Geostatistical interpolation of runoffJon Olav Skøien and Günter Blöschl

Institute of Hydraulic and Water Resources EngineeringVienna University of Technology

Conclusion

Spatio-temporal geostatistical interpolation offers an alternative to regionalisation of rainfall-runoff model parameters for ungauged catchments (PUB)! Method uses real catchment boundaries! No information in addition to runoff data and catchment boundaries needed! To do:

- Consider other aggregation schemes

- Preserve variance in a better way

Motivation

& Common practice to use deterministic models for prediction of runoff in hydrology& Stochastic models have been used for regionalisation of model parameters

C We suggest a geostatistical interpolation method for runoff taking into account:- Runoff from neighbouring catchments

- True catchment boundaries - Network structure

Estimating variance between catchments

Time

T1

A1

T1

A1 T2

A2

T2

A2

hths

hths

hthshs

1500

350

100

25

6

1.5

0.5

0

0 5 12 20 35 55 80 120 200 300

0

0.0003

0.0006

0.0009

0.0012

0.0015

0.0018

0.0021

0.0024

0.0027

0.003

Temporal lag(hours)

Spatial lag (km)

Spatio-temporal sample variogram of runoff

- Variogram cloud of the catchment pairs

- Binned temporal distances

=>Point variogram found by joint fitting of the spatio-temporal variances between catchments

Type of variability Our approach

Soil and vegetation

Channel network

Euclidian

Euclidian

} Variogram

Tree structure Boundaries

mm/day

ò ò ò òò ò ò ò ----+-+=A T A TA T A T

tstsst dsdrdtdtrsTA

dsdrdtdhtrhsTA

Thah ttgttgg ),(1

),(1

),(2222

Catchment: can be seen as a set of points in space and time

! Variance between catchments - The average of point variances

! Variances dependent on spatial and temporal distancesC Regularisation

100 km2

49 km2

25 km2

9 km2

25 km2

20 km

2100 km

249 km

225 km

29 km

225 km

20 km

0.40

0.28

0.19

0.13

Kriging weights

Block to beinterpolated

EGU General AssemblyVienna, April 25-29, 2005Contact: Jon Olav Skøienskoien@hydro.tuwien.ac.at

÷÷

ø

ö

çç

è

æ--+--

---=

ò ò ò òò ò ò ò

ò ò ò ò

ttgttg

ttgg

dsdrdtdtrsTA

dsdrdtdtrsTA

dsdrdtdtrsTTAA

j j j ji i i i

i i j j

A T A TjjA T A Tii

A T A Tjiji

ij

),(1

),(1

5.0

),(1

2222 ÷÷

ø

ö

çç

è

æ--+--

---=

ò ò ò òò ò ò ò

ò ò ò ò

ttgttg

ttgg

dsdrdtdtrsTA

dsdrdtdtrsTA

dsdrdtdtrsTTAA

j j j ji i i i

i i j j

A T A TjjA T A Tii

A T A Tjiji

ij

),(1

),(1

5.0

),(1

2222

Results

Two interpolated runoff hydrographs as examples! Hourly runoff 1998 interpolated! Only showing 35 day period! Nash-Sutcliffe for all catchments.0.75-0.97

& The catchment filters the signal from the atmospheric forcing - The smoothing is stronger from a large catchment than from a small one

& Also temporal filtering takes place, due to the residence time in the catchmentb

& The residence time simplified as T=aA , A is area and a and b parameters& We assume a simple aggregation within the catchment

- Runoff at outlet is average of runoff generated within the catchment during time T

Regularisation of a variogram in space and time:

Back-calculation of a point variogram:

The influence of support on interpolation

A spatial example - Four neighbour blocks with equal centre-to-centre distance

! Largest catchment will get largest kriging weight! Smallest catchment get smallest weight! Size of interpolated block does not come into account in symmetric case

Block Kriging

! Gamma value in kriging matrix calculated separately for each pair of catchments! Kriging matrix solved as normally! The kriging matrix is only solved once for each ungauged catchment.

