The native resolution DEM is first smoothed using a user-defined filter (Table 2). The “Daly” filter is defined as (D94) 1SEi,j=12Ei,j+18Ei+1,j+Ei-1,j+Ei,j+1+Ei,j-1, where SEi,j is the smoothed elevation and Ei,j is the high-resolution elevation at grid point (i, j). D94 computes multiple smoothed DEMs to account for data density changes across the domain, while TIERv1.0 only computes one smoothed DEM. Once the smoothed DEM is calculated, the slope aspect (0–360∘) is computed and facets are defined. There are five (5) facets in TIERv1.0: (1) north (aspect >315∘, aspect ≤45∘); (2) east (45∘< aspect ≤135∘); (3) south (135∘< aspect ≤225∘); (4) west (225∘< aspect ≤315∘); and (5) flat (D94). Flat aspects are areas with terrain gradients (slopes) less than the user-specified minGradient (m km-1, Table 2). After the facets are defined, small facets are merged together with neighboring facets using the minimum size model parameters (Table 2). Flat regions that are very narrow are considered ridges and behave like neighboring facets. These are merged into the neighboring facets on the west or south slopes depending on their orientation (D94).

The user defines a land–ocean mask field in the input grid file. This mask defines which grid points are large bodies of water (ocean points) and are used in the coastal proximity calculation. The distance to the coast is computed using the great circle distance assuming a spherical earth for every grid cell within a user-defined distance threshold.

To identify the atmospheric layer (Sect. 2.2.4) of a grid point, the local topographic position of a grid point is computed first. The topographic position calculation uses the high-resolution DEM. Following D02, for each grid point, the following steps are taken.

The minimum elevation within a user-defined local search radius (r, Table 2) is found. D02 suggest a search radius of 40 km. 2EMi,j=minEi-r:i+r,j-r:j+r, where Ei-r:i+r,j-r:j+r denotes the DEM elevations within ±r grid points at valid land points in the i and j directions, and EMi,j is the local minimum elevation.

The topographic position is then estimated as 3Ti,j=Ei,j-EMi-r:i+r,j-r:j+rNg, where Ng is the number of land grid points within the search radius (Ng≤r2).

Following the determination of the topographic position, each grid cell is placed into the first (boundary or inversion layer) or second layer (free atmosphere) of the idealized two-layer atmosphere. The height of the inversion layer is defined by the user (Table 2) and added to the mean elevation computed on the left-hand side of Eq. (3). This defines an inversion height above sea level for all grid points. All grid points where Ei,j is less than the inversion height are placed into layer 1, while all other grid points are placed into layer 2 (D02).

After the input domain grid file has been processed, the pre-processing routine generates station metadata files for all precipitation and temperature stations that will be used in the regression model. Each station is assigned the closest grid point value of the smoothed DEM, facet, topographic position, atmospheric layer, and coastal distance.

For each grid point, a set of stations is used to estimate the precipitation and temperature values. First, all stations within the user-defined search radius are found (nearby stations), up to the maximum number of stations considered (Table 4). From that subset of stations, all stations on the same facet as the current grid point are identified (facet stations). Then, a set of distance-dependent weights and weights for each physical process component described in Sect. 2.2.1–2.2.4 is generated for all nearby and facet stations for each grid point. These component weights are then combined to create the final station weight vector. 4W=WdWfWlWtWp, where W is the final weight vector, Wd is the distance-dependent weights, Wf is the facet weights, Wl is the atmospheric layer weights, Wt is the topographic position weights, and Wp is the coastal proximity weights. All component weights and the final weight vector are normalized to sum to unity (D02). For precipitation, only Wd, Wf, and Wp are used to estimate W, while temperature uses all five component weights in W.

