Comparing  Probabilis/c  Inversion  Techniques  on  a  Highly  Parameterized  Ground  Water  Model  

Sean  A.  McKenna,  Hongkyu  Yoon,  David  B.  Hart    

Geoscience  Research  and  Applica/ons  

Sandia  Na/onal  Laboratories  

SIAM  UQ  2012  Mee/ng  April,  2012  

This  material  is  based  upon  work  supported  as  part  of  the  Center  for  Fron/ers  of  Subsurface  Energy  Security,  an  Energy  Fron/er  Research  Center  funded  by  the  U.S.  Department  of  Energy,  Office  of  Science,  Office  of  Basic  Energy  

Sciences  under  Award  Number  DE-­‐SC0001114.      Sandia  Na/onal  Laboratories  is  a  mul/-­‐program  laboratory  managed  and  operated  by  Sandia  Corpora/on,  a  wholly  owned  subsidiary  of  Lockheed  Mar/n  

Corpora/on,  for  the  U.S.  Department  of  Energy's  Na/onal  Nuclear  Security  Administra/on  under  contract  DE-­‐AC04-­‐94AL85000.

Mo/va/on  

•  Quan/fy  uncertainty  in  inverse  parameter  es/mates  – Basis  for  probabilis/c  predic/ve  modeling  – Computa/onally  efficient  and  accurate    – Robust  for  strongly  heterogeneous  media  

•  Culebra  dolomite  as  a  mo/va/ng  example  

Faster,  Cheaper  and  At  Least  As  Good!  

Culebra  Dolomite  •  The  Culebra  Dolomite  near  the  WIPP  site  

(NM,USA)        •  Predic/ve  performance  measure  is  advec/ve  

transport  /me  to  a  prescribed  boundary  •  Observa/on  data  include  two  decades  of  

steady-­‐state  head  measurements  and  transient  pumping  test  results.  

n  Exis/ng  Mul/ple  Star/ng  Point  (MSP)  results  (Hart  et  al.,  2009)    n  Start  with  a  large  number  of  equally  likely  

indicator  simula/on  “seed”  fields  n  200  seed  fields  serve  as  the  start  of  200  

independent  calibra/ons  

~ 30 km  

4  

Heterogeneous  Field  Parameteriza/on  

•  “Pilot  Points”  (similar  to  a  kernel-­‐based  approach)  •  Choose  loca/ons  in  the  model  domain  and  update  their  proper/es  to  produce  beder  fit  to  measured  heads  (“calibra/on  points”)  

•  Spread  influence  of  each  point  to  neighboring  model  cells  by  using  the  spa/al  covariance  func/on  as  a  weigh/ng  scheme  

5  

Pilot  Point  Example  

Simple  example  of  five  pilot  points  used  to  update  a  log10  transmissivity  field  

6  

Pilot  Point  Example  (Cont.)  

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No  Pilot  Points   5  Pilot  Points  

Pilot  points  and  variogram  provide  a  means  of  calibra;ng  the  heterogeneous  field  to  match  pressure  data  

Parameter  Es/ma/on  

y = Xp+!

! = (y" Xp)TQ(y" Xp)

p = (XTQX)!1XTQy

!p = (XTQX)"1XTQr

Xij =!yi!pj

Linear  Model  

Non-­‐Linear  Model  

Assumes  that  the  inverse  of:  XTQX  exists  and  can  be  uniquely  es/mated      If  #  parameters  (n)  >  #  observa/ons  (m),  this  is  not  the  case  

Regulariza/on  

XTQX ! (XTQX +! 2TTST ) Tikhonov  

XTQX ! (V1E1V1T ) Truncated  SVD  

XTQX ! V1V2[ ]E1 00 E2

"

#$$

%

&''V1V2[ ]T =V1E1V1T +V2E2V2T

Hybrid  Tikhonov-­‐TSVD  Regulariza/on:  

