We present our wind speed error modelling approach using a case study of the oil and gas-producing region of the Canadian province of British Columbia (BC). BC has been a leader in leveraging sustained provincial-scale measurement campaigns (British Columbia Energy Regulator, 2022) to track emissions, guide regulation, and achieve low methane intensities (e.g., Conrad et al., 2023c; Johnson et al., 2023). Moreover, oil and gas production in BC's Montney and Horn River basins is located immediately downwind of the Rocky Mountains for the prevailing westerlies (Fig. 1), with complex topography and terrain that provides an especially challenging wind modelling example.

The pink polygon in Fig. 1 identifies the approximately 122 000 km² region of interest (ROI) defined for this case study that bounds upstream oil and gas production in the Northeast of the province. The bounds of the ROI were defined as a convex hull around active and shut-in upstream facilities and wells during early 2023 (dilated with a 25 km radius circular kernel in the NAD83 UTM 10N coordinate system) intersected with provincial boundaries (northern and eastern sides). The ROI contains approximately 1083 oil and gas facilities and 10 441 wells.

Anemometer-measured, “ground-truth” wind speeds were acquired from 35 active weather stations within the case study ROI that each report a scalar wind speed and direction on an hourly basis (Fig. 1). As summarized in Table 1, these included Environment and Climate Change Canada's (ECCC) surface weather stations (ECCC, 2025), British Columbia's Ministry of Environment and Parks' Air Quality Network stations (BC ENV, 2025; BC Ministry of Environment and Parks, 2015), and British Columbia's Ministry of Forests, Lands, and Natural Resource Operations' Wildfire Management Branch's weather stations (BCWS, 2025). Two additional ECCC weather stations within the ROI were excluded as they are within 10 km of a station (Fort Nelson Airport; World Meteorological Organization ID 71945) that inputs data to the NAM12 NWP model used for the case study demonstration of our methodology (see Sect. 2.1.3). Station-measured wind data were obtained for May to October (i.e., non-snowy months when aerial surveys can reliably be performed in the ROI) during the calendar years of 2021–2024. Critically, these data are not inputs to the evaluated NWP model and thus constitute an independent “ground-truth” data set in the subsequent analysis.

Data available from the individual networks include hourly reported wind speed magnitudes and directions; however, given the potential for poor resolution and high uncertainties in anemometer-measured wind direction, we ignore direction data and analyze wind speed magnitudes directly. Recommended installation height for anemometers is usually at 10 m above ground level (e.g., Burt, 2012; Manfredi et al., 2008; Oke, 2004), and wind speeds are “most often [averaged over a period of] 10 min” (Burt, 2012). However, installation heights and averaging periods for these weather station networks are either not explicitly stated, ambiguous, or non-standard. For example, documentation for the ECCC weather stations (Environment and Climate Change Canada, n.d.) notes that installed anemometers are usually located at 10 m above ground level and report one, two, or ten minute-averaged wind speeds during the period ending at the top of the hour. While the BC-FLNRO network explicitly states a targeted height range for each installed anemometer, there is no available information regarding averaging periods for the BC-ENV and BC-FLNRO networks nor anemometer height for the BC-ENV network. Thus, anemometers for the BC-ENV network are assumed to be at 10 m above ground level and wind speeds reported by the BC-FLNRO network are scaled to 10 m above ground level from the stated anemometer height using the logarithmic wind profile in Johnson et al. (2021). Finally, throughout this analysis the averaging period is assumed to be 10 min for all weather stations, while noting that this will likely provide a model with larger dispersion relative to measurements over longer averaging periods – refer to the Limitations section (Sect. 4) for further discussion.

For the case study, we assess the North American Mesoscale Forecast System (NAM; National Centers for Environmental Prediction (NCEP) et al., 2025) NWP model, which uses the U.S. National Oceanic and Atmospheric Administration's (NOAA) Non-Hydrostatic Multi-scale Model (NMM). This NWP model is appealing for methane inventory development in Canada as it provides spatiotemporally resolved two-component wind speeds at 10 m above ground level, is freely available from multiple sources online, and provides near-complete coverage of Canada's provinces. Wind speed estimates are available for a forecasted period of 84 h following each data assimilation cycle that occurs every 6 h. These data are provided on NCEP's #218 grid, which has a 12 km resolution at 35° N and is underlaid in yellow in Fig. 1. Wind speed data on this 12 km resolution grid (hereafter called “NAM12” data) were sourced (see “Code and data availability” section) from the U.S. National Centers for Environmental Information's https server (up to 15 September 2021) and NOAA's Open Data Dissemination program on Amazon Web Services (from 16 September 2021, onwards). Two-component wind speeds were acquired for each hour during the calendar years of 2021–2024 using the latest data assimilation cycle at each time point to minimize forecast lag; data at 00:00, 06:00, 12:00, and 18:00 UTC therefore represent analyses rather than forecasts. Component wind speeds were summed in quadrature to provide wind speed magnitudes on the 12 km grid for further processing and comparison with ground-truth, weather station-measured, 10 m wind speeds.

