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Penumbra: A spatially-distributed shade percent and incident ground-level irradiance 4

model for simulating landscape scale solar energy 5

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Jonathan J. Halama1,4, #a ¶, Robert E. Kennedy2¶, James J. Graham3¶, Robert B. McKane4, #a ¶, Brad 8

L. Barnhart4, #a ¶, Kevin Djang5&,#a, Paul B. Pettus4&,#a, Allen Brookes4&,#a, Patrick C. Wingo4& 9

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1 Environmental Science, Oregon State University, Corvallis Oregon, USA. 11 12 2 College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis 13

Oregon, USA. 14 15 3 Department of Environmental Science and Management, Humboldt State University, Arcata 16 California, USA. 17 18 4 U.S. Environmental Protection Agency, Corvallis, Oregon, USA. 19 20 5 CSRA, Corvallis Oregon, USA. 21 22 #aCurrent Address: Western Ecology Division, U.S. Environmental Protection Agency, Corvallis, Oregon, United 23 States of America 24 25

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* Corresponding author 27

E-mail: [email protected] 28

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¶These authors contributed equally to this work. 31

&These authors also contributed 32

mailto:[email protected]

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Abstract 33

Landscape irradiance is a significant environment driver, yet can be complicated to model 34

well. Several solar radiation models simplify the complexity of light by estimating it: at discrete 35

point locations, over averaged zonal areas, or along polylines. These modeling approaches have 36

appropriate applications, but do not provide spatially-distributed or temporally dynamic 37

representations of solar radiation throughout entire landscapes. We created Penumbra, a landscape-38

scale incident solar radiation model, to address this deficiency. Penumbra simulates spatially-39

distributed ground-level shade and incident solar irradiance at flexible timescales by modeling 40

local and distant topographic shading, vegetative shading, and the influence of cloud coverage. 41

Spatially resolved inputs of a digital elevation model, normalized digital surface model, and 42

landscape object transmittance are used to estimate spatial variations in incident solar irradiance 43

at user-defined temporal time steps. We test Penumbra’s accuracy against several independent 44

irradiance datasets. Penumbra is a dynamic, spatially-distributed ground-level solar incident 45

irradiance model to be used for producing spatial data of shade percentage and irradiance. Output 46

data is intended for estimating effects of solar energy patterns on the health and resilience of 47

aquatic, terrestrial, and upland ecosystems. 48

Introduction 49

Solar energy drives most of Earth’s ecosystems, and must be well characterized in 50

environmental models of those systems. Solar energy is generally the largest energy input into 51

ecosystems [1]. While ecological models each have their own unique foci, spatially-distributed 52

solar radiation is rarely represented, even though it is often a dominant ecosystem driver. Solar 53

energy entering the atmosphere varies predictably with time of day and geo-location, but 54

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estimating the energy that reaches the ground surface requires explicit modeling of how objects 55

intercept it to create shade. In this paper we present Penumbra, a model developed to provide 56

ground-level net-solar energy in a spatially-distributed form. 57

Shade, or the removal of portions of solar radiation projected toward the earth’s surface, 58

has high spatial and temporal variability. Important components that generate shade at a given 59

location include both proximal and distal topography, surface structures such as buildings and 60

vegetative canopies, and cloud coverage. For some regions of the world, cloud coverage can have 61

significant impacts on the resulting solar energy reaching the earth’s surface [2]. 62

For ecological modeling purposes, spatiotemporally inaccurate representations of solar 63

energy may cause compounding effects and introduce unknown modeling uncertainty. For 64

example, site specific climate station data is often homogeneously used to represent large spatial 65

extents, though small-scale forest or stream system microclimates may greatly vary over short 66

distances. Accurate representations of environmental drivers are especially crucial at scales for 67

which processes are sensitive to microclimate variability [3]. No matter the modeling method 68

taken, dynamic spatially-distributed shade representation requires a balance between 69

computational runtime, light complexity, and the volume of model output [4]. Stakeholders 70

focusing on landscape restoration and management need to simulate ground-level irradiance across 71

a wide range of both spatial and temporal scales in a computationally tractable fashion. 72

In this paper, we discuss existing solar energy modeling approaches, then describe our new 73

shade and ground-level irradiance model called Penumbra. We present stand-alone Penumbra 74

results using the 3D visualization tool VISualizing Terrestrial-Aquatic Systems (VISTAS) [5]. To 75

test the model, we compare Penumbra results against the Department of Energy (DOE) National 76

Solar Radiation Data Base (NSRDB) Salem, Oregon, USA station, two sites from the 77

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Environmental Protection Agency (EPA) Oregon Crest-to-Coast Environmental Monitoring 78

transect (O’CCMoN) dataset, and two sites from the Snow Telemetry (SNOTEL) network [6-8]. 79

We conclude with an example of Penumbra simulating the Mashel River Watershed, Washington, 80

USA to demonstrate Penumbra’s the intended use. 81

Current Approaches 82

First it is critical to briefly address some existing approaches ecological models take to 83

represent solar energy. Simulation methods used to account for solar energy in environmental 84

models range from simplistic to complex (Fig 1). The simplest solar energy simulation approach 85

is to use a global irradiance model. For soil temperature modeling this approach is taken by both 86

Visualizing Ecosystem Land Management Assessments (VELMA) and the Soil & Water 87

Assessment Tool (SWAT) [3, 9]. Solar energy is represented as an averaged irradiance value per 88

time step. This global approach has negligible computational modeling cost, but oversimplifies 89

spatial and temporal solar energy by omitting topographic and object shading. 90

Fig 1. Continuum of shade modeling approaches. A representation of the shade modeling 91

approaches in a continuum, with corresponding models that account for solar energy to some 92

varied degree. 93

A similar ecological modeling approach is to utilize monitored solar energy at a point 94

location monitored with a pyranometer, quantum sensor, pyrheliometer, or processed 95

hemispherical imagery. These instruments and imagery can provide detailed data describing local 96

environmental shading and radiation, but for only a single location per instrument. When these 97

instruments are manually operated, the data can only represent short temporal periods due to the 98

burden of collecting field data by personnel. This approach also forces spatially-distributed models 99

to treat these data as spatially homogeneous. 100

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A few ground-level solar energies models exist to overcome the limitations mentioned 101

above. SHADE2 is a field photo processing model that represents solar energy across streams per 102

photo captured location [10]. This approach assesses riparian canopy and overhanging tree 103

shrubbery through image processing and predicted solar angle assessment through time. Image 104

scene orientation is assessed in the x and z dimensions in order to simulate stream surface shading 105

from the overhanging understory and riparian canopy. SHADE2 provides excellent agreement to 106

observed irradiance readings, yet mainly for East to West running streams [10]. SHADE2 107

limitations include image collection being burdensome on personnel, and model results do not 108

address the broader methodological issues of solar data extrapolation through space and time. 109

Beyond site-specific observations, many ecological models require both spatial and 110

temporal solar energy as a data driver. Some stream network models utilize a solar energy metric 111

to characterize fish and vertebrate habitat, evaluate the thermal loading, or assess water quality 112

metrics on individual streams and the network as a whole [11]. Yet solar data are rarely monitored 113

at a sufficient level for large landscape scale modeling. This data shortage forces some models to 114

extrapolate observed data throughout the model space. To overcome this issue, some models like 115