XTQX !V1(ZTQZ +! 2DTSD) Where:   D = TV1

Tonkin  and  Doherty,  2009,  Water  Resources  Research    

y =Gp Goal:  G    

Super  Parameters  

XTQX ! V1V2[ ]E1 00 E2

"

#$$

%

&''V1V2[ ]T =V1E1V1T +V2E2V2T

Resolu;on  matrix:  R  

y =Gp

G = V1E1V1T( )XTQ

p =GXp+G!Super  Parameters  

Goal  

Goal  achieved  

Mapping  from  base  parameters  to  super  parameters  

Sampling  Null  Space  p! p = !(I ! R)p+G! Unknown  error  between  true  and  

es/mated  parameters  

C(p! p) = !(I ! R)C(p)(I ! R)T +GC(!)GT

Use  informa/on  in  C(p)  and  C(ε)  to  define  C(p-­‐    )  

Parameter  error  from  calibra/on  null  space  (V2)  

Errors  of  es/mates  in  solu/on  space  (V1)  driven  by  measurement  noise  

Moore,  C.  and  Doherty,  J.,  2005.    The  role  of  the  calibra;on  process  in  reducing  model  predic;ve  error,  Water  Resources  Research,  Volume  41,  No  5.  Tonkin  M.,  and  Doherty,  J.,  2009.    Calibra;on-­‐constrained  Monte  Carlo  analysis  of  highly  parameterized  models  using  subspace  techniques,  Water  Resources  Research,  45,    

(p ! pST ") =V2V2T (p ! pST ) Project  stochas/c  parameter  differences  that  

have  zero  impact  on  calibra/on  into  null  space    

10  

p

Ques/ons  •  1)  Can  NSMC  approach  approximate  ensemble  predic/ons  obtained  with  MSP  runs?    

•  2)  Given  a  set  of  previously  run  models,  what  is  an  effec/ve  means  of  expanding  the  predic/ve  ensemble?  

•  3)  In  the  absence  of  exis/ng  calibra/ons,  can  a  mean-­‐field  representa/on  serve  as  an  ini/al  star/ng  point?  

NSMC  =  Null  Space  Monte  Carlo  

Three  Property  Fields  •  Simultaneous  es/ma/on  of  three  spa/ally  correlated  property  fields  Transmissivity   Anisotropy   Stora/vity  

Loca;ons  in  blue  are  fixed  values  and  red  are  loca;ons  where  parameters  are  adjusted.    For  transmissivity,  2  fields  (Zones  0  and  1)  are  es;mated,  then  combined    1380  observa;ons,  ≈  1200  parameters   12  

MSP  Calibra/on  Process  

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PEST Iteration Number

Phi (

sum

of s

quar

ed

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dual

s)

200  star/ng  fields  are  calibrated  to  1380  steady-­‐state  head  and  transient  drawdown  observa/ons      Computa/on  /me  is  several  months  

Approach  1:  MSP  

n  Stochastic Inverse Modeling (~ Ramarao et al (1995, WRR))

• • •

Random seed fields  

Calibrate model(s) to observed data  

• • •

Prediction with calibrated fields  

Pro

babi

lity

Prediction  

Observed  

Predicted  

EXPENSIVE!!!!  

n  Conceptual model is stochastic – poorly known location and extent of highly fractured zones. Leads to multiple, equally probable starting points (seed fields for inversion). Calibrate each seed field.

Up  to  50  itera/ons  

Advec0ve  Travel  Times  

Approach  2:  NSMC  

n  Null-Space Monte Carlo Method (~ Tonkin & Doherty (2009,WRR))

Single field  

Calibrate  

• • •

Prediction (may involve 1-2 iterations for inverse modeling)  • • •

Calibration-constrained random fields  

Pro

babi

lity

Prediction  

Observed  

Predicted  

Not  so  expensive:  1-­‐2  itera/ons  

n  Shift the approach: Calibrate one seed field and then modify that field to create probabilistic distribution of results

