HYDROGRAV:Improving Hydrological Model Calibration with Ground-Based and Space-Borne Time-lapse

Gravity Surveys

Peter Bauer-Gottwein, Silvia Leiriao and Xin He

Institute of Environment & Resources, Technical University of Denmark

Ole B. Andersen

Danish National Space Centre, Technical University of Denmark

The state of the art in hydro-gravityGravity Hydrology

Hydrological dataConceptual hydro model

Parameter estimates

GRACE inter-satellite range data

Geophysical inversionspherical harmonics (e.g. Wahr et al., 1998)

or MASCON (e.g. Rowlands et al., 2005 )or wavelet expansion (Seitz et al., this meeting)

Correction for climate and oceansInfinite sheet conversion

Numerical hydrological modelCalibration with hydro dataGrid or catchment based

Simulated water storage changes

GRACE-derived water storage changes

Comparison and Conclusionse.g. Winsemius et al., 2007, Andersen et al., 2005, Rodell et al., 2004, Seneviratne et al., 2004

The HYDROGRAV ApproachHydrological data

Conceptual hydro modelParameter estimates

Hydro-Geophysical inverse problem

Hydro observationsGravity observations

Numerical hydrological modelGrid or catchment based

Hydrological forward problem

Simulated gravity changes

Simulated water storage changes

Geophysical forward problem

The hydrological forward problem

• Surface water, unsaturated zone and groundwater components

• Physically based and lumped parameter models• Fully distributed (grid based) and catchment

based models• Modeling approach is scale dependent and

constrained by data availability• Geologic heterogeneity is a problem• Variety of well-established and thoroughly tested

modeling tools (e.g. USGS MODFLOW, MIKE SHE, SWAT, Hydrogeosphere, Hydrus, HEC-RAS)

Geophysical forward problem: Ground-based gravity observations

( )

( ) ( ) ( )( )3/22 2 2( , , ) m

m m m

z zg x y z dzdydxx x y y z z

γ ρ+∞+∞+∞

−∞−∞−∞

− −∆ = ⋅ ∆ ⋅− + − + −

∫ ∫ ∫∆g: Hydrological gravity signal (ms-2)γ: Gravitational constant (6.67·10-11 Nm2kg-2)∆ρ: Density variation (kgm-3)xm: x-coordinate of gravity measurement locationym: y-coordinate of gravity measurement locationzm: z-coordinate of gravity measurement location

xm,ym,zm

Hydrological density variations

Saturated Zone w SS hρ ρ∆ = ⋅ ⋅ ∆

ρw: water density (1000 kgm-3)SS: specific storage (m-1)∆h: head change (m)

Unsaturated Zone wρ ρ θ∆ = ⋅ ∆

ρw: water density (1000 kgm-3)θ: water content (-)

Rivers, Lakes, Overland Water

w fSρ ρ∆ = ⋅ ∆

ρw: water density (1000 kgm-3)Sf: flooding state: 1 if

flooded, 0 else

Vertical integration over anomalous region

Surface water bodies and unconfined aquifers: ∆ρ is constant in the region between the initial and the final water table and 0 elsewhere.

xm,ym,zm

hi

hf

wat

er le

vel

( , , ) ( , ) ( , , )

1 for ( , ) ( , )( , , )

0 else

w

i f

x y z Sy x y f x y z

h x y z h x yf x y z

ρ ρ∆ = ⋅

⎧ < <⎪⎪= ⎨⎪⎪⎩

hi: initial water table elevationhf: final water table elevationSy: specific yield, 1 for surface

water, 0.01-0.3 for unconfined aquifers

Vertical integration of anomalous region

( )

( ) ( ) ( )( )

( )

( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( ) ( )( )

3/22 2 2

3/22 2 2

1/222 2

1/22 2 2

( , ) ( , , )

( , )

( , )

f

i

mw

m m m

hm

wh m m m

m f

i

m mw

m m m

z zg Sy x y f x y z dzdydx

x x y y z z

z zSy x y dzdydx

x x y y z z

x x y y zSy x y

x x y

h

hy z

γ ρ

γ ρ

γ ρ

+∞+∞+∞

−∞−∞−∞

+∞+∞

−∞−∞

−

−

− −∆ = ⋅ ⋅ ⋅ =− + − + −

− −= ⋅ ⋅ =− + − + −

⎡ ⎤− + − + −⎢ ⎥⎢ ⎥== ⋅ ⋅ ⎢− − + − + −⎢⎣ ⎦

∫ ∫ ∫

∫ ∫ ∫

dydx+∞+∞

−∞−∞⎥⎥

∫ ∫

Surface convolution of the density distribution with the inverse square distance weight

Geophysical forward problem: Space-based gravity observations

• Spherical coordinates• Distance sensor-target much larger• Assumption of 2D density variation usually

ok• Inter-satellite range data• Spherical harmonic coefficients• MASCONs• ...