Data

! Interpolation of 17 stations from Innviertel, Austria- marked with black circles, numbered C1-C17

! Yearly runoff relatively homogenous (small green numbers)! Also other neighbouring catchments used for interpolation

Litz

Alfenz

Alfenz

Alvier

Lutz

Frutz

Ehbach

Dornbirnerach

Rheintalbinnenkanal

Bregenzerach

Bregenzerach

Subersach

Weißach

RotachRuggbachLeiblach

ZürsbachLech

Schwarzach

Gillbach

Mühlbach

Bizauerbach

Ehbach

Lech

Hornbach

Lech

Isar

Vils

Loisach

Schalklbach

Inn

Inn

Trisanna

RosannaRosanna

Inn

Salvesenbach

Inn

RofenacheVenterGurgler

Ötztaler

Ötztaler

Ötztaler

InnInn

Obernberger

Schmirnbach

Gschnitzbach

SillNavisbach

Sill

WattenbachWeerbach

Inn

Tuxbach

Ziller

ZillerGerlosbach

Ziller

Inn

Brandenberger

Walchentaler

Kelchsauer

Brixentaler

Inn

Weißache

Kitzbüheler

AschauerKitzbühelerFieberbrunner

Gerlosbach

Zamserbach

Sanna

Pitze

ZirmbachLängentalbach

Gurglbach

Leutascher

Stampfangerbach

Gießenbach

Windauer

Sill

Ruetz

Vils

Loisach

Leutasch

Tiroler

Großache

SalzachObersulzbach

Untersulzbach

Mühlbach Salzach

Felber

Salzach

Fuscher

Fuscher

Hüttwinkelache

Rauriser

Dientenbach

Naßfelder

Gasteiner

Großarler

Kleinarler

Fritzbach

LammerLammer

TorrenerSalzach

Tauglbach

Almbach

Berchtesgad.Ache

Salzach

Glanbach

Saalach

Urschlaubach

Leogangbach

Saalach

Loferbach

Saalach

Salzach

Moosache

Kraftwerk

Saalach

Habach

Hainbach

Enns

Taurachbach

MurMur

Taurach

Weißpriach

TaurachMur

Thomatalbach

Zederhausbach

Pladenbach Oichtenbach

Obersulzbach

Salzach

Forstaubach

Schwarzbach

WeißenbachSchwarzbach

Mur

Stuhlfeldnerbach

Göriachbach

Lessachbach

Löhnersbach

Eugenbach

Moosache

Moosache

MühlbergerSchwemmbach

Mattig

Waldzeller

Mettmach

Lochbach

Ach

Oberach

Rieder

Osternach

Inn

Pram

PfudabachPfudabachDiersbachPram

Ranna

Große

Große

Steinerne

Innbach

InnbachTrattnach

StillbachTrattnach

Innbach

Dürre

Aschach Großer

Waldbach

Traun

Langbathbach

Griesler

Äusserer

Vöckla

Frankenburger

Vöckla

Dürre

Vöckla

Aurach

Staiger

TraunSchwaiger

Alm

Aiterbach

Grünbach

Schmiedleitnerbach

Sipbach

Krems

Krems

Ipfbach

Gaflenzbach

Reichramingbach

Enns

Steyr

KrummeSteyr

Steyr

Teichl

DambachTeichl

Teichl

Steyrling

Steyr

Paltenbach Krumme

Krumme

Enns

Feldaist

Rettenbach

Waldaist

Aist

Große

Kleine

Naarn

Schwemmbach

SchwemmbachSchalchener

Pram

Aschach

Sarmingbach

Messenbach

Ottnanger

LaudachSchwemmbach

Pram

Enns

Reichramingbach

Gusen

Dürre

Weißenbach

Pyhrnbach

Alm

Vorderer

Hintersteinerbach

Maltsch

Große

Feldaist

Ramingbach

Hörschinger

Mitterweißenbach

Enknach

Stambach

Pettenbach

GroßeDiestlbachWald-(Schwarze)