Advec0ve  Travel  Times  

Ground  Water  Parameter  Es/ma/on  n  Calibrated parameters (>1200 parameters in total) transmissivity (T), horizontal hydraulic anisotropy,

storativity (S), and a section of recharge n  200 multiple random seed fields are calibrated to

observed data, and then 100 best fields are selected for travel-time analysis

• • Pilot points

Calibration-based five fields  

Travel time-based five fields  

Mean field of 200 random seed fields  

Best/fastest   Worst/slowest  25th  Percentile in 200 calibrated multiple fields  

50th   75th  

NSMC Analysis (3 groups -11 fields)  

NSMC generation of 200 random fields. From each calibrated field  

Travel  Time  and  Obj.  Func/on  Objec/ve  Func/on  Selec/ons   Travel  Time  Selec/ons  

5  calibrated  fields  selected  with  each  criterion  100  NSMC  realiza/ons  shown  for  each  calibrated  field  

Travel  Time  Distribu/ons  

00.10.20.30.40.50.60.70.80.91

1000 10000 100000

Prob

ability

Advective-­‐only  Travel  Times  (yr)

100 Selected Fields from multiple seed fields100 Selected Fields from NSMC fields with r102100 Selected Fields from NSMC fields with r880100 Selected Fields from NSMC fields with r151

00.10.20.30.40.50.60.70.80.91

1000 10000 100000

Prob

ability

Advective-­‐only  Travel  Times  (yr)

100 Selected Fields from multiple seed fields100 Selected Fields from NSMC fields with r120100 Selected Fields from NSMC fields with r515100 Selected Fields from NSMC fields with r910

Objec/ve  Func/on  Selec/ons   Travel  Time  Selec/ons  

3  calibrated  fields  selected  with  each  criterion  (low,  median,  high)  Original  MSP  calibra/on  results    100  NSMC  realiza/ons  shown  for  each  calibrated  field  

Mean  field  of  200  mul/ple  seed  fields  can  be  used  to  get  a  calibrated  model  for  NSMC  method  

Stochastic Inverse Method NSMC method with mean field

Measured steady head (m)

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Prob

ability

Advective-­‐only  (no  dispersion/diffusion)  Travel  Times  (yr)n  =  0.16;  b  =  4m

Multiple  initial  seed  fieldsNSMC  with  5  calibrated  models  (Phi)NSMC  with  5  calibrated  models  (Travel  Time)NSMC  with  mean  field

Comparison  of  travel  /mes  for  different  methods  shows  the  effec/veness  of  the  NSMC  method  

Effects of high T channel

~ 20x speed up

~ 80x speed up

40 RFs from each calibrated model  

100 Selected fields  

Te  has  a  similar  distribu/on  for  both  fields,  but  S.D.  of  Te  distribu/on  is  quite  different  

100 selected Fields MSP (Multiple Starting Point)

100 selected Fields (NSMC method with the mean field)

Te  distribu/on  along  three  transects  capture  calibrated  Te  trends  

Calibrated model with mean field NSMC 100 selected fields

Change  Trunca/on  Threshold?  

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Decrease  threshold  from  1.0E-­‐04  to  1.0E-­‐05,  ~70%  increase  in  number  of  super  parameters  

Conclusions  n  Can  NSMC  approach  approximate  ensemble  predic/ons  obtained  with  MSP  runs?    n  Yes,  but  the  calibra/on  constraint  will  bias  es/mates  and  predic/ons  to  values  proximal  to  the  ini/al  calibra/on  

n  Given  a  set  of  previously  run  models,  what  is  an  effec/ve  means  of  expanding  the  predic/ve  ensemble?  n  Select  final  ensemble  from  larger  set  of  NSMC  realiza/ons  using  calibra/on  quality  and  non-­‐uniform  sampling  from  NSMC  realiza/ons    

n  Without  exis/ng  calibra/ons,  can  a  mean-­‐field  representa/on  serve  as  an  ini/al  star/ng  point?  n  Yes,  NSMC  realiza/ons  provide  good  match  to  observa/ons  and  reasonable  approxima/on  of  MSP  predic/ve  distribu/on  

Ques/ons  

samcken@sandia.gov