Prior to model fitting, alignment of the measured and NAM12 wind speed datasets was required. Fortunately, all datasets for the case study ROI reported at identical timepoints, the top of each hour. Wind speeds at the specific coordinates of the weather stations were assigned using a simple nearest-neighbour interpolation of the NAM12 grid points within the NAD83 UTM 10N projected coordinate system. Missing data in the measurement datasets were stricken. Similarly, recognizing that wind speeds over a finite averaging period cannot be exactly zero but may be below the detection limit of an anemometer, we also struck zero-valued wind speeds in the measurement dataset. Following pre-processing, we have five m×1 vectors containing case study data from all of the 35 weather stations (m = 470 014); these are the ground-truth (u [m s⁻¹]) and NWP model-estimated (ũ [m s⁻¹]) 10 m wind speeds across all weather stations, the corresponding eastings (sx) and northings (sy) of each measurement in projected coordinates [km], and the corresponding time of each measurement in Coordinated Universal Time (t [d]).

One logical and easily implemented approach to circumvent these challenges leverages a statistical tool known as a copula. Copulas were first presented by Sklar (1959) and are discussed in abundant detail in the literature (see, for example, Nelsen, 2006). In this section, we simply present the formal definition and relate it to the present challenge. Sklar's theorem states that the joint CDF of n arbitrary random variates Hx1,…,xn can be represented exactly as a function of each xi's univariate marginal CDF Fixi through an n-dimensional function called a “copula” C (e.g., Nelsen, 2006), that is 3Hx1,…,xn=CF1x1,…,Fnxn or in terms of PDFs (e.g., Arrieta-Prieto and Schell, 2022) 4πx1,…,xn=cF1x1,…,Fnxn×∏i=1nπixi where c… is the copula density. This formulation is useful as it separates independent information about each xi (defined by their marginal distributions) and the dependence structure of the xi (defined by the copula). Stated differently, the copula maps the marginal CDFs to the joint CDF (Nelsen, 2003, 2006; Sadegh et al., 2017).

In the present context, introducing our variables of interest, the desired CDF (Eq. 2) has the form 5Hu|ũ,sx,sy,t=CF1u1|ũ1,sx,1,sy,1,t1,…,Fnun|ũn,sx,n,sy,n,tn;sx,sy,t

Here, we have noted that the marginal distributions for the true wind speeds are conditional on the NWP model-estimated wind speeds and locations and timepoints of the remote survey and that the copula (i.e., the dependence structure or autocorrelation) is similarly functional on the locations and timepoints. The key advantage of this approach is that the copula-based model of the desired joint probability can be parsed into two tractable problems. This two-step modelling process requires: (i) a wind speed error model for the marginal distributions at each location and timepoint, Fiui|ũi,sx,i,sy,i,ti and (ii) the copula C…;sx,sy,t that joins them to give the joint CDF. However, we can further simplify the modelling process by deriving a univariate region-averaged wind speed error model with CDF Fu|ũ that is representative of the average error in NWP-estimated wind speeds over the ROI (i.e., dropping the location and timepoint dependence of the marginal distributions). Since this error model would be assumed applicable within the ROI regardless of location/timepoint, the marginal distributions Fi in Eq. (5) are all equivalent Fi≡F∀i∈{1,…,n}. This gives the format of the joint CDF that we ultimately seek to model 6Hu|ũ,sx,sy,t=CFu1|ũ1,…,Fun|ũn;sx,sy,t

Numerous types of copulas exist for modelling complex dependence structures (see, for example, Nelsen, 2006). In this work, we employ the Gaussian copula, which has the form 7CFu1|ũ1,…,Fun|ũn;sx,sy,t=Φn(Φ-1Fu1|ũ1,…,Φ-1Fun|ũnT;0,Σsx,sy,t) where Φ-1x is the inverse CDF (quantile function) of the standard univariate normal distribution and Φnx;μ,Σ is the joint CDF for the multivariate normal of arbitrary dimension n with n×1 mean vector (μ, defined here to be the zero vector, 0) and n×n covariance matrix (Σ), which is parameterized by locations and timepoints (sx, sy, and t, all n×1). We choose the Gaussian copula for this work because it reduces the modelling to that of a covariance function or covariogram for a Gaussian field, Σsx,sy,t, and algorithms to generate random numbers from a multivariate normal distribution are readily available in mathematical software applications.

