. 14
( 14)

for (i=1;i<=lattice_size_x;i++)
if (fabs(u[i])>0.00001) {lap[i]=-2*u[i];lap[iup[i]]=u[i];lap[idown[i]]=u[i];}}

for (i=1;i<=lattice_size_x;i++)
if (fabs(lap[i])>0) u[i]+=fac*lap[i];}
for (i=1;i<=lattice_size_x;i++)
fprintf(fp1,"%f %f %f\n",(i-lattice_size_x/2)/10.0,u[i],-v[i]);
Appendix 7

Codes for modeling stochastic processes

A7.1 Fractional-noise generation with Fourier-¬ltering method

The following code generates a 1D fractional noise (i.e. time series) using the Fourier-¬ltering method.
The user is prompted for all of the information used to generate the noise. The code will produce
noises with Gaussian or log-normal distributions depending on directions from the user. The random
number seed idum generates a pseudo-random sequence of numbers. In order to generate distinct
output, different values of idum must be used.

{ float *precyear,beta,sum,sddevcomp,mean,sddev;
int length,year,format;
int idum;
char outputfile[30];
FILE *fp;

printf("\nName of output file: ");
printf("\nLength of desired noise: (factor of two, please) ");
printf("\nGaussian or Log-normal dataset? (1 for gaussian 2 for l-n): ");
printf("\nMean: (if log-normal this mean is that of the final,
log transformed data) ");
printf("\nStandard Dev: ");
printf("\nBeta: ");
printf("\nRandom number seed: (any negative integer) ");
/*we contruct a vector of length two times the input length so that we
may cut off the ends to eliminate the periodicity introduced by filtering*/

for (year=1;year<=2*length;year++)
for (year=3;year<=2*length;year++)
for (year=length/2;year<=length+length/2;year++)
for (year=length/2;year<=length+length/2;year++)
/*below we rescale the amplitudes to restore the desired moments*/
for (year=length/2;year<=length+length/2;year++)
if (format==2) if (mean!=0.0) precyear[year]=

if (format==2) for (year=length/2;year<=length+length/2;year++)
for (year=length/2;year<=length+length/2;year++)
fprintf(fp,"%d %f\n",year-length/2,precyear[year]+(float)(year-length/2)/(length));

The following code generates a 2D Gaussian fractional noise (i.e. raster dataset) using the Fourier-
¬ltering method. The user is prompted for all of the information used to generate the dataset.
{ float fact,*precyear,betax,sum,mean,sddev,sddevcomp;
int lengthx,lengthy;
int *nn,i,j,idum,index;
char outputfile[30];
FILE *fp;

printf("\nName of output file: ");
printf("\nLength of desired noise: (factor of two, please) ");

printf("\nMean: ");
printf("\nStandard Dev: ");
printf("\nBeta: ");
printf("\nRandom number seed: (any negative integer) ");
for (i=1;i<=lengthx;i++)
for (j=1;j<=lengthy;j++)
for (j=1;j<=lengthy;j++)
for (i=1;i<=lengthx/2;i++)
for (j=1;j<=lengthy/2;j++)
for (i=1;i<=lengthx/2;i++)
for (j=1;j<=lengthy/2;j++)
for (i=1;i<=lengthx/2;i++)
for (j=1;j<=lengthy/2;j++)
for (i=1;i<=lengthx/2;i++)
for (j=1;j<=lengthy/2;j++)


A7.2 Stochastic model of Pleistocene ice ages

The following code is the basis for the stochastic-resonance model of ice ages. The model outputs
four columns with the form: t (yr), T , v , and B h .
#define maxoutputlength 10000000

{ FILE *fp1;
int n,t,t2,idum,duration,nsteps;
float tau,per,drift,eta,*T,*B,Bh,*v,deltaT,timestep;

timestep=100; /* yr */
duration=3000000; /* yr */
for (t=1;t<=nsteps;t++)
{if (T[t]<-1) deltaT=-T[t]-3.2*sqrt(-1-T[t])+eta*gasdev(&idum)+drift;
else deltaT=-T[t]+eta*gasdev(&idum)+drift;
for (t2=t-2*tau;t2<t;t2++)
if (t2>1) Bh+=-(T[t2]-1)*exp(-fabs(t-t2)/tau)/tau;
if (T[t]<-1) v[t+1]=fabs(B[t+1]-B[t]); else v[t+1]=0;
fprintf(fp1,"%d %f %f %f\n",t*100.0,T[t],v[t+1],Bh);}