Spatial Stream Network (SSN) model, statistically extrapolate monitored site solar data across 116

point locations [12]. SSN geo-statistically interpolates point location data across a polyline 117

network, making the site-specific solar energy data spatially implicit (Fig 1). Spatial models can 118

apply this approach to extrapolate solar energy across landscape polygons or gridded framework 119

as well. However, when models extrapolate point data they ignore spatial shadowing effects from 120

landscape objects or topography between the point locations containing observed data. The 121

assumption that one location can be a surrogate for another introduces modeling uncertainty [13]. 122

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These models are often limited by the unavailability of data, such as stream bank width, land use, 123

or riparian coverage [14]. 124

Ray tracing models take a complex, spatially-distributed approach by simulating light rays 125

directly. Ray tracing accounts for the complex temporal and spatial nature of solar energy by 126

tracking discrete light path reflection, refraction, transmission, and absorption for each discrete 127

simulation time step [15]. This method is computationally expensive due to the number of light 128

rays and their potentially large number of landscape object interactions. Even over a small spatial 129

extent at a small spatial grain, the computational requirements to run a simulation can become 130

burdensome [16]. Demonstrated by the iLand model, highly accurate light representations are 131

feasible using ray tracing, but their applications have been limited to relatively small (

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Solar energy directly impacts forest and agricultural photosynthesis, stream temperature, 145

and habitat suitability. Process-based, or mechanistic, models can effectively simulate the mixed 146

interactions of such environmental processes across landscapes [19]. Well calibrated process-147

based models can be particularly helpful for policy making and land management of forest or 148

agriculture based on vegetation developmental trends due to resource availability [3]. Yet, process-149

based models are often burdened by complexity resulting in restrictions to simulation runtime and 150

memory allocation [19]. 151

Niche in Landscape Scale Solar Model 152

Overall, spatial models simulating stream networks or entire landscapes must balance 153

promptness, temporal capability, and spatial extent when simulating shade or light. The simplest 154

methods tend to misrepresent the spatial complexity of solar energy, while complex methods 155

introduce excessive runtime and memory costs over large landscapes. Therefore, an environmental 156

light simulation model is needed that can address the limitations of current light simulation 157

methods. With moderate landscapes in mind (> 100 km2), an ideal landscape-scale irradiance 158

model would address the following needs: 159

Produce comprehensive ground-level irradiance and shade percent estimates as a function 160

of spatiotemporal variations in tropospheric incident radiation and shadowing by 161

topography, landscape objects, and cloudiness. 162

Minimize user burden through relatively limited model inputs and parameterization. 163

Provide flexible temporal runtime parameters to simulate landscape solar energy at user-164

required temporal and spatial resolutions (minutes to days; plots to regional landscapes). 165

Develop flexible model coupling (loose and tight) for external model integration. 166

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The resulting model should be able to assess the major components that affect net solar energy 167

across spatially heterogeneous landscapes in a spatially-distributed manner. The model must 168

perform within reasonable simulation runtimes to avoid burden on a coupled model. 169

Research Goal 170

To address these needs, we developed Penumbra, a new shade and ground-level irradiance 171

model. Penumbra is a direct solar energy reduction model, meaning the quantity of incoming solar 172

energy is reduced due to the spatial structure of the environment being simulated. The resulting 173

shade and irradiance data are the percent-shade and ground-level irradiance for a wide range of 174

spatial and temporal scales in a computationally tractable fashion. The output ground-level net-175

irradiance, being produced from the reduction of direct solar energy via shadowing, is therefore 176

implied indirect solar energy. 177

The model incorporates tropospheric incident radiation, which varies based on time and 178

local atmospheric conditions, topographical shading due to local and distant terrain features, and 179

object shadowing due to buildings and vegetative canopies. Penumbra can be utilized as a 180

standalone tool or integrated into other models, and falls within the current spectrum of available 181

radiation modeling approaches (Fig 1). 182

Penumbra is designed to quantify radiative variability across diverse landscapes and 183

atmospheric conditions through time to help scientists and stakeholders understand the potential 184

consequences of land management on human populations, aquatic environments, and terrestrial 185

habitats. Penumbra accomplishes simulating these solar energy tasks under either stand-alone 186

mode or model-coupled mode. Running stand-alone Penumbra provides dynamic solar energy 187

across a temporally static landscape. Running in coupled mode an ecosystem model provides 188

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Penumbra updates of a changing landscape (e.g. vegetation height), and Penumbra in return 189

provides updates of shade percent and irradiance reaching ground and water body surfaces. 190

Penumbra in can produce spatial data representing shade percentage, ground-level 191

instantaneous irradiance, averaged irradiance over time, and incident power. These data can 192

provide users with an assessment of their landscape’s current energy state. This information can 193

be especially useful for reference in decisions regarding vegetation management, stream 194

management, and overall landscape health. 195

Penumbra Model 196

Penumbra is a spatially-distributed, gridded model capable of processing landscapes based 197

on user-defined spatial inputs and temporal parameters. Penumbra’s approach is to simulate 198

landscape scale irradiance as simply as possible while balancing that landscape’s complex spatial 199

structure to maintain a suitable accuracy. Penumbra maintains a suitable level of accuracy through 200

a spatially-distributed gridded assessment of the landscape; any one cell’s topography and object 201

has the potential to cast shade upon any other cell. 202

The cell resolution of a simulation dictates how detailed objects can be represented in the 203

length-width (X-Y) dimensions. An object’s vertical z-dimension is an implicit representation of 204

reality, meaning the object’s structure is a voxel with equal light transmittance from top to bottom. 205

Penumbra can simulate an object’s vertical z-dimension as either a single voxel, or as two stacked 206

voxels with varied transmittance. The stacked voxel option allows the user to represent forest 207

canopy and forest understory as two uniquely different environments. Thus, at fine scales, 208

Penumbra simulations can be viewed as modeling a tree’s light transmittance, or at coarser scales 209

be viewed as modeling a forest canopy’s light transmittance. 210

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Penumbra is a large landscape scale model that simulates the shade and net-solar energy 211

reaching the ground-level. Penumbra does so by relying on the input data characterizing the 212

landscape. These data per pixel encapsulate the solar energy reduction caused by the corresponding 213

landscape objects, no matter the data grid resolution. 214

The model spatially computes total potential solar energy through three independent 215

phases: global irradiance reaching the landscape, landscape object shadowing, and landscape 216

topographic shadowing. All components follow equivalent simulation time steps and data 217

aggregate to form the final shade and irradiance outputs. These calculations are performed across 218

a gridded landscape for each grid-cell independently. Penumbra assumes that for any given time 219

step the environmental lighting is received from a single distant source which is spatially uniform 220

[20] (Fig 2, eq. 12). This assumption allows Penumbra to assess light ray interference as a 221

reduction of the total available irradiance due to spatially-distributed topographic shade, landscape 222

object shade, and daily atmospheric conditions. 223

Fig 2. Spatially-distributed Shade Modeling. Penumbra's solar ray approach to shade modeling 224

for both object and topological shadowing. Equation numbers correspond to equations described 225

under the Processing Steps. 226

Inputs 227

Penumbra requires a small set of model parameters and spatial input data to run (Table 1). 228

