function: 19J(α)=χobs(α)+λχprior(α), with a scalar weighting factor λ and the two cost function components associated with departures from prior information and differences between observations and simulations: 20χprior(α)=∑k=18(αk-α‾k)2(σkα)2+∑k=1Nimages∑j=1Nspj(βju,k)2+(βjv,k)2(σβ)221χobs(α)=∑k=1Nimages∑x∈SARk⋅(NRCSsim(Usim(βu,βv,αwake))-NRCSobsk)2(σNRCSk)2, where the squared differences between the observed and simulated NRCS are summed over all SAR image points x, which fulfil the criteria described in Sect. . In total, about 40×106 SAR image pixels from Nimages=30 images were used in the inversion. For λ=1, the parameters σα,σβ,σNRCS can be interpreted as error standard deviations of the respective control and observation vector components. We will however not use this interpretation rigorously because the structure of the cost function implies some oversimplifying assumptions, particularly concerning the spatial independence of observation errors, and the σ parameters are instead used to control the relative weighting of different components in the cost function. To avoid the dominance of SAR images acquired at high wind speeds and large NRCS values, we chose 22(σNRCSk)2=VAR(NRCSk)×105; i.e. the deviations between simulations and observations are scaled with the standard deviation of each image with index k. For the weighting of the prior terms, we chose 23σβ=1ms-124σ1α=σ2α=144251/(σkα)2=0fork=3,…,8; i.e. the errors in the background wind field are assumed to be of the order of 1 ms-1. The error variance for the Fitch correction parameters is chosen such that a 50 % correction leads to about the same increase in the cost function as a 2 ms-1 background wind field correction over a mesoscale patch, i.e. about 100 km. As a prior, we use α‾1=1 and α‾2=1. Technically, α1 and α2 were rescaled in the forward model such that the cost function has the standard form commonly used for the L-curve analysis . Very little is known about the remaining control vector components, and they are not regularised at all. We come back to the choice of the weighting factor λ later on in the section.

For each choice of λ, the cost function (Eq. ) defines a nonlinear least-squares minimisation problem that was solved numerically using a Gauss–Newton method , which would probably be called the incremental 4D-Var method in the context of data assimilation. In this iterative technique, a linearised version of the problem is solved in each iteration step. The equivalent linear system to be solved in each step for the correction Δα of the control vector has the form 260=∂J∂Δα=[MADHTG-1(HM(αi)-y)]T+MADHADG-1HTLMTLΔα. Here, MAD and MTL denote the adjoint and tangent linear models of the advection diffusion model described in Sect. , respectively. The observation operator H and the respective tangent linear and adjoint operators HTL and HAD refer to the CMOD backscattering model. The tangent linear and adjoint models were hand-coded and tested using the dot-product test . Both the advection/diffusion and the CMOD imaging models are differentiable, with the exception of the TVD advection scheme, which contains switches. The challenges with respect to the adjoints of advection schemes have been discussed in previous studies . The challenge of the linear problem (Eq. ) is that the matrices are not available in explicit form but only as operators in the form of program subroutines; i.e. direct solvers are not practical. The standard approach in such situations is to use a conjugate gradient (CG) method, which is an iterative solver for symmetric systems . With a solution Δα of the system (Eq. ), the next iteration of the control vector α is given by 27αi+1=αi+ρΔα, where ρ=1 usually leads to a cost function reduction. If that is not the case, smaller step sizes ρ=0.5,0.25,… are tested until a decrease is observed.

All model components and the inversion scheme were implemented in FORTRAN90 and parallelised on a Linux cluster computer. A 2D domain decomposition with 4 by 4 domains is used for the advection/diffusion model and the respective adjoint and tangent linear models. All 30 SAR images are inverted in parallel; i.e. the program requires 480 processors. The model trajectory required in the adjoint model is stored in memory at full temporal resolution, i.e. at 20 s time steps.