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avalanche, 244 Amargosa Valley coupled evolution with alluvial
calculatealongchannelslope, 239 example of dust emission from channels
computeflexure, 249 playas, 13, 53 models for, 101--103
fillinpitsandflats, 235 Lathrop Wells cone introduction to, 4
fourn, 258 age of, 79 plucking, 4
hillslopediffusioninit, 244 diffusive evolution of, 51 sediment ¬‚ux driven model
hillslopediffusion, 229 study site for contaminant behavior of in block-uplift
malloc.h, 222 dispersion model, 79--83 example, 94
math.h, 222 study site for ¬‚ood hazard code for 2D implementation,
nr.h, 222 assessment, 73 248
nrutil.h, 222 study site for potential ¬‚uvial introduction to, 6, 94
realft, 256 system contamination, 75 role in model of perisistent
setupgridneighborsperiodic, 224 study site for radionuclide mountain belt topography,
setupgridneighbors, 223 dispersion in soils, 46 105--107
setupmatrices, 249 Andes, 119 sediment ¬‚ux driven model
stdio.h, 222 best-¬t ¬‚exural parameter of, 113 application to the Sierra
stdlib.h, 222 Cordillera Real Nevada, 96--101
3DEM, 223 glacial erosion and exhumation stream power model
of, 28 application to the Sierra
advection equation example of 1D ¬‚exure solution, Nevada, 96--101
analytic methods for, 88--90 113 as a type of advection equation,
contrasting behavior to diffusion example of 2D ¬‚exure solution, 87
equation, 31, 87 115 behavior of in block-uplift
introduction to, 87 example of curvature effects in 2D example, 94
method of characteristics, 88 ¬‚exure, 116 code for 2D implementation,
numerical methods for, 90--93 example of foreland basin model, 243
Lax method, 91 119--120 introduction to, 5
stability criteria, 91 landslide distributions from, 209 role in model of persistent
Two-step Lax--Wendroff method, use as a paleo analog for western mountain belt topography,
91 US, 1 105--107
upwind differencing method, Antarctic ice sheet Bessel function
92 introduction to, 20 code for numerical integration,
alluvial channels model reconstruction of, 138 232
diffusion equation model of, Appalachian Mountains, 101 code for summing series solutions
8, 102 arid cycle of erosion, 169 for volcanic cones, 233
introduction to, 4 arroyos, 169 use of for modeling diffusive
alluvial fan ASHPLUME plume deposition model, evolution of cinder cones, 49
cut and ¬ll cycles on, 38 83 boundary condition
introduction to, 9 introduction to, 34
windblown sand transport across, balance velocity method, for ice boundary conditions
17 velocities, 136, 149 periodic vs. ¬xed, 223
Alternating Direction Implicit (ADI) basal sliding Brooks Range
method introduction to, 20 glacier model reconstruction of,
code for 2D implementation of Basin and Range, 1--2 146
diffusion equation, 229 Beartooth Mountains
code for 2D implementation to cirques of, 27 channel head
solve ¬‚exure equation, 259 bedload transport, 8, 171 de¬nition of, 4
introduction to, 63 bedrock channels channel meandering, 164
use of to solve the 2D ¬‚exure abrasion, 4 introduction to, 164
equation, 122 cavitation, 4 model of, 165--166