In a previous work, we described a simple method to probabilistically model true wind speeds from NWP model-estimated wind speeds (i.e., a wind speed error model, πu|ũ) based on controlled release experiments assessing a provider of remote methane detection (Conrad et al., 2023b). This model fit the ratio u/ũ, which we denoted as the relative error ratio (RER), to candidate probability distributions such that the wind speed error model had the form 8πRERu|ũ=πcanduũ;θũ where πcandx;θ is a candidate probability distribution with non-negative support. This previous model, which assumes the RER follows the same distribution regardless of the NWP model-estimated wind speed, fit the limited experimental data well but is overly simplistic here where we have years of hourly wind speed data across 35 unique sites and need not constrain the RER in this way.

In the present study, we use a more-general model for region-averaged wind speed error that uses the true wind speed (u) as the random variate (i.e., not the RER) and allows the distribution parameters to be functional on the NWP model-estimated wind speed (i.e., θũ) such that 9πu|ũ=πcandu;θũ

We considered nine candidate distributions with non-negative support – Burr Type XII, Gamma, Inverse Gaussian, Log-Logistic/Fisk, Lognormal, Nakagami, Rayleigh, Rician, and Weibull – all of which have two parameters except the Rayleigh distribution, which has just one. Moreover, we permitted each parameter θiũ to be a constant a, linear function bũ+c, or offset power-law dũe+f giving a total of 75 candidate models for πu|ũ. For each candidate model, we calculated a weighted maximum likelihood estimate using MATLAB's fmincon (constrained minimization) function and chose the model that minimized the Akaike Information Criterion (Akaike, 1974; essentially a parameter count-adjusted negative log-likelihood) as optimal. Data were weighted to account for biased spatial clustering of weather stations using a custom implementation of Deutsch and Journel's (1997) “DECLUS” cell declustering algorithm, which weights station data by the inverse of station count on a regular but randomly perturbed two-dimensional grid. Following recommended practices (e.g., Deutsch, 1989), this algorithm was implemented with a cell size of 147 km, which minimized the declustered mean of ground-truth wind speeds. Declustering weights, normalized to the unit interval, for the 35 weather stations of the BC case study are shown by colour in Fig. 1.

To model the covariogram within the ROI for use in the Gaussian copula, we employ geostatistical variography techniques on the m case study data. There is myriad literature regarding the evaluation, characterization, and simulation of autocorrelated geostatistical data including the seminal work of Cressie (1993). Here, we outline our approach to modelling Σsx,sy,t following published techniques within this field. We begin by transforming our case study data to follow a normal distribution via a normal score transform, which gives an m×1 vector z corresponding to the random variate in the multivariate CDF in Eq. (7) (here, of length m). We employ a transformation that leverages our optimized region-averaged wind speed error model – each element of z is 10zi=Φ-1Fui|ũi and can be interpreted as a function of location and timepoint via zi=Zsx,i,sy,i,ti. The vector z represents a random sample from a multivariate normal distribution with zero-mean and some covariance structure, the latter of which we can now simulate using the corresponding location and timepoint vectors. However, to avoid bias in the empirical estimation of covariance (e.g., Isaaks and Srivastava, 1988), we characterize the covariance of z by instead modelling its semivariogram.

Like covariance, the semivariogram γ is a means to characterize how similar and different zi are in the near- and far-fields, respectively – i.e., the spatiotemporal autocorrelation of z. In the most general sense, the semivariogram is functional on two specific positions in space and time and is one-half of the variance of the difference of the zi at these positions. Letting the vector ri=sx,i,sy,i,tiT, the semivariogram is 11γr1,r2=12varZr1-Zr2

This formulation can be simplified greatly. Referring to Cressie (1993) for further detail, if the random field Z is intrinsically stationary, such that it has a constant mean over all values of r and the variance varZr1-Zr2 depends only the spacing (lag) between the positions h=r2-r1, then the semivariogram can be restated as 12γr1,r2=γh=12EZr1-Zr1+h2 where E⋅ is the expectation operator. Furthermore, if the random field Z is second-order stationary such that the autocovariance covZr1,Zr2 depends only on the spacing between the positions, then the covariance of two positions separated by h is easily derived from the semivariogram via 13covh=cov0-γh where cov0 is the variance over the entire field, which is equivalent to the sill of the semivariogram (see below). Thus, with a semivariogram model, the elements of a covariance matrix can be easily calculated as a function of spatiotemporal lag.