A station's relevance to the current grid point decreases as the station distance increases (e.g., Shepard, 1968); thus, this component station weighting decreases with increasing distance. Here, we generally follow the synagraphic computer mapping (SYMAP) algorithm of Shepard (1968, 1984) and develop inverse distance weights that are further modified by including direction information. Direction information is used to downweigh stations that are in a similar direction but further distance than other stations, as their influence has been “shadowed” by the nearer station (Shepard, 1968). The distance weighting function of Barnes (1964) is used: 5I=exp-dys, where I is the inverse distance-dependent weights, d is the station distance vector, and y and s are the user-defined Barnes exponent and scale factor, respectively (Table 4). The angle-dependent weights are then computed as (Shepard, 1984) 6Ts=∑s≠qIq1-cosAs-Aq, where Ts is the station angle weight for station s, and subscripts s and q denote station subscripts for stations 1:nx, where nx is the maximum number of stations considered for each grid point (Table 4). The final distance-dependent weights are then computed as 7Wd,s=Is21+Ts∑s=1n∑Ts, where n is the number of stations considered at the current grid point. Wd is then normalized to sum to unity.

Stations on the same facet type as the current grid point receive an initial facet weight of 1. D02 introduces a method to reduce the weight of stations on the same facet type but with intervening facets of different types between the station and grid point (D02 Eq. 5). This is not implemented here, and all stations on the same facet type as the current grid point receive the same weight. This could be a decision considered for exploration in a future TIER release. The distance-dependent weights already account for this implicitly, but the explicit inclusion of additional weight decreases for stations on the same facet type will increase the localization of the TIER station weighting even further. This would increase the small-scale features of TIER.

The atmospheric layer weight function is defined as 8Wl,s=Δls=01Ei,j-Esa,Δls=1, where Δls is the layer difference between the grid point and station s, the elevations are defined using the high-resolution DEM and station elevation (Es), and a is the user-defined layer weighting exponent (Table 4). Following D02, stations in the same atmospheric layer as the current grid point receive an initial weight of 1. D02 includes an additional check to see if the station–grid elevation difference is smaller than some threshold value for a station. If this is true, a station in a different layer and grid cell may still receive a weight of 1. TIERv1.0 does not include the additional conditional statement and only stations in the same atmospheric layer receive an initial weight of 1. The vertical elevation difference is then used to weigh the remaining stations. The atmospheric layer weighting is applied only to temperature variables in TIERv1.0.

The topographic position weights are computed following D07: 9Wt,s=1,Δts≤Δtm0,Δts>Δtx1Δtsz,Δtm<Δts≤Δtx, where Δts is the topographic position difference between the current grid cell and station s, Δtm and Δtx are the user-defined minimum and maximum topographic position differences, and z is the topographic position weighting exponent (Table 4). The topographic position weight enhances identification of stations that lie in similar topographic areas (e.g., valleys) and is applied only to temperature variables in TIERv1.0.

Using the computed distance to the coast, the coastal proximity weights are computed as 10Wp,s=1,Δps≤11Δpsc,Δps>1, where Δps is the absolute difference between the current grid cell and station s distance to the coast values, and c is the user-defined coastal proximity weighting exponent (Table 4). D02 computes coastal proximity weights using the same inverse distance function but also includes a threshold (Δpx), which, if Δps>Δpx, is set to zero. This weighting factor highlights stations with similar coastal proximity to the current grid cell.

Once nearby stations are selected and the final weight vector is computed (Eq. 4), a base grid point estimate is developed using the weighted average of all nearby stations: 11μ^bo=∑s=1nYs⋅Ws, where μ^bo is the grid point meteorological variable estimate, and Ys and Ws are the observed station value and the station weight for station s, respectively. The uncertainty of this value is estimated as the standard deviation of the leave-one-out estimates, which is all possible combinations of nr-1 stations, nrnr-1, which in this case are nr possible combinations. 12σ^bo=∑i=1nrμ^b,-12nr-1, where nr is the subset of stations that are both within the distance threshold and on the same facet as the current grid cell, σ^bo is the estimated standard deviation of μ^bo, and μ^b,-1 is the estimated value when the ith station is withheld.