We now come back to the choice of the parameter λ in Eq. (), which controls the weighting of the prior and the observation terms. A classical method to determine a reasonable value for λ is the L-curve analysis , where the so-called L-curve defined as 28Γ(λ)=(log⁡(χobs(α(λ))),log⁡((χprior(α(λ))))=:(ρ^(λ),η^(λ)) is considered, with α(λ) denoting the solution of the minimisation problem for a given λ, and χobs and χprior are defined in Eqs. () and (). The L-shaped curve obtained for the inversion problem described in the previous section is shown in Fig. a, omitting the logarithm in Eq. (). The triangle and square symbols correspond to the smallest and largest considered λ values, respectively. The general idea is that it does not make sense to look at solutions which are very far on the left or very far on the right because they are either replicating noisy observations or simply reproducing prior knowledge. It can be justified that a choice of λ that maximises the curvature 29κ(λ)=ρ^′η^′′-ρ^′′η^′((ρ^′)2+(η^′)2)3/2 of Γ is a reasonable compromise between these two extremes . The computed curvature for our case is shown in Fig. b, indicating a pronounced maximum at around λmax=0.004791. The wake model parameters estimated for different choices of λ are summarised in Fig. b–d. The relatively smooth dependence of the solution vector on λ indicates a stable performance of the inversion procedure. The red asterisks indicate the solutions corresponding to the maximum curvature of Γ. A plot for α1,α2 is not shown because the departures from the first guess were negligible. Our general interpretation of the curves in Fig. b–d is that the inversion scheme uses the wake model to compensate for errors in the background wind field if the prior is weighted more strongly. As these background errors typically have a larger spatial scale than the OWF wakes, the inversion tends to diffuse the simulated wakes, e.g. by increasing the horizontal diffusion coefficient νH or by making the wakes longer by decreasing α3. Figure a shows the standard deviation for the magnitude of the wind speed corrections as well as the mean systematic correction of the u and v components as a function of λ. One can see that the magnitude of the systematic corrections is below 5 cms-1 and that the non-systematic corrections start to grow quite rapidly below λmax.

As a first application of the empirical model Wake2Sea, OWF wakes were added to a complete year of DWD atmospheric model data. Figure  shows respective maps of estimated deficits at 10 m height for the year 2020. The wake model was applied in the same way as in the inversion; i.e. hourly model data were used as input, and a time step of 20 s was used in the explicit scheme. The only adjustment that was necessary is related to very high wind speeds above 30 ms-1 that occurred during some short periods in that year. Although we used a cut-out wind speed of 25 ms-1, above which the deficit production is switched off, spurious deficits generated at previous time steps can still exist and lead to instabilities. For this reason, model wind vectors were scaled down to 30 ms-1 in those situations.

Figure a shows the mean percentage deficits averaged over the entire year. It can be seen that the maximum average deficits with values around 8 % are concentrated in a small area around the OWFs. Averages of the absolute wind speed reductions are shown in Fig. b. One can see that the wind farm cluster in the southwesterly part causes a reduction of at least 0.2 ms-1 within an area of almost 100 km × 100 km. The deficit standard deviation displayed in Fig. c illustrates that the area where wakes can occur from time to time is significantly larger than suggested by the mean deficit in Fig. a. To get an impression of stronger shadowing effects, Fig. d shows the 90 % deficit percentile, i.e. deficit values that are exceeded in 10 % of the cases. One can see that there are larger areas, in particular between and inside the OWFs, with deficit values of 15 % and above.

To provide an independent evaluation of Wake2Sea beyond the SAR-based inversion, we compare modelled 10 m wind speeds with airborne campaign measurements collected over and downstream of offshore wind farms. The airborne data are not used in the inversion and are used only for evaluation. Collocated comparisons are performed along 36 flight segments (Fig. a). Because strict temporal coincidence between aircraft and satellite observations is rarely achievable, we allow a temporal tolerance of 2 h relative to Sentinel-1 overpasses when selecting transects for the validation. This represents a compromise between data availability and maintaining comparable mesoscale conditions during the validation period.

Wake2Sea is evaluated against airborne wind measurements collected during the WIPAFF and XWAKES field campaigns . Airborne wake observations from WIPAFF have been widely used as an independent benchmark to evaluate wake representations in mesoscale simulations with wind farm parameterisations (including WRF) and wake-recovery behaviour in the German Bight . The selected transects span multiple dates and wind farm clusters, providing a geographically and temporally diverse validation dataset. To enable a physically consistent comparison with modelled near-surface winds, the aircraft observations were adjusted to a common reference height of 10 m using a bulk-stability-corrected logarithmic profile based on Monin–Obukhov similarity theory (bulk MOST), with stability information derived from the flight measurements.