channel oscillations introduction to, 15 application to Amargosa Valley,
code for implementation, 270 role in trapping windblown dust, 54--57
model for, 169--173 15, 194 introduction to, 14
observations of, 169, 173 role of parent-material texture in,
cinder cones 15 elasticity, see ¬‚exure
use of diffusion equation for, deterministic models entrainment
49--51 limitations of, 188 introduction to, 14
cirques, 27 diffusion equation eolian dunes
climate oscillations analytic methods for, 34--57 barchans
ice ages as one part of a more complex how they form, 19
model for, 210--220 model, 32 introduction to, 18
role in modulating sediment model for hillslope evolution longitudinal
production from drainage introduction to, 3 how they form, 19
basins, 11 requirements for applicability, relationship to ripples and
role of in controlling dust 30, 33 megadunes, 18
emission, 194 nondimensionalization of, 37 role of grain size in, 18
coherence resonance use in foreland basin modeling, role of sand supply and
introduction to, 211 116 wind-direction variability in,
model for ice ages with stochastic noise, 191--193 18
code implementing the model, Diffusion-Limited Aggregation (DLA) spacing of, 168
277 application to drainage network Werner™s model, 166--169
compaction, see deformable porous evolution, 198 code for implementation, 267
media diffusivity introduction to, 19
complementary error function typical values for western US, 33 equilibrium line altitude (ELA)
de¬nition of, 37 units of, 30 introduction to, 26
conservation of mass discretization late Miocene lowering of, 29
role in the diffusion model of introduction to, 57 localized erosion beneath, 27
landform evolution, 9, 30, 170 divides map of in the western US, 121
contaminant dispersion in ¬‚uvial no-¬‚ux boundary condition, 2, 34 role in model of cirque formation,
systems drainage density 27
introduction to, 74 de¬nition of, 4 erosion rates
previous work on, 75 role in producing sediment-¬‚ux determined by cosmogenic
role of bed scour in, 77 variations through time, 10 isotopes, see cosmogenic
cosmogenic isotopes role of in contaminant dispersion isotopes
erosion rates in Sierra Nevada, 93 model, 77 relationships with elevation and
role in dating fan terraces, 10 use of to distinguish channels relief, 101, 103
role in understanding hillslope from hillslopes in 2D error function
evolution, 3 drainage network evolution de¬nition of, 37
Crank--Nicholson method, 62 modeling, 96 use of for modeling radionuclide
droughts, 191 concentrations in soil, 48
D∞ ¬‚ow routing, 68 drumlins use of in solutions for scarp
debris ¬‚ow controls on geometry, 182 evolution, 43
rheology of, 22 introduction to, 23, 174 Eulerian vs. Lagrangian methods,
deformable porous media, 174 role of sediment deformation in 90
delivery ratio, sediment, 119 creating, 174 excitable media, 185
delta progradation stratigraphy of, 182 explicit vs. implicit numerical
diffusion equation model for, use of to reconstruct ice ¬‚ow methods, 58, 62
51--53 directions, 23 extension
dendrochronology dust cycle, 53, 194 role in Basin and Range
use of to infer channel bank importance of, 14 topography, 1
migration rates, 166 introduction to, 13
desert pavement open system nature of, 13 fault scarps
coevolution with dust deposition, dust transport and deposition application of the diffusion
15 model of equation to, 42--45

Finger Lakes, 181 failure of for the advection Glen™s Flow Law
drumlin ¬eld, 182 equation, 90 1D pro¬le modeling of ice body
introduction to, 23 introduction to, 57 subject to, 128
model for formation of, stability criterion for, 59 introduction to, 22
157--158 why use it?, 59 Great Lakes
model of ice ¬‚ow over, 144 Fourier ¬ltering method as outliers in the frequency-size
observations of, 159 code for 1D generation of distribution of
¬rn, 20 fractional noises, 274 glacially-carved lakes, 25
¬‚exure code for 1D implementation to model for, 154
coherence method solve ¬‚exure equation, 256 role of ¬‚exure in controlling, 154
use of to map elastic thickness, code for 2D generation of Greenland ice sheet
111 fractional noises, 275 model reconstruction of, 137
elastic thickness code for 2D implementation to use in code implementing the
de¬nition of, 111 solve ¬‚exure equation, 258 sandpile model, 265
equation use of for solving the ¬‚exure Greenland ice sheet (LIS)
introduction to, 111 equation, 112--116 introduction to, 19
forebulge use of to generate fractional groundwater ¬‚ow
de¬nition of, 112 noises, 190 as a type example of an
integral method for, 113 Fourier series method advection-diffusion process,
introduction to, 109 for solving the diffusion equation, 32
series solutions, 112--116 35
solution methods in 1D, 111--113 use of for linear equations only, Hanaupah Canyon
solution methods in 2D, 113--116 36 application of ADI method to
wavelength fractal scaling modeling evolution of, 63
comparison of values for rock of drainage networks, 196 example of terrace evolution in,
vs. ice loading, 112 of dust accumulation, 195 38
role in controlling glacial of landslide distributions, 205--210 type example of a ¬‚uvial system,
erosion, 152 of soil moisture ¬elds, 207 1--12
typical range, 109 of topographic transects, 207 use of for illustrating
¬‚ood-envelope curve fractional noises multiple-direction
role of in contaminant dispersion geoscience applications of, 190 ¬‚ow-routing methods, 68
model, 78 how to construct, 190 hanging valleys, 26
¬‚ow routing methods introduction to, 190 hillslope processes
ρ8, 67 bioturbation, 33
D∞, 68 Gaussian plume creep, 33
D8, 66 de¬nition of, 15 mass movements, 33
DEMON, 68 glacial buzzsaw hypothesis, 28 model for hillslope evolution
introduction to, 66 glacial erosion with, 60, 61
multiple ¬‚ow direction (MFD) Hallet™s model for, 21, 151 models for, 205--210
routing method, see Multiple introduction to, 21 model for tephra transport at
Flow Direction (MFD) routing modeling ¬‚exural response to Yucca Mountain, 75
method unloading, 120--123 rain splash, 33
range of applications for, 66 glacially-carved lakes rilling, 33
what is best?, 69 frequency-size distribution of, 24 slope wash, 33
¬‚ux-conservative equations, 88 introduction to, 24 model for hillslope evolution
foreland basins model for, 153 with, 33, 59
modeling of, 116--120 near ice margins, 157 those that are diffusive, 33
Forward Time Centered Space (FTCS) glaciers Horton™s Laws
method 1D pro¬le modeling of introduction to, 196
code for 1D implementation of uniformly-sloping bed, 127 statistical inevitability of, 197
nonlinear diffusion model, wavy bed, 127 hummocky moraine, 178
227 difference from ice sheets, 25 Hurst phenomenon, 189
code for 2D implementation of introduction to, 20 hydrologically-corrected DEM
diffusion equation, 225 role in sediment production, 12 code for implementation, 235