Next, the variable-elevation linear regression coefficients are solved for 13β^=AsTWAs-1AsTWY, where β is the vector of linear regression coefficients βo,β1, As is the nr×2 design matrix, W is the nr×nr diagonal weight matrix populated with the final weight vector W, and Y is the vector of observed station values. In D94, D02, and D08, these coefficients determine the grid point estimate as 14μ^=β^0+β^1Ei,j,β1M≤β^1≤β1X, where β1M and β1X are the user-defined minimum and maximum valid regression slopes (Table 4). Note that slope here is in physical units per distance (e.g., mm km-1 or K km-1), which is also referred to as the lapse rate in atmospheric science. In TIERv1.0, we have chosen to use the base grid point estimate, μ^b, as the intercept value in the variable-elevation regression equation. This is done because when β^1 falls outside of the bounds in Eq. (14), a default slope value is used, but β^0 is not modified. Thus, the base estimate of a variable for a grid cell is sometimes derived from an equation the system considers invalid. Therefore, we fully disassociate the intercept and slope estimates. Here, we provide an initial assessment of this choice in Sect. 3, but this methodological choice should be examined in more detail in future work. Subsequently, we then also modify the elevation used in the regression equation to be the difference between the high-resolution DEM elevation and the W weighted station elevation using the smoothed DEM station elevations. The switch to an elevation difference is required as we are effectively correcting the base estimate to the DEM elevation, and the base estimate has an intrinsic elevation associated with it. Therefore, the TIERv1.0 grid point estimate is 15μ^=μ^bo+β^1ΔE,β1M≤β^1≤β1X, where ΔE is the difference between the smoothed DEM elevation and the W weighted station elevation using the smoothed DEM station elevations. When β^1 is invalid, the default slope is used, and when the initial β^1 is valid, the uncertainty of β^1 is estimated in a similar manner to that of μ^bo using Eq. (12). Note that for temperature variables only, the user can define a spatially variable default lapse rate (Table 3, Fig. 4). The standard deviation of all valid slope estimates from the leave-one-out estimates, nrnr-1, is used as the uncertainty estimate of β^1: 16σ^β1=∑i=1nrβ^1,-12nr-1, where σ^β1 is the estimated standard deviation of β^1 and β^1,-1 is the estimated value when the ith station is withheld.

D02 define the method to adaptively adjust the station search radius until the minimum number of needed stations is met. Here, we do not adjust the search radius and instead attempt the regression and uncertainty estimation when nr≥nm, where nm is the user-specified minimum number of stations required for the regression (Table 4). When nr<nm, the regression is attempted for 2≤nr<nm, and the default slope is used when nr<2. Additionally, for nr<nm, Eq. (16) is never applied and there is no direct uncertainty estimate of β^1 for those grid cells.

Finally, D94 found that normalizing the precipitation lapse rate (km-1) after performing the regression reduces the large spatial variability in precipitation lapse rates due to the large spatial variability in the underlying precipitation amounts. The normalization is done at each grid cell as 17B^1P=β1PYP‾, where YP‾=1nr∑s=1nrYP,s, where YP‾ is the mean precipitation (mm) of all stations considered for the regression for the current grid point, β^1P is the estimated slope in physical units (mm km-1), and YP,s is the station precipitation (mm) at station s. Accordingly, σ^B1P is 18σ^B1P=∑i=1nrB^1P,-12nr-1. The normalization allows for the bounds in Eqs. (14)–(15) to be broadly applicable for precipitation, as well as for a reasonable default lapse rate to be applied to grid points where a valid regression slope cannot be found. Temperature lapse rates are computed in physical units (K km-1), as there is little variability in temperature lapse rates.