A start Julian day, stop Julian day, and year range of simulation define the temporal grain and 229

extent. Penumbra functions under two temporal frames: the frequency of solar angle assessments 230

and the regularity of data aggregation per output. The sun’s daily position is defined by the 231

temporal frequency (Daily-Grain) of shadowing assessments and frequency of data aggregation. 232

Penumbra’s temporal grain can be set to any span from one minute up to a full day. Penumbra 233

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tracks shade and irradiance per temporal grain so that model outputs can be first internally 234

aggregated to any temporal extent equal to or greater than the temporal grain. Solar energy can be 235

output as the average or accumulated per data output frequency. 236

Table I: List of Penumbra inputs, parameters, and outputs. 237

Required Inputs Optional Inputs Outputs

Start/Stop Julian Day and Year Forced Start/Stop Times Ground-level Irradiance

Temporal Grain Daily Transmittance Map Total Shade %

Temporal Aggregation Daily % Cloud Coverage Object Shade %

Digital Elevation Model (DEM) a Area of Interest Mask Topographic Shade %

Normalized Digital Surface Model

(nDSM) a Cell Data Writer Position Cell Data Values

Object Transmittance Model (OTM) a Pseudo Height Adjustment aTechnically optional: Penumbra can simulate only terrain or only object shading. If the DEM 238

or nDSM are excluded, any shadowing that would occur is being excluded. DEM and nDSM 239

are required for complete (topographic and object) shading to be simulated. 240

A simulation’s spatial resolution is defined by the digital elevation data (DEM), which 241

represents the landscapes topographic features. A corresponding normalized digital surface model 242

(nDSM) represents landscape object heights, and is coupled with an object transmittance model 243

(OTM) representing a light transmission reduction factor scaled to 0 through 1. The nDSM heights 244

can be obtained through several GIS activities. Aerial LiDAR data can be utilized to obtain fine 245

resolution DEM and nDSM data, but Penumbra can also simulate lower resolution nDSM data 246

obtained from species biomass or age relationships. When lower resolution nDSM data are used, 247

the corresponding OTM should reflect the upscaling representation of the actual environment. The 248

appropriate light reduction due to landscape objects (trees, shrubbery, etc.) is currently ongoing 249

research with strong promise due to approaches like the Light Penetration Index, and similarly the 250

Laser Penetration Index [21-22]. Currently, spatial input data must be provided in the ESRI ASCII 251

Grid format [23]. When both a DEM and nDSM are provided, topographic and object shadowing 252

are simulated; when only a DEM or nDSM is provided, only topographic or object shadowing will 253

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be simulated, respectively. Conversely, topographic and object percent shade can be simulated 254

simultaneously, yet exported individually as topographic or object shade data. This unique model 255

data output flexibility allows Penumbra simulations to isolate the effect of topographic shadowing 256

and object shadowing. 257

Each aspect of model irradiance reduction can and should be calibrated by the user. All 258

calibration parameters are scaled from one to zero, where a value of one applies no change and 259

any lower value retains that percent of influence by that variable. All calibration parameters are 260

applied to data in a uniform and multiplicative manner. 261

The following calibration parameters exist to improve Penumbra simulation performance, 262

yet not all have a strong influence on model performance. The TICALI (Total Irradiance) and TRCALI 263

(Transmittance) are essential calibration parameters. A TICALI calibration adjusts the landscapes 264

extraterrestrial irradiance. A TRCALI calibration uniformly shifts the OTM. The degree to which 265

the GRCALI (Ground Effect calibration) and CECALI (Cloud Effect calibration) improve model 266

accuracy is terrain-specific and atmosphere-specific, and each may provide only a minor 267

improvement in overall simulation exactitude. 268

Processing Steps 269

Penumbra processes the landscape by traversing the gridded data in one direction per time 270

step through a method called the Walking Algorithm. The Walking Algorithm was developed from 271

the Marching Squares and Marching Cubes approaches [24]. The Walking Algorithm traces a light 272

ray’s path from each DEM surface cell toward the sun using the sun’s azimuth and altitude. Shade 273

percentage for both object and topographic shade are tracked in parallel. The global solar energy 274

sub-model computes the extraterrestrial irradiance per time step for a given location based on the 275

study area’s centroid latitude and longitude. Sky conditions reduce this global irradiance to a net 276

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global irradiance. The topographic and object shade percentage arrays are multiplied against the 277

net global irradiance to calculate a ground-level spatially-distributed net irradiance. 278

Solar Angles 279

Solar azimuth and altitude are calculated at time intervals dictated by the simulation input 280

Daily-Grain. Altitude (α) and azimuth (γ) are calculated by: 281

γ = arcsine ((cos(φ) * cos(δ) * cos(Ω)) + (sin(φ) * sin(δ))) (1) 282

α = arccosine (EST * (ΔJDay) - λ) (2) 283

where latitude is φ, longitude is λ, declination is δ, hour angle is Ω, Earth sidereal time is EST, and 284

ΔJDay is the current Julian day minus the January 1st, 2000* Julian day (Fig 3, Table 2) [25]. 285

Fig 3. Solar Angles and the Walking Algorithm. Solar altitude (α) and azimuth (γ) are first 286

calculated to control the direction the Walking Algorithm traverses the landscape. The Walking 287

Algorithm azimuth rise (Y) and run (X) conversions with example sun positions. 288

Table 2: Azimuth and altitude angles conformed to a Cartesian coordinate system. 289

(Conditional Equations 3 – 10)

If azimuth is: 0 > α ≥ 90: If azimuth is: 180 > α ≥ 270:

(Eq. 3) Run = X = | (sin α * v) | (Eq. 7) Run = X = -1 * | (sin α * v) |

(Eq. 4) Rise = -1 * | (cos α * v) | (Eq. 8) Rise = | (cos α * v) |

If azimuth is: 90 > α ≥ 180: If azimuth is: 270 > α ≥ 360:

(Eq. 5) Run = | (cos α * v) | (Eq. 9) Run = X = -1 * | (cos α * v) |

(Eq. 6) Rise = | (sin α * v) | (Eq. 10) Rise = -1 * | (sin α * v) |

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Walking Algorithm 291

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Shade and net irradiance are individually computed by the Walking Algorithm from each 292

cell to a termination event. The Walking Algorithm itself is a set of methods that control the 293

direction and distance of shadowing evaluation (Fig 3). A termination event occurs when the 294

Walking Algorithm determines the solar ray’s path encountered topographic shadowing or the 295

edge of the simulations spatial extent. 296

At each time step the azimuth and altitude of the sun are applied to derive the rise, run, and 297

sun ray elevation as discrete distances. The azimuth is applied from a nadir perspective to calculate 298

the rise and run direction. The rise and run are calculated once per time step using the following 299

equation sets under the four Cartesian angular conditions (Table 2), where α is the current solar 300

azimuth and v is the assessment step distance. The variable v is defaulted to one quarter the 301

simulation’s cell resolution. The benefit of the rise/run/elevation relationship over direct use of the 302

azimuth and altitude is a reduction of repeated trigonometric calculations, which if repeatedly 303

performed are computationally expensive. 304

As the Walking Algorithm traverses the landscape, each encountered cell is assessed for 305

both topographic and object shadowing upon the origin cell. The sun’s altitude determines the 306

solar ray vertical rise (VerticalSOLAR) based on the distance traveled from the origin cell where γ is 307

the current solar altitude and XTOTAL is the Walking Algorithms current assessed distance from the 308

origin cell to the current cell encountered. 309

VerticalSOLAR = tan(γ) * XTOTAL (11) 310

Terrestrial Irradiance 311

The terrestrial solar irradiance (TSI; W/m2) is the quantity of solar radiation reaching the 312