ice sheets lava ¬‚ow moraines
basal shear stress 2D radially-symmetric model application of the diffusion
observations of, 132 code for implementation, equation to, 40--42
cold vs. warm-based, 21 263 mortality
ice-albedo feedback, 214 modeling of role of particulate matter in
role of in controlling bistability of temperature-dependent causing, 14
the climate system, 215 viscous behavior of, 130-- Multiple Flow Direction (MFD)
implicit method 132 routing method
computational advantage of, rheology of, 22, 125 code for implementation, 236
62 LIDAR, 67, 71 code for implementation in ¬‚ood
introduction to, 62 linear stability analysis, 165 hazard analysis, 239
stability of, 62 application to channel introduction to, 67
isostasy meandering, 164 mathematical de¬nition of, 67
Airy limit, 113 introduction to, 161 use of for improving USGS DEMs,
in the evolution of the Sierra of oscillating alluvial channels, 71
Nevada, 97 171 use of in contaminant dispersal
introduction to, 109 lithology model, 78
role in ice sheets role in controlling bedrock use of in estimating ¬‚ood hazards,
introduction to, 23 channel erosion, 5, 103 72--74
model of, 127, 139 load-accumulation feedback, 216, 218
role in prolonging the mountain load-advance feedback, 217 New York State drumlin ¬eld, 181
belt denudation, 109 longitudinal pro¬le Newton iteration, 130
role in relief production, 28 analytic model for coupled non-Newtonian ¬‚uids
role in shaping Basin and Range bedrock-alluvial channels, analytic solutions for, 126--129
topography, 1 103 introduction to, 22
role in the formation of Great code implementing 1D coupled numerical solutions for, 129--132
Lakes, 25 bedrock-alluvial model, 242 rheology of, 22
role of in 2D bedrock drainage in¬‚uence on glacial erosion, 21 nonlinear advection equations, 90
network modeling, 94 model for steady-state bedrock nonlinear transport on hillslopes
time-dependent response of, channel, 5 introduction to, 3
111 Lorentzian spectrum, 212 Numerical Recipes
use with codes in this book,
Kardar--Parisi--Zhang (KPZ) equation, Mars 222
207 application of ¬‚ow routing
kei(r) methods to, 69 palimpsests, 25
series expansions for, 257 eolian dunes on, 168 perfectly plastic model
use as a basis function for series ¬‚uvial activity on, 12 application to climate modeling,
solutions to the ¬‚exure spiral troughs 216
equation, 114 code for 1D model application to ice sheets and
Kelvin functions, see kei(r) implementation, 271 glaciers, 22
knickpoints model of, 183--187 application to thrust sheet
introduction to, 6 method of characteristics, see mechanics, 147
modeling of using advection advection equations generalization of to
equation, 90 Microsoft threshold-sliding behavior,
Excel, 223 127
Langevin equations, 191 Visual C++, 222 introduction to, 22
Lathrop Wells cone, see Amargosa Mohr--Coulomb criterion, 207 limitations of, 22
Valley moisture use of in modeling 1D ice sheet
Laurentide ice sheet (LIS) role in controlling particle pro¬les, 125
impact on global climate, 215 entrainment persistent mountain belt paradox
introduction to, 23 introduction to, 14 introduction to, 101
model of, 138 role of in triggering mass role of piedmont deposition in,
lava channels movements, 205 101--103
meandering of, 164 Monte Carlo methods, 188 piedmont, see alluvial fan