Following Eq. (12), the semivariogram can be empirically estimated from the case study data by 14γ^hx,hy,ht=12Mhx,hy,ht∑i=1Mhx,hy,ht(Zsx,i,sy,i,ti-Zsx,i+hx,sy,i+hy,ti+ht)2 where positive real-valued scalars hx, hy, ht are the lags in two-dimensional cartesian space and time and Mhx,hy,ht is the number of pairs of data in z that are separated by these lags. The (empirical) semivariogram can be simplified by assuming that the semivariogram is isotropic in space. Letting hs=hxx^+hyy^ be the spatial lag vector and hs=hx2+hy2 be its positive, real-valued magnitude, the empirical semivariogram simplifies to 15γ^sths,ht=12Mhs,ht∑i=1Mhs,ht(Zsi,ti-Zsi+hs,ti+ht)2 where si=sx,ix^+sy,iy^, x^ and y^ are the cartesian unit vectors, and we introduce the subscript “st” to identify that the semivariogram describes space and time. Practically, for a finite dataset, there are limited realizations of the continuous positive variables hs and ht. Consequently, empirical estimation of semivariograms requires discretization of the data by lags with a defined tolerance – see, for example, Deutsch and Journel (1997) for further discussion.

In theory, one could interpolate the empirical semivariogram to estimate our desired covariance matrix, Σsx,sy,t for arbitrary locations in space and timepoints. However, careful modelling of the semivariogram is required to ensure that it is valid for the simulation of covariance. We must develop a semivariogram model using appropriate functions (or a superposition of appropriate functions) that are conditionally negative semidefinite, which in turn ensures a positive definite covariance matrix Σsx,sy,t (e.g., Curriero, 2006; Pyrcz and Deutsch, 2006). For spatiotemporal applications with an isotropic correlation in space, one useful approach to make modelling of the semivariogram more tractable is to separate it into space and time components. We adopt the product-sum model of De Iaco et al. (2001) in this work 16γst∗hs,ht=γst∗hs,0+γst∗0,ht-k∗γst∗hs,0γst∗0,ht where γst∗hs,ht is the optimal model of the semivariogram, γst∗hs,0 is a marginal spatial semivariogram model at zero temporal lag, and γst∗0,ht is a marginal temporal semivariogram model at zero spatial lag, which are linked by an optimized parameter k∗.

We first model the spatial semivariogram. For each available timepoint of the case study data, we calculate an empirical semivariogram for lags at every 25 km, with a bin tolerance of ± 25 km (e.g., the first bin is 0–50 km, the second is 25–75 km, and so forth). Rather than taking the arithmetic mean of the data within each bin (as indicated by Eq. 15), here we calculate a weighted mean for each bin using weights via Richmond's (2002) two-point declustering method. Like the cell declustering described in Sect. 2.2.2, this method weights the kth data pair based on the inverse of the number of pairs that originate and terminate in the same cell as the kth pair; our implementation of this method uses the same cell size for the earlier cell declustering, 147 km. As in the region-averaged wind speed error model, this space-informed weighting was executed to provide what we believe is a better average empirical semivariogram across the ROI. We then fit candidate semivariogram models to this empirical semivariogram using a weighted least squares algorithm where the weights are proportional to the number of valid data informing each lag bin of the empirical semivariogram.

We consider 21 unique monotonic semivariogram models as candidate fits. Firstly, as a baseline, the constant model which assumes zero spatial correlation. We also consider semivariograms of the form 17γst∗hs,0=b1+b2-b1γ10hs;b4+b3-b2γ20hs;b5 where γkoh;b is one of four monotonic semivariogram models standardized between zero and one and parameterized by a fitted parameter b>0 (the range of the model):

Exponential: γkoh;b=1-exp⁡-aexphb

Next, we model the temporal semivariogram. For each weather station, we calculate the empirical semivariogram for integer-valued hours up to two weeks (336 h). Due to the large amount of temporal data in the case study, we derive these empirical semivariograms iteratively by pulling a subset of random pairs of data and terminating when more than 10⁴ data pairs with temporal lags less than or equal to two weeks have been obtained. We then estimate an ROI-averaged empirical semivariogram by taking the weighted average across the weather stations for each bin, weighted by the product of the cell declustering weights shown in Fig. 1 and the amount of data at each lag. We continue by fitting candidate semivariogram models using weighted least squares with weights corresponding to the those used in the empirical semivariogram calculation, summed over all weather stations.