Earth’s atmosphere on a given Julian day. TSI is calculated using the following equation [26], 313

where Ssun is the solar constant and n is the Julian day. 314

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TSI = Ssun * 2π * (n ⁄ 265.25) [12] 315

Cosine Effect and Calibration on Terrestrial Irradiance 316

The terrestrial irradiance is reduced to account for the solar altitude angle using the 317

following equation, where Θ is the solar altitude and TSI (Eq. 12). Calibration parameter TICALI 318

and CECALI are applied here. 319

TSIcos = cos(Θ) * TSI * TICALI * CECALI [13] 320

Solar Ray Height 321

As the Walking Algorithm traverses the landscape cell by cell, the ray’s height is calculated 322

using the Walking Algorithm’s altitude (γ) and total distance assessed (XTOTAL) (Fig 2). The Solar 323

Vertical Rise Component (VRCSOLAR) represents the absolute height from the initial cell, while 324

the current Solar-ray Absolute Height (SolarAH) represents the relative height between the ray’s 325

current height and the corresponding ground elevation (Fig 2). 326

VRCSOLAR = |tan(γ)| * XTOTAL (14) 327

SolarAH = Origin Cell Elevation + VRCSOLAR (15) 328

Object Shade Reductions 329

As the Walking Algorithm encounters a new cell, that cell’s landscape object interaction is 330

assessed by subtracting the object’s absolute elevation by the SolarAH to obtain z-dimensional 331

difference between object height and solar ray (Fig 2). 332

Object Delta Height (ODH) = SolarAH - Object Height (16) 333

If the ODH is positive, the solar ray passed over the object. If the ODH is negative, the solar 334

ray penetrated through the object. As each landscape object’s ODH is encountered, that object’s 335

associated transmittance value is applied to the accumulated solar transmittance in a multiplicative 336

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manner. When the Walking Algorithm is terminated, the resulting accumulated solar transmittance 337

value is the origin cell’s percent illumination. The multiplicative of the all encountered cell object 338

transmittance reductions is assigned to the origin cell (Fig 2). 339

Object Light Deduction (OLD) = ∏ Object Transmittance𝑛𝑖=0 (17) 340

Topographic Shade Reductions 341

Topographic shadowing may occur upon the origin cell. Like object shadowing, as the 342

Walking Algorithm encounters a new cell the potential interaction is assessed by subtracting the 343

elevation from the SolarAH to obtain a delta representing the z-dimensional difference between the 344

terrain and ray height (Fig 2). 345

Terrain Delta Height (TDH) = SolarAH - elevation (18) 346

If TDH is positive, the solar ray is above the ground. If TDH is negative, the solar ray 347

penetrated the ground and the Walking Algorithm is terminated. Unlike object shadowing, the 348

aggregation of topographic shadowing does not occur since direct light cannot attenuate through 349

terrain. Instead the Topographic Light Deduction (TLD) is calculated based on an inverse 350

relationship between the number of cells walked (XTOTAL), plus the ground calibration factor 351

(GRCALI) is applied here. 352

Topographic Light Deduction (TLD) = (1/ XTOTAL) * GRCALI (19) 353

If XTOTAL equals one (meaning only one cell was walked), the TLD is equal to GRCALI, the 354

maximum topographic shade. Any number of additional cells walked would generate a weakening 355

of topographic shade due to the inverse relationship of XTOTAL. The inverse effect accounts for 356

scattering of indirect. A larger XTOTAL results in reduced topographic shadowing over greater 357

distances. 358

Graphic Processing Unit (GPU) 359

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The use of a GPU for computational processing is an option in Penumbra. The GPU module 360

replaces the Walking Algorithm, which is the most processing intensive aspect of Penumbra. The 361

GPU approach is based on a computer graphics process known as shadow mapping (term shadow 362

coincidental), which simulates obfuscation of a surface or object from a light source by rendering 363

a scene from the light source’s viewpoint, and generating a lookup table from the subsequent 364

render results [27]. The lookup table is known as a shadow map, and each point in the scene can 365

evaluate against the map by testing its distance from the light source versus the shortest distance 366

recorded in the map [28]. 367

As previously mentioned, Penumbra assumes that light rays are traveling along parallel 368

paths; therefore, an orthographic projection can be used for the sun’s viewpoint, guaranteeing that 369

each ray into the scene only travels through a single row of pixels. The GPU utilizes “shaders” 370

(GPU term “shaders” and Penumbra meaning of “shade” are homonyms) to both quickly render 371

the shadow map and to perform the shadow lookup for each grid point in a massively parallel 372

fashion [27]. However, there is an implicit overhead to moving data to and from the GPU memory. 373

When processing small datasets, the central processing unit (CPU) bound Walking Algorithm will 374

often exceed the GPU shadow mapping approach in runtime performance. When processing 375

landscapes with many spatial or temporal units, the parallel nature of the GPU allows for the 376

shadow mapping approach to produce superior runtime performance. For landscapes characterized 377

at high resolution or representing large spatial extents, and simulations set to a small temporal 378

grain, the GPU is a viable option to shorten simulation runtime requirements. 379

Outputs 380

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Penumbra can provide representations of landscape illumination as: object shade, terrain 381

shade, total shade (object and terrain shade combined), or net solar energy. All these forms can be 382

output in isolation or collectively. 383

Object and Topographic Shade 384

Shade outputs are all percentage based. Penumbra internally tracks the deduction of light 385

as a percentage in the Object Light Deduction (OLD) and Topographic Light Deduction (TLD) 386

arrays; not the percentage of shadowing by topography or objects. Spatial shade data are generated 387

per aggregation output by inverting the OLD and TLD data to create Topographic Shade (TSHADE) 388

and Object Shade (OSHADE), where 1 represents complete shade and 0 represents no shade. 389

TSHADE = 1 – TLD (20) 390

OSHADE = 1 – OLD (21) 391

Total shade (ShadeTOTAL) is calculated by taking the multiplicative of TLD and OLD. 392

ShadeTOTAL = TSHADE * OSHADE (22) 393

Net Ground-Level Solar Energy 394

Energy units are defaulted as Watts per meter2, but micromoles/m2/second (µmoles/m2/s) 395

is an option. The net energy (EnergyNET) is the multiplicative outcome of ShadeTOTAL (Eq. 22) and 396

TSICos (Eq. 12). EnergyNET is the result of the spatially-distributed shadow reductions and all 397

irradiance reductions (Fig 2). 398

EnergyNET = ShadeTOTAL * TSIcos (23) 399

A standard Penumbra simulation will focus on the total shadowing effect, though object 400

and topographic shading can be isolated. EnergyNET relies on TSICos, which incorporates the sun 401

cosine effect, atmospheric reductions, along with ShadeTOTAL representing all terrain shading. 402

EnergyNET ultimately represents the final spatially-distributed ground-level energy. 403