Berndorfer

LochbachMoosbach

Hainbach

Mühlheimer

Laussabach

Kleine

Trattenbach

Grünbach

Antiesen

Sulzbach

Hinterer

Schwemmbach

Kleine

Donau

Donau

Donau

Donau

DonauDonau

March

March

Donau

DonauIsper

Ybbs

Kleine

Ybbs

Urlbach

Große

Ötscherbach

Lassingbach

GroßeGamingbach

Jeßnitz

Kleine

Erlauf

Weitenbach

Mank

Melk

Pielach

Pielach

Sierningbach

Krems

Fladnitz

Traisen

Gölsen

Traisen

ZwettlKamp

KampPurzelkamp

Purzelkamp

Kamp

Perschling

Große

Schmida

Göllersbach

Wien

Schwechat

Mödling

Triesting

Triesting

LiesingSchwechat

Kalter

Fischa

Fischa

Fischa

Piesting Piesting

Piesting

Piesting

Fischa

Reisenbach

Fischa

Rußbach

Rußbach

Lainsitz

Braunaubach

Reißbach

Deutsche

Thaya

Pulkau

Thaya

Weidenbach

Schwarza

NaßbachSchwarza

Sierning

SierningSchwarza

GroßerFeistritz

Pitten

Pitten

Große

Leitha

WarmeWarme

Leitha

Leitha

RomaubachThaya

Zaya

Melk

Große

Erlabach

Rußbach

Leitha

Warme

Taffa

Petersbach

Liesing

Türnitztraisen

Hochberggraben

Laimbach

Schopfgraben

Schliefaubach

Hagenbach

Maisbirbaumer

Staffbach

Halbach

Zaya

Weidlingbach

Schwechat

Perschling

Lainsitz

Spitzer

Pulkau

Unrechttraisen

Schlattenbach

Ois

Kleine

FrauenbachProssetbach

Mährische

Gablitzbach

TaxenbachDeutsche

Mauerbach

Liesing

WienHalterbach

Leitha

Rabnitz

Rabnitz

Stoob

Wulka

Wulka

Wulka

Stögersbach

Lafnitz

Pinka

Pinka

Tauchenbach

Tauchenbach

Pinka

Strem

Strem

Zöbernbach

Güns

Wulka

Eisbach

Lafnitz

Pinka

Raab

Strem

Nodbach

Rabnitz

Augstbach

Ödenseetraun

Unterthaler

Enns

Gulling

EnnsPaltenbach

Enns

Enns

Erzbach

Salza

Radmerbach

Salza

Moderbach

Weizbach

Raab

Raab

Lafnitz

Lafnitz

Safenbach

Feistritz

Ilzbach

Feistritz

Mur

Mur

Pusterwaldbach

Pöls

Mur

LiesingbachMur

Vordernberger

Mürz

Mürz

Thörlbach

MürzLamingbach

Mur

Kainach

Kainach

Schwarze

Weiße

Laßnitz

Sulm

Schwarzaubach

Mur

Gnasbach

Krumbach

Mur

Raab

Prätisbach

Saifenbach

Übelbach

Saggaubach

Laßnitz

Lafnitz

Raab

Raab

Mur

Feistritz

Granitzenbach

Mur

Pinka

LaßnitzGleinzbachWildbach

Liesingbach

Stainzbach

Stanzbach

Mur

Rabnitzbach

Stiefing

Sulm

Rittscheinbach

Ingeringbach

Feistritzbach

Schwarzenbach

Drau

IselIslitz

GschlößbachTauernbach

Isel

Schwarzach

DorferbachTeischnitzbach

Isel

Isel

Winkeltalbach

Drau

DrauMödritschbach

Drau

Möll

Mallnitzbach

Möll

Möll

Lieser

Hochalmbach

Malta

Lieser

Gail

Gail

Gail Gösseringbach

GailGail

BodenbachLoiblbach

Vellach

GurkGurk

Gurk

Metnitz

Görtschitz

Görtschitz

Glan

Steiner

Glan Gurk

Lavant

Lavant

Treffner

Möll

GurkRieger

Drau

Lieser

Drau

Tiebelbach

Wimitz

Wimitz

Metnitz

Weißenbach

Glan

Moosburger

Silberbach

Lavant

Lavant

Wölfnitzbach

Gurk

Olsabach

Zickenbach

Goldbach

Inn

Inn

Inn

Radurschlbach Pitze

Fagge

Ötztaler

Fundusbach

Alpeinerbach

Längentaler

Kraspesbach

Melach

Ruetz

Dürrach Grundache

Mühlbach

DrauVillgratenbach

0

0

1

1

0

0

0

0

0

0

0

0

1

000

00

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

11

1

1

1

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0

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0

1

1

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0

0

0

0

0

0

1

0

00

0

10

0.0464

0.053

0.0415

0.0377

0.0631

0.0658

0.048

0.0562

0.0268

0.0696

0.0605

0.0757

0.0354

0.04380.03660.0307