The initial precipitation normalized slope estimates are used to recompute the default slope if the user specifies (Table 4). In this case, all grid points with valid regression slopes are used to compute the domain mean normalized precipitation slope. This value is then substituted at all grid points with default slope estimates. Next, a 2-D Gaussian filter is applied to the normalized slopes to reduce noise and smooth the artificial numerical boundaries in slope values and is taken as the final precipitation slope estimate (Fig. 5a). The parameters of this spatial filter (size and spread) are specified in the TIER model parameter file (Table 4).

After the slope estimates have been finalized, the precipitation is field is recomputed using Eq. (15) and then a feathering process is applied to smooth any remaining very large gradients (e.g., D94; Fig. 5a). The feathering routine operates on the normalized precipitation slopes and searches for grid cell–grid cell gradients in the normalized slope larger than a user-specified value (Table 4). If a large gradient is found, the slope of the grid cell with less precipitation is increased until the gradient falls below the maximum allowable value. The feathering routine iterates over the grid until there are no remaining large gradients and is an additional smoothing step for precipitation in TIER. Also, the feathering routine only runs for grid cells with larger elevation changes than a user-specified minimum gradient (Table 4), which effectively ignores flat areas (D94), and in TIERv1.0 the feathering routine only operates on grid cells above a user-specified minimum elevation.

Finally, uncertainty estimates are recomputed for the entire grid, first for the base estimate and slope components, then for the total uncertainty of Eq. (15) (Fig. 5a). For those grid points with no initial σ^B1, the nearest neighbor estimate is used. Then the same Gaussian filter applied to the normalized precipitation slopes is applied to the gridded σ^bo and σ^B1. The final uncertainty contribution due to uncertainty in the precipitation slope at a grid point in physical units (mm) is then computed as 19σ^βP=σ^β1Pμ^PabsΔE, where σ^βP is the final uncertainty (mm) due to uncertainty in the precipitation slope, σ^β1P is the final slope uncertainty (mm km-1), and μ^P is the final precipitation estimate (mm). The filtered σ^boP field is used as the final base precipitation estimate, σ^bP. The total uncertainty is estimated as the combined standard deviation of the two component estimates: 20σ^P=σ^bP+σ^βP+2covσ^bP,σ^βP, because the covariance between the two component uncertainties is sometimes nonzero. The covariance is computed locally at each grid point using a user-defined 2-D window of points (Table 4) around the current grid point.

Post-processing for temperature is simpler than that for precipitation because the temperature lapse rates are in physical units. The initial valid temperature slope estimates are used to recompute the default lapse rate if the user specifies (Table 4) when there is no spatially varying default temperature lapse rate information provided (Table 3). Again, the mean of all valid regression slope estimates is used as the updated default temperature lapse rate for this case. As for precipitation, a 2-D Gaussian filter is then applied to the slopes to reduce noise and smooth the artificial numerical boundaries in slope values and is taken as the final temperature slope estimate (Fig. 5b). Then, the final temperature estimate is computed using these updated lapse rate values and Eq. (15).

As for precipitation, the component and total uncertainty estimates are then finalized for temperature. The base temperature estimate uncertainty and slope uncertainty are smoothed using the 2-D Gaussian filter to estimate the final component uncertainties, σ^bT and σ^β1T, respectively. Then, the final temperature uncertainty contribution due to temperature lapse rate uncertainty is computed using Eq. (18), and Eq. (19) is used to compute the total uncertainty of the temperature estimate, substituting subscript Ts for Ps in both.