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Calibration Process 404

Penumbra calibration is a multistep process. The first run is a baseline for comparison, and 405

each subsequent run focuses on one additional calibration parameter. A Penumbra initial run has 406

GRCALI which is set to 0.5 and all other calibration parameters set to 1.0. Using the initial 407

simulation output, Penumbra’s peak irradiance is calibrated to the observed peak irradiance using 408

TICALI (Eq. 24). The TICALI calibration calculation adjusts the percent difference between the 409

Observed Peak Irradiance (ObsPI) and the Simulated Peak Irradiance (SimPI). The observed value 410

to derived (ObsPI) would be the peak irradiance value within the observed data. TICALI helps 411

account for regional atmospheric irradiance deductions caused by air particulate that reduce 412

irradiance. 413

TICALI = 1 – ((SimPI - ObsPI)/ ObsPI) (24) 414

For Penumbra runs two and three, the TRCALI and GRCALI calibration parameters are 415

adjusted to improve topographic shading and object shading. Topographic and object shading can 416

be collective in nature, causing an interactive influence within the observed data. This interaction 417

can make sequential calibration difficult due to TRCALI or GRCALI shifts impacting the opposing 418

topographic or object shade results. General site knowledge can help guide the adjustment of these 419

two calibrations parameters. The dominant influence should be calibrated first. 420

The Penumbra Cloud sub-model is a simplistic daily percent cloud coverage, where the 421

percent of clouds observed is a direct reduction in available TSICOS. Cloud coverage can be 422

provided as a comma separated value (csv) file. Cloud coverage data can be obtained from many 423

weather stations or assimilated by processing satellite imagery from data products captured by 424

platforms such as Geostationary Operational Environmental Satellite (GOES). CECALI allows the 425

model user to adjust the cloud coverage impact on TSICOS to better represent the types of clouds 426

20

their study area generally contains. Here we do not present simulations utilizing the cloud coverage 427

option due to cloud coverage being non-spatial in the current version of Penumbra. 428

Model Testing 429

Penumbra’s solar position, extraterrestrial irradiance, and simulated net irradiance were all 430

tested individually to vet solar energy assessments in isolation. Each phase marks simulation 431

moments where error can propagate through downstream calculations. Penumbra’s first three 432

calibration parameters were utilized during the testing of net irradiance. CECALI was not modified 433

due to the observed data being all clear sky days. 434

Penumbra was tested against observed solar energy data to guide model calibration and 435

assess final model performance. The solar angles were tested using U.S. Navy Observatory data. 436

The global irradiance model was tested using the NSRDB data. Two datasets containing measured 437

solar energy were utilized for model testing: the O’CCMoN dataset, and the Snow Telemetry 438

(SNOTEL) dataset. Two datasets were used to provide different examples of ground-level solar 439

energy testing for the model. 440

Modeled versus Observed Solar Angle 441

Penumbra’s solar angle sub-model was derived from research aimed at simplifying solar 442

azimuth and altitude calculations [25]. To understand any Penumbra error due to these simplified 443

calculations, Penumbra solar angle azimuth and altitude data for position 44.915960°N -444

123.001439°W were output for every hour of June 21st, 1990. The modeled data accuracy was 445

characterized using bivariate correlation against matching U.S. Navy Observatory matching time 446

and location data [29] (S1 Fig). Penumbra calculations and the U.S. Navy Observatory data for 447

azimuth and altitude agreed well with an r2 of 0.9923 and 0.9991, respectively. 448

21

Modeled versus Observed Global Irradiance 449

Penumbra’s irradiance calculation starts as extraterrestrial irradiance derived from a global 450

irradiance model [26]. To understand any Penumbra error due to these calculations, TSICOS (Eq. 451

12) was characterized using bivariate correlation against the Department of Energy (DOE) 452

NSRDB data for the Salem Oregon NSRDB station at 44.915960°N -123.001439°W for the year 453

1990 [6]. As an airport, this site had no topographic or object shading to interfere with the observed 454

data. These hourly data agreed well with Penumbra hourly TSICOS with a r2 of 0.9577 (S2 Fig). 455

Model Calibration and Validation 456

The O’CCMoN monitoring project provides continuous Photosynthetically Active 457

Radiation (PAR) observations at hourly increments for eight pairs of forest and open sites 458

extending along a 200-km transect from the crest of the Oregon Cascade Range to the Pacific 459

Coast. Forest sites are mature stands with well-established canopies. Open sites are former mature 460

forest stands that had been clear-cut harvested shortly before installation climate station sensors. 461

The forest versus open sites allow contrasting the two dramatically different environment types. 462

High resolution representation of landscape objects at these sites was required to test Penumbra’s 463

ability to model object shadowing from individual trees. Airborne LiDAR data provided these 464

representations, but of eight potential O’CCMoN paired sites, only the Moose Mountain and Fall 465

Creek O’CCMoN transect locations had existing LiDAR coverage [30]. For both locations, 466

modeled open site and forest site PAR predictions were evaluated against observed hourly data for 467

July 8th through July 11th, 2008. Final results for Moose Mountain and Fall Creek were the result 468

of following the calibration procedure. 469

Moose Mountain 470

22

The Moose Mountain Forest site is a predominantly Douglas-fir forest located in the 471

western Oregon Cascade Range at an elevation of 658 meters above sea level on a southeasterly 472

slope. The Moose Mountain Open site is in a clear-cut 460 meters to the southwest at an elevation 473

of 668 meters (Fig 4). PAR is monitored with a LI-COR LI-190SL instrument with the forest site 474

sensor height being 297 centimeters (cm) and the open site sensor height being 282 cm from 475

ground level [7]. Hourly Penumbra results demonstrate model’s ability to simulate varying 476

environments and fine detail (Figs 5 and 6). 477

Fig 4. Moose Mountain Open site and Forest site. Sites are part of the Oregon Crest-to-Coast 478

Environmental Monitoring transect (O’CCMoN) dataset. Figure is the gridded 3D representation 479

of the site generated from LiDAR data. 480

Fig 5. Results for Moose Mountain Open site PAR. The O’CCMoN Moose Mountain Open site 481

simulated versus observed PAR (µmoles/m2/s). Note, figure 6 y-axis scale is 0.1x to this figures 482

y-axis scale scales. 483

Fig 6. Results for Moose Mountain Forest site PAR. The O’CCMoN Moose Mountain Forest 484

site simulated versus observed PAR (µmoles/m2/s). Note, figure 5 y-axis scale is 10x to this figures 485

y-axis scale scales. 486

The initial, uncalibrated Moose Mountain simulation yielded moderate agreement for the 487

open site with a percent error of 0.511 and a root mean square error (RMSE) of 506.0 488

(µmoles/m2/s) (Table 3). The overall simulation error over four days revealed that Penumbra on 489

average was off by 286.1 (µmoles/m2/s) in this environment. The initial, uncalibrated performance 490

for the forest site was poor due to no calibration being applied, and heavily overestimated terrain-491

level solar energy. Simulated vs. observed PAR percent error was 19.2, and RMSE was 515.7 492