Figure a gives an overview of the flight tracks used in the comparisons. Figure b and c summarises the statistical agreement between airborne observations and the original DWD 10 m winds (Fig. b) as well as the same winds after applying Wake2Sea (Fig. c). The comparison demonstrates a clear improvement when using Wake2Sea relative to the DWD background wind field alone. In the 2D histograms, the Wake2Sea estimates are more tightly clustered around the diagonal than the corresponding DWD values, indicating both reduced systematic deviation and lower scatter with respect to the aircraft observations. The error metrics corroborate this improvement. The mean bias decreases in magnitude from -0.93 ms-1 (DWD) to -0.30 ms-1 (Wake2Sea), the standard deviation decreases from 0.85 to 0.67 ms-1 and the root-mean-square error declines from 1.26 to 0.74 ms-1. These results indicate that the empirical wake parameterisation adds substantial predictive skill and captures the wake-affected offshore flow more realistically than the background meteorological forcing alone.

As an independent validation of the inversion results, comparisons were performed with measurements taken at the research platform FINO-1. We emphasise that FINO-1 wind measurements are not assimilated and were not used to tune any Wake2Sea parameters; they are used only for an independent in situ evaluation of the results. The location of the platform is indicated by the black triangle in Fig. a. There are two main challenges in this comparison. (1) The lowest FINO-1 measurement is at 34 m height, i.e. significantly above the 10 m reference level used in this study. (2) The FINO-1 platform is located in the centre of a large wind park cluster; i.e. the wakes measured by the platform are more inner-park wakes rather than external wakes in nature. In this context, it is important to remember that the interior of wind parks was excluded in the inversion because of radar reflections from the turbine structures; i.e. the empirical model is not optimised for inner-wind-park wakes.

The 34 m FINO-1 measurements were extrapolated down to 10 m assuming a neutral wind profile U(z)=u*log⁡(z/z0)/κ in combination with the Charnock equation: 30z0=0.015(u*)2/g with gravitational acceleration g, roughness length z0, friction velocity u* and a Karman constant κ.

The empirical model Wake2Sea was used to add OWF wakes to a complete year of DWD model data, resulting in a dataset for wind speeds U10 with hourly sampling. The comparison of the original DWD data for the year 2020 with the extrapolated FINO-1 measurements is shown in Fig. a. One can see a significant bias of about 1 ms-1, which is consistent with the missing OWF wakes in the model data. The corresponding comparison with the model data including wakes can be found in Fig. b. In this case the bias is reduced by a factor of 2, and the standard deviations and the RMSE are reduced significantly as well. Considering the remaining bias of 0.54 ms-1, one can now argue that the assumption of a neutral boundary layer is not realistic, particularly inside a wind park cluster. Inside the wakes, turbulence and vertical mixing are increased, and one can expect that the wind speeds at 10 and 34 m are in closer agreement than suggested by the theoretical log profile. Following this argument, a second comparison was performed in which the 10 m wind was assumed to be equal to the 34 m FINO-1 measurement for those cases where the wake model indicated a wind deficit of at least 3 %. These estimates were then compared to model data with and without wakes as before. The results shown in Fig. c and d indicate that the bias is now reduced by a factor of 10, with a remaining value of 0.05 ms-1, and the standard deviation is also slightly improved.

In any case, the wake model improves the agreement of the model data with the FINO-1 observations, despite the fact that the model was never tuned to these interior wind cluster conditions. We are not claiming that our procedure for introducing additional vertical mixing is highly accurate, but the analysis shows that the reasonable assumption of additional mixing leads to further reductions in bias. The order of magnitude of the respective deviations is now close to error levels associated with assumptions about the vertical profiles, and this calls for more dedicated in situ measurements in offshore wind farm areas.