0.06510.0609

0.0438

0.0269

0.0189

0.0482

0.0397

0.0561

0.063

0.0453

0.0373

0.0414

0.0326

0.0274

0.0283

0.0324

0.0239

0.03870.036

0.0125

0.0414

0.0295

0.04640.04470.0519

0.039

0.0378

0.035

0.0290.0292

0.0335

0.0311

0.0385

0.03250.0268

0.0297

0.02980.0316

0.0304

0.0293

0.0446

0.04350.0427

0.0413

0.0316

0.0386

0.0454

0.0394

0.0347

0.0324

0.0269

0.0364

0.03440.03540.0433

0.045

0.0463

0.0284

0.0193

0.01450.0394

0.0258

0.0313

0.0272

0.0113

0.0419

0.0217

0.042

0.0458

0.0269

0.0335

0.0365

0.0423

0.03350.0560.0518

0.0330.0398

0.0423

0.0433

0.0473

0.037

0.037

0.0404

0.0368

0.0217

0.0475

0.0349

0.0361

0.025

0.03230.0453

0.05050.039

0.0323

0.0454

0.0425

0.0394

0.0415

0.0341

0.0362

0.0251

0.0355

0.0477

0.0386

0.0392

0.0125

0.0546

0.0342

0.0476

0.0109

0.0335

0.0515

0.03420.0261

0.0343

0.0327

0.02880.0251

0.019

0.0239

0.0105 0.0185

0.0805

0.0175

0.0268

0.1112

0.02270.3855

0.0247

0.0358

0.0259

0.0307

0.038

0.0287

0.0188

0.0122

0.01230.0126

0.0107

0.0224

0.017

0.0087

0.017

0.0148

0.0156

0.0142

0.028

0.0143

0.01690.01550.0134

0.0145

0.0227

0.0242

0.0216

0.0146

0.0126

0.01110.0148

0.01390.0123

0.0119

0.0115

0.0141 0.0157

0.0816

0.0549

0.0313

0.0352

0.0714

0.0266

0.0204

0.0215

0.0196

0.0198

0.0272

0.0159

0.03910.01

0.0406

0.01050.0084

0.0202

0.0146

0.0338

0.0209

0.0103

0.0211

0.0367

0.0328

0.0608

0.14730.0828

0.0531

0.0612

0.02760.0368

0.0446

0.0259

0.0486

0.0278 0.0305

0.0308

0.0344

0.0086

0.0494

0.0117

0.0104

0.0127

0.0122

0.0119

0.0142

0.00540.0284

0.0166

0.0135

0.0131

0.0144

0.0148

0.01710.0085

0.0144

0.0317

0.04

0.0087

0.0144

0.0329

0.0336

0.0544

0.0598

0.0166

0.0105

0.0097

0.0082

0.0208

0.0039

0.0576

0.004

0.0303

0.182

0.01280.0190.0137

0.0216

0.01310.012

0.0095

0.017

0.04

0.0158

0.0313

0.0089

0.0142

0.0147

0.0745

0.0094

0.0086

0.0181

0.0113

0.0226

0.0118

0.01130.0134

0.0041

0.0042

0.0221

0.01550.0129

0.0378

0.0272

0.0272

0.0133

0.0266

0.0402

0.0323

0.03470.0258

0.022

0.0164

0.0245

0.007

0.0126

0.0094

0.0371

0.0211

0.0068

0.0055

0.0023

0.0243

0.0158

0.0183

0.00730.0083

0.01190.0089

0.0069

0.0056

0.0052

0.0057

0.0012

0.001

0.0092

0.0086

0.0041

0.0103

0.0096

0.00530.0078

0.0231

0.0836

0.0219

0.034

0.019 0.0107

0.0083

0.0077

0.016

0.0067

0.0142

0.0015

0.0035

0.008

0.0069

0.0081

0.0062

0.0051

0.0009

0.003

0.0016

0.0183

0.02170.0203

0.0109

0.00830.0038

0.01210.0108

0.0101

0.0085

0.0342

0.0024

0.00190.0015

0.0057

0.0053

0.00830.0039

0.0013

0.013

0.0327

0.0093

0.0032

0.002

0.0235

0.0028

0.0031

0.0104

0.0293

0.1404

0.0123

0.0121

0.0209

0.0505

0.0242

0.011

0.0197

0.0015

0.0046

0.0121

0.0074

0.0094

0.003

0.0021

0.0248

0.0046

0.0386

0.004

0.01070.0032

0.0041

0.0066

0.00520.0049

0.006

0.0051

0.00720.009

0.004

0.0052

0.004

0.0039

0.0032

0.0022

0.0028

0.0051

0.0068

0.0076

0.0053

0.0048

0.004

0.0051

0.0036

0.0036

0.0064

0.0057

0.0029

0.0034

0.0079

0.0085

0.0078

0.0037

0.0023

0.0041

0.0515

0.0617

0.0404

0.0335

0.0294

0.03010.0244

0.0306

0.0304

0.0381

0.0261

0.036

0.0344

0.0101

0.0113

0.0082

0.0079

0.0098

0.0084

0.0057

0.0125

0.0058

0.0096

0.0205

0.0192

0.0258

0.0205

0.0193

0.0220.0175

0.0183