(µmoles/m2/s). 493

23

Table 3: Moose Mountain parameters, initial results, and calibrated results. 494

Open Site

Initial

Run

Calibrated

Run

Calibration

Parameters

Initial

Settings

Calibrated

Settings

Percent

Agreement

0.52 1.03 Irradiance

1.0 0.94

RMSE 506.0 224.6 Transmittance 1.0 0.945

Mean Error 286.1 -17.3 Topographic 0.5 0.2

Sensor ht. (cm) 282.5 Clouds 1.0 1.0

Forest Site

Percent

Agreement


1.0 0.94


Mean Error -10.0 -15.6 Topographic 0.5 0.2

Sensor ht. (cm) 297.2 Clouds 1.0 1.0 495

Calibration of Penumbra greatly improved PAR predictions for both the open site and 496

forest site (Table 3). The Moose Mountain Open site final calibration provided an excellent 497

observed versus simulated PAR agreement with a percent error of 1.029 and a RMSE of 224.5 498

(µmoles/m2/s). The overall simulation error for the four days revealed that Penumbra on average 499

was off by only -17.27 (µmoles/m2/s). The Moose Mountain Forest final calibration provided a 500

fair agreement with a percent error of 1.84 and a greatly reduced RMSE of 53.78 (µmoles/m2/s). 501

The overall simulation error for the forest site over four days was on average off by -15.57 502

(µmoles/m2/s) (Figs 5 and 6) (Table 3). The VISTAS Moose Mountain static frame spatially 503

represents the calibration run results (Fig 7) and video portrays model dynamic (S3 Video). 504

Fig 7. Static Results for Moose Mountain Open and Forest sites. Still-frame from S3 Video of 505

July 06th, 2008 at 2:30pm. All shade reductions are scaled from 1.0 to 0.0, where 1.0 is no 506

reduction and 0.0 is full reduction. Sub frames: (A) net solar energy as PAR (µmoles/m2/s), (B) 507

Total shade reduction (0.0 – 1.0), (C) Object shade reduction (0.0– 1.0), (D) Topographic shade 508

reduction (0.0 – 1.0). For all (0.0 – 1.0) shade reductions 1.0 is no reduction and 0.0 is full 509

reduction. See S3 Video for full animation of this simulation. 510

24

Falls Creek 511

The Falls Creek Forest site is a predominantly Douglas-fir forest. The PAR sensor location 512

is at an elevation of 528 meters on a slightly northern slope (Fig 8). The Falls Creek Open site is 513

in a recovering clear-cut 328 meters to the west at an elevation of 534 meters. PAR is monitored 514

with a LI-COR LI-190SL instrument with the forest site sensor height being 334 cm and the open 515

site sensor height being 353 cm from ground level [7]. Hourly uncalibrated and calibrated results 516

demonstrate Penumbra’s ability to simulate varying environments and fine detail (Figs 9 and 10) 517

(Table 4). 518

Fig 8. Falls Creek Open site and Forest site. Sites are part of the Oregon Crest-to-Coast 519

Environmental Monitoring transect (O’CCMoN) dataset. Figure is the gridded 3D representation 520

of the site generated from LiDAR data. 521

Fig 9. Results for Falls Creek Open site PAR. The O’CCMoN Falls Creek Open site simulated 522

versus observed PAR (µmoles/m2/s). Note, figure 10 y-axis scale is 0.1x to this figures y-axis scale 523

scales. 524

Fig 10. Results for Falls Creek Forest site PAR. The O’CCMoN Falls Creek Forest site 525

simulated versus observed PAR (µmoles/m2/s). Note, figure 9 y-axis scale is 10x to this figures y-526

axis scale scales. 527

Table 4: Falls Creek parameters, initial results, and calibrated results. 528

Open Site

Initial

Run

Calibrated

Run

Calibration

Parameters

Initial

Settings

Calibrated

Settings

Percent

Agreement


1.0 1.0


Mean Error -59.96 148.1 Topographic 0.5 0.25

Sensor ht. (cm) 353 Clouds 1.0 1.0

Forest Site

Percent

Agreement


1.0 1.0

25


Mean Error -59.90 -10.4 Topographic 0.5 0.2

Sensor ht. (cm) 334 Clouds 1.0 1.0 529

The initial, uncalibrated Falls Creek Penumbra simulation yielded an excellent open site 530

agreement with a percent error of 1.09 and RMSE of 252.6 (µmoles/m2/s) (Table 4). The overall 531

simulation error for the four days revealed that Penumbra on average was off by -59.96 532

(µmoles/m2/s). The initial, uncalibrated performance at the forest site had poor agreement with the 533

observed PAR data, with a percent error of 19.2 and the RMSE of 515.7 (µmoles/m2/s). We show 534

these uncalibrated results to highlight how a simplified irradiance model performs when object 535

and topographic shading are not assessed. To that effect, this initial uncalibrated Penumbra 536

simulation is already reducing solar energy by roughly half, seen by comparing the open site to 537

forest site results (Figs 9 and 10). 538

The Falls Creek Open site final calibration yielded a good observed versus simulated PAR 539

agreement with a percent error of 0.77 and the RMSE of 296.8 (µmoles/m2/s) (Table 4). Initial run 540

agreement was higher than the final calibration run due to mid-afternoon PAR reductions. The 541

reduced agreement was the result of calibration compromises made to improve the forest site 542

performance. The overall simulation error for the four days revealed that Penumbra on average 543

was off by 148.1 (µmoles/m2/s). 544

The calibration compromise between the open and forest site greatly improved the final 545

forest site PAR simulated agreement with only minor loss to open site PAR simulated agreement. 546

The Falls Creek Forest final calibration provided good agreement with a percent error of 1.51 and 547

a low RMSE of 34.2 (µmoles/m2/s) (Table 4). The overall simulation error for the forest site four 548

days was on average off by -10.2 (µmoles/m2/s). A VISTAS static frame represents the Falls Creek 549

calibration run (Fig 11) and video portrays model dynamic (S4 Video). 550

26

Fig 11. Static Results for Falls Creek Open and Forest sites. Still-frame from S4 Video of July 551

06th, 2008 at 2:30pm. All shade reductions are scaled from 1.0 to 0.0, where 1.0 is no reduction 552

and 0.0 is full reduction. Sub frames: (A) net solar energy as PAR (µmoles/m2/s), (B) Total shade 553

reduction (0.0 – 1.0), (C) Object shade reduction (0.0– 1.0), (D) Topographic shade reduction (0.0 554

– 1.0). For all (0.0 – 1.0) shade reductions 1.0 is no reduction and 0.0 is full reduction. See S4 555

Video for full animation of this simulation. 556

SNOTEL Site Calibrations 557

The SNOTEL Network of sites monitor and record an assortment of meteorological data 558

in mountainous snow-zone locations in the western United States and Canada [8]. SNOTEL sites 559

are maintained by the U.S. Department of Agriculture Natural Resources Conservation Service 560

(NRCS) to ensure the highest practicable data quality and continuity [31]. NRCS has added 561

pyrometers to select SNOTEL sites through third party interests [32]. These sites are potentially 562

influenced by local and distant topographic shading and by object shading from surrounding forest 563

vegetation. Unlike the O’CCMoN dataset, which specifically provides monitored forest sites, there 564

is no guarantee all SNOTEL site irradiance data captures object shadowing. SNOTEL sites are 565

still of value for Penumbra calibration because each site’s pyrometer captures the region’s 566

atmospheric irradiance reduction and any topographic shadowing. SNOTEL data sites are also 567

distributed throughout the Pacific Northwest, which provides solar irradiance testing data over a 568

larger geographic extent than the O’CCMoN data alone. 569

We identified two SNOTEL sites that had appropriate pyrometer data and publicly 570

available LiDAR data needed to represent landscape objects for calibration and testing. These sites 571

are Meadows Pass and Mount Gardner (S5 Fig). The high resolution spatial inputs were derived 572

from LiDAR project bare earth and highest hit gridded data [33]. We aggregated these data to 3-573