A rough idea about the shape of the wakes produced with the Wake2Sea model can be obtained by assuming that the background wind field u is constant, neglecting lateral diffusion and considering the stationary case with ∂D∂t=0. If we further assume α4=0, the solution for D is given by 31D(x)≈D0exp⁡(-α32x/u), where D0 is the initial deficit, and x is the distance from the wind farm. If the deficit dependence in the sink term (see Eq. ) is expanded to quadratic order, i.e. 32∂D∂t≈…-Dα32(1+2α4D)(1-α5ΔT)2+…, the stationary solution is given as 33D(x)≈ξ1exp⁡(α‾3x/u)-2α4ξ1 with 34α‾3=α3(1-α5ΔT) and 35ξ1=D01+2α4D0. Figure a shows a comparison of the linear and quadratic approximation of the stationary deficit profiles and the solution obtained by numeric integration of the complete sink expression for neutral conditions; i.e. ΔT=0. One can see that the quadratic approximation is in very good agreement with the full nonlinear solution and that the non-linearity leads to a slightly slower decay of the wake near the wind farm compared to the linear solution.

The distance at which the deficit has dropped to half of the initial value is then given by 36D1/2=uα‾32log⁡(2ξ1(D0-1+α4)). Figure b shows this parameter as a function of air–sea temperature difference and initial deficit D0 for a wind speed of 8 ms-1. According to Eq. (), a simple linear scaling can be applied to obtain D1/2 for other wind speeds. One can see that there is a pronounced stability dependency and a relatively weak impact of the initial deficit D0.

The estimated lateral diffusion νH leads to an across-wake expansion that can be analysed analytically based on some simplifying assumptions. The wake expansion has been addressed in previous studies, e.g. ; however this is the first time this effect is estimated for the near-surface deficits using satellite observations. If we assume that the across-wake profile has a top-hat shape as suggested by at a distance x=0 km from the wind farm and furthermore that the deficit is advected by a constant velocity u, then the evolution of the across-track deficit profile as a function of x can be described as 37D(y)=D(x)2(erfy+a4νHx/u-erfy-a4νHx/u), where erf denotes the error function, y is the across-track coordinate with y=0 corresponding to the centre and D(x) is the along-wake deficit profile. To separate the deficit decay due to the lateral diffusion from the effects caused by vertical diffusion, we simply assume D(x)=const. Figure c and d shows examples of across-wake profiles at x=0 km (solid black line) and at x=50 km (red dashed line) for u=8 ms-1. The initial wake width is 10 km in Fig. c and 30 km in Fig. d. One can see that the lateral diffusion has a smaller effect on the maximum deficit if the wake is wider because it takes more time for the diffusive lateral fluxes to change the shape of the profile.

Another interesting question is the maximum deficit that can be reached in very large wind parks. As the wind speed reduction caused by the wake is included in the source term in Eq. (), the deficit cannot grow above 100 %, and there is a well-defined saturation limit. If the linear approximation is used for the vertical diffusion, i.e. α4=0, and furthermore a constant CT value is assumed, the following solution is obtained inside the wind park: 38D(x)=D∞(1-exp⁡(-x(α32/u+c1))), where x=0 defines the upstream boundary of the park, and 39c1=0.5NCTA/dZ40D∞=c1uα32+c1u. The distance after which half of the saturation deficit D∞ is reached is then given by 41x1/2=log⁡(2)α32/u+c1. Figure a and b shows x1/2 and D10m∞≈α72D∞ as a function of wind speed U10 and the normalised rotor area N⋅A for CT=0.8. One can see that the internal wakes require longer distances to build up under high-wind-speed conditions and in OWFs with lower turbine density. The model predicts that for slightly higher wind speeds, the deficit can grow well beyond 30 % in very dense wind parks. We have to emphasise again that the model was not tuned to internal wake measurements, and it is implicitly assumed that the vertical deficit diffusion can be described with the same parameterisation inside and outside OWFs. Furthermore, this analysis does not include the horizontal diffusion discussed in the previous section (Sect. ); i.e. conceptually it refers to an infinitely wide wake. The comparison of the Wake2Sea model results with FINO-1 data in Sect.  suggest at least that there is no obvious overestimation of deficits inside of wind park clusters.