0.0307

0.0189

0.015

0.01620.0168

0.0173

0.0126

0.0124

0.0231

0.0176

0.0129

0.0141

0.0079

0.0153

0.0074

0.0302

0.0166

0.0091

0.0076

0.0086

0.0143

0.0138

0.0177

0.0101

0.0098

0.0094

0.0208

0.0216

0.0161

0.0175

0.0122

0.01540.01370.011

0.0297

0.0121

0.0115

0.0171

0.0077

0.0072

0.0172

0.007

0.021

0.0227

0.0227

0.0234

0.04990.0561

0.07670.0685

0.0395

0.0325

0.05010.036

0.0325

0.0374

0.0261

0.0287

0.0280.023

0.0266

0.0195

0.0418

0.0239

0.0081

0.0182

0.0593

0.0135

0.017

0.0287

0.0298

0.0306 0.0245

0.03190.0327

0.02350.0402

0.0304

0.01810.0163

0.0134

0.011

0.0133

0.0118

0.0092

0.0075

0.0116 0.0114

0.014

0.0124

0.0128

0.0259

0.01110.0172

0.0293

0.0195

0.027

0.0196

0.0099

0.0093

0.0133

0.0124

0.0102

0.0104

0.0064

0.0137

0.0122

0.0122

0.0117

0.0103

0.0052

0.0053

0.0279

0.03

0.0311

0.0332 0.0514

0.05

0.0356

0.034

0.0493

0.0421

0.0397

0.0199

0.0338

0.0445 0.0312

0.0336

0.02040.0289

200048

200071

200089

200097

200105

200154

200162

200204

200220

200246

200253

200287

200303

200311200345200352

200360200378

200428

200469

200485

200493

200527

201012

201053

201087

201095

201111

201152

201160

201178

201194

201236

201251201277

201293

201301

201319

201350201368201376

201392

201418

201434

201459201525

201533

201558

201566

201574201582

201624

201657201665

201681

201723

201749

201756201772

201780

201806

201822

201830

201848

201863

201889

201897

201913

201921201939201947

201970

201996

202036

202044

202077202093

202101

202127

202135

202143

202218

202275

202283

202291

202309

202317

202341

202382

203026203034203042

203067203075

203083

203125

203133

203141

203158

203166

203174

203190

203208

203224

203232

203257

203265203307

203315203323

203331

203349

203364

203398

203422

203463

203471

203489

203497

203505

203521

203539

203547

203554

203570

203596

203638

203703

203737

203745203752

203760

203778

203786203794

203810

203828

203836 203844

203893

203901

203919

203927

203935203943

203976

203984

203992

204008

204016

204024

204529

204545

204602204628

204677

204685

204693

204701

204719

204727

204735

204768

204776

204784

204826204834204842

204859

204875

204891

204917

204925

204958

204966204974

204990205021

205039

205047

205054 205096

205120

205153

205229

205294

205336

205377

205385

205393

205401

205419

205435

205443

205468205476

205500

205534205559

205583

205609

205625

205641

205690

205732

205740

205757

205765

205773205781

205799

205807

205815205823

205831

205856

205864

205872 205880

205898

205922

205948

205955

205971

205989

205997

206003

206029

206052

206078206086

206102

206169

206227

206268

206284

206292206318

206326

206342

206367

206375

206383

206425

206466

206508

206524

206532

206557

206573

206581

206599

206607

206615

206623

206631

206649

206656206664206680

206698

206706206722

206730

206748

206763

206771

206789

206797

206805

206813

206821

206839

206854

207035

207126

207134

207175

207274207282

207308

207324

207357

207399207613

207654

207662

207688

207696

207704

207712

207720

207738207746

207787

207795

207803

207811

207829

207837

207845

207852

207860

207878

207886

207894

207902

207910

207936207944

207951207969

207977