27

meter cell resolution. Each site was compared to SNOTEL observations for four clear sky days. 574

Penumbra temporal grain was set to 10 minutes. Output data aggregations were set to 60 minutes 575

to match the monitored SNOTEL irradiance data. The data from the grid cell matching the 576

SNOTEL site latitude and longitude coordinates was queried per time step. SNOTEL sensors were 577

not at ground-level (Table 5). For Penumbra to simulate irradiance at the stations irradiance meter 578

height, a pseudo-height elevation option was built into the model to adjust the queried cell data 579

height to match the SNOTEL sensor’s true height. 580

Table 5: SNOTEL 3-Meter results for four-day site calibration sequence. 581

Initial

Run

Calibrated

Run

Calibration

Parameters

Initial

Settings

Calibrated

Settings

Meadows Pass Site

Percent Agreement 0.4657 0.6854 Irradiance 1.0 0.766

RMSE (Watts/m2) 288.66 34.2 Transmittance 1.0 1.0

Mean Error (Watts/m2) 174.67 66.46 Topographic 0.25 0.25

Sensor ht. (m) 5.48 Clouds 1.0 1.0

Mount Gardner Site

Percent Agreement 0.7754 0.6743 Irradiance 1.0 0.7335

RMSE (Watts/m2) 411.2 265.2 Transmittance 1.0 1.0

Mean Error (Watts/m2) 292.5 158.0 Topographic 0.25 0.25

Sensor ht. (m) 5.48 Clouds 1.0 1.0 582

SNOTEL observed data was suitable for a partial calibration of Penumbra. Meadows Pass 583

site observed versus simulated data showed a 22% increase in percent agreement, a RMSE drop 584

from 288 to 34.2 (Watts/m2). The calibrated results also reduced the four-day mean error from 585

174.67 to 66.46 (Watts/m2) (Table 5). Mount Gardner site observed versus simulated data showed 586

an 10% decrease in percent agreement, though a RMSE drop from 411.2 to 265.2 (Watts/m2). The 587

calibrated results also reduced the four-day mean error from 292.5 to 158.0 (Watts/m2) (Table 5) 588

with video portraying model dynamics (S6 Video). 589

Landscape Scale Simulation 590

28

For a visual comparison at regional scales, we demonstrate a case study within the Mashel 591

River Watershed, a watershed located west of Mount Rainer, Washington. This watershed provides 592

a prototypical Penumbra application due to the spatially diverse topography, land management 593

practices, and stakeholder interests within the watershed. The Mashel River Watershed area is 209 594

km2. Penumbra input data and parameters are listed in Table 6. Aboveground plant biomass values 595

(g carbon/m2) at a spatial resolution of 30 m across the entire watershed were obtained from the 596

LandTrendr model for the year 1990 [34]. The nDSM (surface objects) were derived from a 597

Douglas-fir biomass-to-height relationship [35]. The objects light transmittance was provided as a 598

uniform grid matching the O’CCMoN Falls Creek site TRCALI value, while the TICALI was obtained 599

from the nearby Meadows Pass SNOTEL calibration (Table 5). 600

Table 6: Penumbra setup for Mashel River Watershed. 601

Temporal Parameters Spatial Input Data

Temporal Grain 10 minutes DEM Mashel_30m_DEM.asc

Temporal

Aggregation Daily nDSM Mashel_30m_Objects.asc

Start Year 1990 Transmittance Mashel_30m_Object_Transmittance.asc

Start Julian Day 001 Center Latitude 46.501747

End Year 1990 Center Longitude -122.060215

End Julian Day 365 GPU false

Calibration Parameters

TICALI 0.766 TRCALI 0.965

CECALI 1.0 GRCALI 0.25 602

Here we present summer solstice to signify the peak energy for the Mashel River 603

Watershed (Fig 12) with video portraying model dynamics (S7 Video). This simulation illustrates 604

Penumbra’s ecological modeling purpose. At any moment in time, the watershed’s landscape solar 605

energy is the result of the time of year, topographic shading due to the sun’s altitude, and object 606

shading from the dominant Douglas-fir forests that grow over much of this watershed’s landscape. 607

29

Ignoring any component that influences the final net-irradiance across any landscape is an 608

incomplete representation of solar energy. 609

Fig 12. Upper Section of Mashel River Watershed Summer Solstice Irradiance. Still-frame 610

from S7 Video of June 21st, 1991. Ground-level solar energy as Watts/m2. See S7 Video for full 611

animation of this simulation. 612

Discussion 613

Penumbra can simulate shade and irradiance using standard inputs commonly used in 614

landscape-scale analyses. Penumbra is one of many models for simulating environmental solar 615

energy. Simplistic solar energy models provide computationally simple irradiance, but at the loss 616

of spatial heterogeneity. These models are ideal when solar energy is not a significant variable in 617

the larger context of the research. Ray tracing models have tackled the complexity of light 618

interaction within detailed landscape representations, but still today are burdened by computational 619

limitations, especially for large spatial extents. We acknowledge true objects like trees are more 620

complex than Penumbra is representing. Trees have complex structures that allow light through 621

breaks in the tree crown and tree tops. But, at large landscape scales the current data dilemma is 622

that only airborne LiDAR data is detailed enough to represent tree canopy structure. Even with 623

accurate tree height, LiDAR does not fully represent tree understory. Penumbra provides a 624

landscape perspective and modeling approach that fills the niche between the simplistic to complex 625

irradiance modeling approaches. 626

Spatially-distributed landscape scale representations of solar energy allow for the 627

development and improvement of ecological processes that are sensitive to variations in incident 628

solar energy. Solar energy directly drives air temperature and provides energy for leaf 629

photosynthesis, both important drivers of evapotranspiration rates. Solar energy directly warms 630

30

surface soil and surrounding air temperature. That surface energy transfers down into subsurface 631

soils. The quantity of solar energy available to warm subsurface soils is dependent on the landscape 632

coverage (topography and vegetation) that blocks direct solar radiation. During rain events, heat 633

exchange between soil and water in surface and subsurface runoff occurs, further influencing 634

vertical and lateral soil temperature profiles. Vertical and horizontal groundwater energy transfers 635

may indirectly impact stream water temperature, in addition to direct solar radiation impacting 636

stream water temperatures. By quantitatively describing the relationship between canopy 637

openness, incident ground-level irradiance and soil temperature, the Penumbra simulation results 638

presented here point to an approach for dynamically simulating effects of upslope disturbances on 639

groundwater and stream temperatures. This achievement could occur by linking Penumbra with 640

an ecohydrological model (e.g., VELMA, Regional Hydro-Ecologic Simulation System 641