In experiment 1, the parameter “distanceWeightExp” (Table 4), which is the exponent in the distance-dependent weighting function (Eq. 5), is modified from 2 (default) to 1.75 (modified) for a spatial simulation of Tmin⁡. This decreases the negative slope of the inverse distance weighting function such that stations further from the considered grid point receive more weight in the modified case than the default. The resulting Tmin⁡ distributions and difference field are given in Fig. 9. The spatial distributions are very similar throughout most of the domain as the observation network is relatively high density across most of the domain (Fig. 6). Where the station density decreases along the eastern side of the domain, differences increase in magnitude east of 119∘ W. Notably, there are also pockets of differences outside of ±1 ∘C in areas with high station density along the coast and between 40–42∘ N and 121–123∘ W. These locations contain complex terrain, specifically large elevation gradients, and the modified station weights result in different estimated temperature lapse rates in addition to changes in the base estimate, resulting in the different temperature estimates. However, the calibration sample statistics are not significantly different at the 90 % confidence level than the default parameter set (Table 6), suggesting that the changes in the gridded field are not able to be differentiated in a meaningful way.

For experiment 2, we examine precipitation and modify the “coastalExp” (Table 4) parameter from 0.75 to 1 to examine the influence of changes to the coastal proximity weighting. Qualitatively, the two precipitation distributions are identical to the overall precipitation pattern, remaining essentially unchanged (Fig. 10a–b). The difference fields show that there are shifts in the precipitation placement throughout the domain through the alternating positive/negative difference patterns, particularly across the complex terrain (Fig. 10b), but essentially there is no net precipitation change with a total relative difference of 0.2 % between the two estimates. Absolute differences can be as large as 46 mm in areas of large total accumulations; however, the relative differences in those areas are generally less than 10 % (Fig. 10b). Correspondingly, dry areas have smaller absolute differences but sometimes larger relative differences, as can be seen along the eastern third of the domain (Fig. 10b). The mean absolute value of the cell-to-cell precipitation gradient is 1.56 mm km-1 versus 1.69 mm km-1 (8.3 % increase) in the base and modified cases, respectively. Increased station localization should be expected to increase high-frequency variability and thus spatial gradients. The calibration sample statistics are not significantly different at the 90 % confidence level from the default parameter set (Table 6). However, in this case, the confidence bounds are almost all non-overlapping, which may suggest increasing the coastal exponent further would improve the model performance.

The total uncertainty is generally increased by a few millimeters across the domain (0.55 mm on average), with a corresponding relative increase in uncertainty of around 5 %–10 % (Fig. 10c). This is due to the fact that increasing this weight exponent decreases the weight of stations more dissimilar to the current grid point, effectively increasing the localization of the weights and increasing the variability of the leave-one-out estimates (Eq. 16).

Finally, we change the “nMaxNear” parameter (Table 4) from 10 to 13, which controls the maximum number of stations used for each grid point the interpolation model. Again, the precipitation pattern is essentially qualitatively unchanged between the two configurations with a 0.2 % domain average change; also see Fig. 11a. However, the relative difference fields highlight larger and more systematic changes to the precipitation distribution than in experiment 2. Areas of the highest accumulation in the base case have less precipitation in the modified case (compare Fig. 10a and Fig. 11a) with 89 % (217/245) of the grid points having precipitation >300 mm in the base case having less precipitation in the modified case. Conversely, 57 % (7686/13 430) of the grid points having <300 mm in the base case have more precipitation in the modified case. This is because generally more stations are included in the estimate for each grid cell, which results in a smoother final estimate through smoothed base and slope precipitation estimates in Eq. (15). The mean absolute value of the cell-to-cell precipitation gradient is 1.56 mm km-1 versus 1.48 mm km-1 (5.5 % decrease) in the base and modified cases, respectively. The calibration sample statistics are statistically equivalent to the base case, but the MAE in experiment 3 is statistically significantly larger than that in experiment 2. This is an expected result given that the final estimate is less localized for any specific station.

Increasing the number of stations considered reduces the estimated uncertainty across nearly the entire domain (Fig. 11b–c). On average, there is a 2.1 mm (16 %) reduction in the domain mean uncertainty, with some grid cells having reductions >25 %. The large decreases are primarily in regions of complex terrain, and this is controlled by changes in the slope uncertainty estimate, σ^βP (Fig. 11c). This change in total uncertainty is slightly larger than but opposite in sign to the parameter modification in experiment 2.