In contrast to the spatial semivariogram, the temporal empirical semivariogram has a clear periodic nature that aligns with the 24 h diurnal cycle (see further discussion and results in Sect. 3.2). Specifically, we observe that wind speed errors are maximally correlated at lags that are multiples of 12 hours, with peak positive correlations at lags that are multiples of 24 hours. Given the observed diurnal variation, we adopt candidate models that capture this “hole effect” of the empirical temporal semivariogram. These models include monotonic semivariograms (γk0ht) as in the spatial semivariogram modelling above superimposed with a periodic semivariogram. The candidate periodic semivariograms are linear combinations of zeroth-order Bessel functions of the first kind J0x as in Ecker and Gelfand (1997) and Ye et al. (2015) and a cosine function. For temporal lag ht in days, the fitted models have the form 18γst∗0,ht=b1+b2-b1γ10ht;b5+b3-b2γ20ht;b6+b4-b3(1-∑j=1JcjJ02πjht+ϕj-dcos⁡2πht)

Here, we scale the Bessel functions' arguments by 2π and introduce phase shifts ϕj to force the (j+1)th peak to occur at a lag of exactly one day; similarly, the cosine's argument has a scaling factor of 2π to capture the diurnal cycle. Fitted parameters bi are constrained such that each component semivariogram causes a net nonnegative effect b4≥b3≥b2≥b1≥0; the ranges of the γko are positive and rank-ordered b6>b5>0; the coefficients of the Bessel basis functions are decreasing c1>⋯>cJ; the coefficient on the cosine is nonnegative d>0; and, like γko, the periodic component is standardized to start from zero and oscillate around one at large lags. The former requires that 19∑j=1JcjJ0ϕj+d=1

Two-hundred and eleven candidate semivariogram models are considered with up to ten (J=10) Bessel basis functions: a constant model b2=b3=b4=0; a constant plus oscillating model (10 total, with b2=b3=0); one monotonic semivariogram plus oscillating models (40 total, with b3=0); and two monotonic semivariograms plus oscillating models (160 total, with all bi>0). We select the optimal model in the same manner as for the spatial semivariogram.

The final step in characterizing the spatiotemporal semivariogram is to optimize for the constant k∗ in Eq. (16). We follow the procedure of De Iaco et al. (2001). First, we calculate a two-dimensional empirical spatiotemporal semivariogram (Eq. 15) using the same lag bins as the independent spatial and temporal semivariograms. Like the spatial and temporal semivariograms, we use a weighted mean in place of the arithmetic mean for Eq. (15) to obtain an ROI-average semivariogram. As in the temporal semivariogram, the weights here are the product of the declustering weights and count of data in each lag bin; however, we use the two-dimensional declustering algorithm here for the declustering weights consistent with the spatial semivariogram. We then take the sum of the weights within each lag bin and perform a relative weighted least squares fit to Eq. (16) to optimize for k∗.

With a full spatiotemporal semivariogram model, we can now define the desired covariance matrix for arbitrary locations and timepoints. First though, we must calculate the sill of the fitted spatiotemporal semivariogram model, which is the value that a semivariogram approaches or oscillates around as the lag(s) go to infinity. For the spatial semivariogram in Eq. (17), the sill is b2 or b3 depending on the form of the optimized model, while for the temporal semivariogram in Eq. (18), the sill is b4. Denoting the sills of the optimized spatial and temporal semivariograms as Ls∗ and Lt∗, respectively, De Iaco et al. (2001) show that the sill of the spatiotemporal semivariogram Lst∗ is 20Lst∗=Ls∗+Lt∗-k∗Ls∗Lt∗ and, under the aforementioned conditions of second-order stationarity, the covariance can be easily computed from the semivariogram as 21covhs,ht=Lst∗-γst∗hs,ht

Re-introducing the vectors sx, sy, and t, the covariance matrix required for the Gaussian copula is therefore 22Σsx,sy,t=covhs,ij,ht,ij,i≠j1,i=j where hs,ij=sx,i-sx,j2+sy,i-sy,j2, ht,ij=ti-tj, and the diagonal of the covariance matrix is fixed at unity to ensure that the marginal distributions of each ui follow the region-averaged wind speed error model.