(RHESSys), Distributed Hydrology Soil Vegetation Model (DHSVM)) that simulates plant-soil-642

water dynamics within hillslope at watershed scales. 643

Penumbra’s modeling approach is to simulate ground-level solar energy as simply as 644

possible, while still providing spatially-distributed shading and net irradiance at a sufficient level 645

of accuracy. Penumbra’s flexible temporal parameters allow for landscape assessments to be 646

performed within the context of the user’s research question. Penumbra’s ability to simulate Sun 647

position at one temporal grain, yet aggregate those time steps to a greater temporal grain allows 648

for flexible irradiance simulations. This flexible aggregation allows for model use within many 649

research projects involving riparian shading, forest disturbances, photosynthesis, or any other 650

biological process where irradiance has an influence. 651

Penumbra is subject to the same limitations as most process based models [19]. Penumbra 652

can have lengthy computational runtime requirements, though at typical watershed resolutions and 653

31

extents Penumbra performs reasonably well. While GPU processing enhancement and direct 654

model integration assist in reducing the runtime burden, further improvements are needed to do 655

large extent assessments at high resolution (e.g., 1m). For some users, the Java platform in which 656

Penumbra was developed may be a drawback. However, the Java language was selected due to the 657

“write once, read anywhere” perspective, and the language works across many computer operating 658

systems. 659

Being a process-based model, Penumbra’s testing and uncertainty characterization is a 660

necessity. We discovered only two vetted Northwest, USA datasets with adequate solar energy 661

readings to perform Penumbra testing, EPA O’CCMoN and SNOTEL. Most field irradiance data 662

are collected for a short period. The long-term irradiance monitoring projects focus on global 663

irradiance, not ground-level irradiance. 664

Penumbra calibration was explained to ensure users are aware this spatially-distributed 665

model’s outputs have the capability to be improved upon beyond initial simulations. Though many 666

locations will not have the observed data needed to perform a rigorous Penumbra calibration, that 667

would also suggest those areas are lacking an understanding of ground-level solar energy in 668

general. Fitting to two current niche needs, Penumbra was developed to meet an ecological need 669

by being capable of functioning within any environment that a DEM and nDSM can be developed. 670

At a conservative minimum, Penumbra shade maps could provide an enhanced understanding of 671

any such area lacking some understanding of solar-energy distributions. This approach also fits 672

well for ecological models producing future predictions since there are no observed data on how 673

our environments will come to be. 674

Penumbra allows for dynamic integration with other environmental models to achieve 675

representations of changing irradiance due to impacts from land-use through time. This can be 676

32

especially useful for simulating scenarios of alternative land management practices across the 677

world. One particularly important example is characterizing the radiative effects of existing forests 678

and anthropogenic land-use on stream temperature. Furthermore, Penumbra can be integrated with 679

dynamic vegetation growth models to determine the time required to increase riparian shade by 680

some percentage. Urban models could also utilize Penumbra’s object shading feature to determine 681

the spatial impacts of urbanization on irradiance and heat loads within urban centers. 682

Conclusion 683

Penumbra provides the capability to represent spatially-distributed shade and terrain level 684

irradiance. These representations allow for improved landscape scale simulations of solar energy 685

within a reasonable simulation timeline without sacrificing the spatial heterogeneity of 686

environmental light. With this model, solar energy interactions within complex and unique 687

environments can be better simulated. 688

Here Penumbra was vetted against three independent data sources each addressing different 689

scales of solar energy. Penumbra’s extraterrestrial irradiance has high accuracy with a r2 of 0.9577. 690

Penumbra’s net irradiance for the high-resolution O’CCMoN sites revealed a calibrated mean error 691

ranging from -10.4 to 148.1 (Watts/m2), while the SNOTEL sited ranged from 66.46 to 158.0 692

(Watts/m2) against the SNOTEL observed. This level of accuracy for a natural phenomenon that 693

is difficult to simulate and not feasible to monitor at landscape scales makes Penumbra remarkably 694

useful for landscape-scale applications. 695

Overall, Penumbra is a novel surface irradiance model which filled an environmental 696

modeling niche by accounting for topographical shading, object shading, and impacts of 697

atmospheric conditions. This information will lead to more integrated modeling systems that better 698

inform stakeholders such as: watershed councils, tribes, local, state, and federal decision makers 699

33

interested in quantifying effects of land use and impacts due to local climate shifts. Current 700

modeling efforts account for no or partial solar radiation reaching the earth’s surface. To better 701

inform future land management decision making, the full consequential impacts of those choices 702

on human and natural systems must be foreshadowed to the best of our abilities. Penumbra has 703

been developed and tested to overcome the current ecological model void existing between the 704

simplistic to complex solar energy modeling approaches. Penumbra can better inform models 705

influencing future land management by making more evident the full consequential impacts of 706

management decisions on human and natural systems. 707

34

Acknowledgements 708

The information in this document has been funded in part by the U.S. Environmental 709

Protection Agency. It has been subjected to the Agency’s peer administrative review, and it has 710

been approved for publication as an EPA document. Mention of trade names or commercial 711

products does not constitute endorsement or recommendation for use. 712

We thank the following for their assistance during the development of Penumbra: the 713

VISTAS group (Dr. Judy Cushing, Nik Stevenson-Molnar, and Taylor Mutch) for endless support 714

with their spatial visualization tool; Scott Pattee (Natural Resource Conservation Service) for his 715

expert knowledge of SNOTEL stations and assistance with providing their precise locations; 716

Laurie Benson, Greg Blair, Sayr Hodgson, Justin Hall, Joe Kane, Paul Swedeen, Dan Stonington 717

for Mashel River Watershed data and forest management information; Cindy Spiry and David 718

Steiner for Tolt River Watershed data and floodplain restoration information. 719

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Supporting Information 815

S1 Fig. Penumbra versus US Navy Observatory azimuth and altitude Angles. Validation of 816

Penumbra simulated azimuth and altitude angles. Estimations compared against the U.S. Navy 817

Observatory for June 21st, 1990. Azimuth agreed with an r2 of 0.9923. Altitude agreed with an r2 818

of 0.9991. 819

http://www.oregongeology.org/dogamilidarviewer

39

S2 Fig. Penumbra versus monitored solar irradiance. Validation of Penumbra simulated 820

extraterrestrial irradiance. Estimations compared against monitored DOE-NSRDB data for 821

Salem, Oregon, USA (44.915960°N, -123.001439°W). Data represents every hour of the year 822

1990. Simulated irradiance (Watts/m2) agreed with an r2 of 0.9577. 823

S3 Video. Moose Mountain. O’CCMoN Moose Mountain site simulation for July 6th, 2008 824

through July 9th, 2008. 825

S4 Video. Falls Creek. O’CCMoN Falls Creek site simulation for July 6th, 2008 through July 826

9th, 2008. 827

S5 Fig. SNOTEL sites. Site locations reference map, along with Meadows Pass and Mount 828

Gardner sites with split views of elevation and landscape object heights. 829

S6 Video. Mount Gardner. SNOTEL Mount Gardner site simulation. The daily time steps are 830

every 10 minutes, per day. Each day of simulation is advanced 7 days between April 24th, 2013 831

through October 23rd, 2013. 832

S7 Video. Mashel River Watershed simulation. The upper Mashel River Watershed daily 833

averaged irradiance for years 1991 and 1992 at 30-meter resolution. 834

1 2 3 penumbra: a spatially-distributed shade percent and...

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