207993

208009

208017

208041

208058

208066

208090

208108

208116

208124

208132208157

208181

208207

208231

208249

208272 208280

208306

208330

208413

208421

208439

208447

208454

208462

208579

208603

208611

208629

208637

208645

208678

208686

208694208702

208736

208744208785

208819208827

208835

208843

208850

208884

208918208926

208991

209007

209064209171

209189

209197

209213

209247

209262

209288

209296

209338

209346

209353

209361

209379

209387

209395

209403

209411

209429

209437

209445

209452

209460

209478

209486

209494

209502

209510

209536

209544

209551

209569

209585209593

209601

209627

209734209742

209890

209908

209916209932

210013

210039

210054

210062

210070

210088

210096

210203

210211

210229

210237

210245

210252

210260

210286

210294

210302

210310

210328

210393

210401

210419

210468

210476

210492

210500

210559

210583

210625

210641

210773

210799210815

210823

210849

210856

210864

210880

210898

210955

210963

210971

210989

210997

211003

211011

211029

211037

211045

211086

211102

211110

211128

211136

211169211185

211193

211227

211243

211250

211268211276

211292

211334

211342

211383

211391

211441

211458

211474

211490

211508

211540

211573

211599

211623

211631

211649

211656

211664

211680

211698

211706

211714

211722

211730

211763

211771

211797211805211813

211821

211854

211862

211870

211888

211896

211904

211912

211920

211938

211946

212027

212043212050

212068212076

212092

212100

212118212126

212167

212183

212225

212316

212324212340

212357

212373

212381

212399

212407

212431

212456

212498

212530

212613

212647

212670 212704

212753212787

212829212837

212852

212860212878

212886

212894

212928

212936

212951

213017

213033 213041

213082

213090

213116

213124

213157213181

213199

213207

213215

213231

213249

213256

213306

213314

213322

213330

213355

213363

213371

213389

213397

213405

215012

215020

230011

230078

230102

230110 230177

230300

230342

230383

230409

230466

230581

230706

230714

230722 230896

230904

231092231100

km

Rieder

Osternach

Inn

Pram

PfudabachPfudabach

Ranna

Große

Steinerne

Innbach

InnbachTrattnach

StillbachTrattnach

Innbach

Dürre

Aschach

Pram

Aschach

MessenbachPram

Kleine

Antiesen

Donau

0

0.0156

0.0142

0.028

0.0143

0.0169

0.0155

0.01340.0145

0.0227

0.0216

0.0146

0.0126

0.01110.0148

0.0139 0.0123

0.0119

0.0115

0.0141

0.0166

0.0135

0.01440.0144

0.0158

0.0142

0.0181

C1

C2

C3C4

C5

C6

C7

C8C9

C10C11

C12

C13

C14

C15C16

C17

Pram

Atmospheric forcing

Hydrographs from some of the neighbours used for interpolation

10

20

30

40

31/10 07/11 14/11 21/11 28/11

Interpolated runoffObserved runoff

10

20

30

40

C4 - Taufkirchen

Interpolated runoffObserved runoff

mm/day

Peak satisfying interpolated despite large variance in peaks of neighbouring catchments

10

20

30

40

31/10 07/11 14/11 21/11 28/11

Interpolated runoffObserved runoff

10

20

30

40

C14 - Neumarkt im Hausruckkreis

Interpolated runoffObserved runoff

Second peak largest in interpolated catchment => underestimation! Method is not able to preserve variance of a time series! Kriging gives the best linear unbiased estimator for each time step

Neighbours

Neighbours

Spatio-temporal point variogram of runoff