playa Rogen moraine, 179 Taylor expansion
introduction to, 12 root-¬nding techniques, 50 use of in discretization, 57
role of hydrology in dust terraces
production salt domes dust deposition on
introduction to, 13 as an example of the introduction to, 13
sand-dominated, 17 Raleigh--Taylor instability, implementation of 2D diffusion
Pleistocene--Holocene transition 162 model, 226
role in triggering terrace saltation introduction to, 9
formation, 10 characteristic distance of, 167 mechanisms for formation, 10
pluvial shorelines introduction to, 13 morphological age dating of, 39
application of the diffusion sandpile method on Mars, 12
equation to, 42--45 code for implementation, 264 relative age indicators of, 10
introduction to, 12 use of in 2D perfectly plastic role in controlling ¬‚ood risk,
polar coordinates model, 135 10
use of for solving diffusive secondary ¬‚ow, 164 transient response to base level fall
evolution of cinder cones, Shreve magnitude, 197 use of diffusion equation to
49 Sierra Nevada model, 38--40, 61--62
positive feedback knickpoint propagation thermal erosion, 166
between channel width and slope introduction to, 6 thermochronology, 29
in alluvial channels, 161 modeling of, 93--101 threshold of critical power, 171
between glaciers, climate, and non-equilibrium landscape of, Tibet
topography, 28 93 archeological ruins of, 45
elevation-accumulation feedback similarity method till fabric, 182
in ice sheets, 20 application to solving diffusion time series analysis
ice-albedo feedback, 213 equations autocorrelation function, 189
in hillslope evolution, 4 introduction to, 36 introduction to, 188
role of in geomorphic instabilities, use in modeling deltaic power spectrum, 189
161 sedimentation, 53 of Pleistocene climate, 212
precision, single vs. double, 222 use of in modeling terrace pro¬le Tokunaga side branching, 197
evolution, 40 topographic inversion, 131
radiation balance sine function topographic steady state
role of in climate models, 213 as functional form of late-stage convergence of stream-power and
radionuclides terraces following base level sediment ¬‚ux driven models
atmospheric nuclear testing as a fall, 40 in, 8
man-made source, 46 sine-generated curve, 166 in hillslopes, 3
dispersion in soils soil moisture in ice sheet modeling, 129
diffusion equation model for, variations in space and time model for hillslope pro¬le during
45--49 modeling of, 206 transient approach to, 37
processes of, 48 observations of, 206 model for hillslope pro¬le in,
random walk solitary waves, 173 34
application to dust accumulation spring sapping slope vs. area curve, 5
on alluvial fan terraces, role of in drainage network use of in foreland basin modeling,
194--196 evolution, 198 117
bounded, 195 Stefan--Boltzmann Law, 212 topologically distinct channel
introduction to, 193 Stokes™ Law, 175 networks (TDCNs), 197
Rayleigh--Taylor instability, 162--164 Strahler ordering, 196 transport capacity
analogy to drumlin formation, stratigraphic completeness, 196 role in controlling bedrock vs.
178 stream function alluvial channel types, 4, 8
recirculation zones use of in ¬‚uid mechanics tridiagonal matrix
role in controlling eolian problems, 162, 175 use of in implicit numerical
deposition, 18, 167 sublimation methods, 62
relief production in glaciated terrain role of in forming spiral troughs, Tule Valley
introduction to, 28 184 shoreline scarps in, 42
RiverTools, 223 sur¬cial geologic mapping, 10 two-step Lax--Wendroff, 173

U-shaped valleys, 26 fossil record of in packrat power spectrum of, 220
US Geological Survey middens, 11
DEMs in¬‚uence on weathering, 3, weathering front
problems with, 70 10 role in hillslope evolution, 3
Uinta Mountains Vialov solution, 128 Wind River Range
glacial erosion in, 27 visualization, 223 glacial erosion in, 26
UNIX cc compiler, 222 Vostok ice core, 214, 219 Wisconsin drumlin ¬eld, 180
Ural Mountains, 101 comparison with coherence
resonance model of ice ages, Yucca Mountain
V-shaped valley, 26 220 volcanic hazard associated with,
vegetation histogram of, 220 75


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