Spherical Harmonic Coefficients

“Primary” GRACE observations: Spherical harmonic coefficients up to N=100 at monthly intervals:

G: Gravity anomaly or geoid height anomaly (ms-2 or m)δCnm, δSnm: Spherical harmonic coefficients (ms-2 or m)λ: Longitudeθ: Co-latitude (i.e. the difference between the latitude and 90° )

Pnm: Associated Legendre function. Normalization:

( )1 0

( ) ( )cos( ) ( )sin( ) (cos )N n

nm nm nmn m

G t C t m S t m Pδ δ λ δ λ θ= =

= +∑∑

2,0

0

(cos )sin 2(2 )mn mP dπ

θ θ θ δ= −∫

( ) ( ) ( )( )/22 20

2 1)( ! 1( ) (2 ) 1 1

( )! 2 !

m nm nnm m n m n

n n m dP x x x

n m n dxδ

+

+⎡ ⎤+ − ⎢ ⎥= − − −

+ ⎢ ⎥⎣ ⎦

Transforming simulated mass distribution into spherical harmonic coefficients

(Wahr et al., 1998, Ramillien et al., 2004, 2005)

For 3-dimensional mass distribution

( )2

cos( ) 1 ( , , , ) (cos ) ( ) sin( )sin2 1( )nm n

nmnnm V

C tr t r P m r drd d

n MRS t

δρ θ λ θ λ θ θ λ

δ

⎧ ⎫⎪ ⎪ ⎛ ⎧ ⎫ ⎞⎪ ⎪⎪ ⎪ ⎟⎪ ⎪⎜ ⎟= ∆ ⋅ ⋅⎜⎨ ⎬ ⎨ ⎬ ⎟⎜ ⎟⎜⎪ ⎪ ⎪ ⎪+ ⎝ ⎠⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭∫∫∫

or, if mass storage is a surface storage

( )

2 cos( ) (1 ) ( , , ) (cos ) ( ) sin( )sin2 1( )nm n

nmnm S

C t z R t P m d dn MS t

δρ θ λ θ λ θ θ λ

δ

⎧ ⎫⎪ ⎪ ⎛ ⎧ ⎫ ⎞⎪ ⎪+⎪ ⎪ ⎟⎪ ⎪⎜ ⎟= ∆ ⋅⎜⎨ ⎬ ⎨ ⎬ ⎟⎜ ⎟⎜⎪ ⎪ ⎪ ⎪+ ⎝ ⎠⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭∫∫

M: mass of the earth (5.97602·1024 kg)R: Radius of the earth (6378 km)zn: Love numbers for the elastic response of a surface-loaded earth

NASA/GSFC GRACE MASCONS • Processed GRACE level 1B Data from July 2002 - Dec 2006• Upgraded atmospheric series, improved ocean tide models, improved

processing. • Mascons solved on a 4ºx4º grid every ten days, where sufficient data

were available to construct a solution.• Apply a spatial & temporal constraint of the form:

where dij and tij are the distance and time differences between the mascons, where T and D are the correlation time and distance .

• Used T=10 days & D= 250 km.• Mascons are computed relative to a mean background field. • The signature in GRACE from a mascon only manifest itself over the

area of overflight.• No global aliasing problem like in spherical harmonic solutions.

exp 2 ij ijd tD T

⎛ ⎞⎟⎜ − − ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Rowlands et al., 2005

Hydro-Geophysical inverse problem

• Joint conditioning of hydrological model parameters with gravity observations and classical hydrological observations (discharge, water levels, soil moisture etc.) See also Werthet al., this meeting.

• Weighted least squares fitting approach using gradient search methods (e.g. PEST), multi-objective evolutionary algorithms, etc.

• Number of degrees of freedom are significantly reduced compared with pure geophysical inversion: From total number of grid cells or spherical harmonic coefficients to number of hydro model parameters

Synthetic Example: Ground Gravity Monitoring of Pump test

100 101 102 103-30

-25

-20

-15

-10

-5

0

Distance from well (m)

Hea

d di

ffere

nce

(m)

1 hour5 hours1 day7 days

100 101 102 103-50

-40

-30

-20

-10

0

Distance from well (m)

Gra

vity

cha

nge

( µ G

al)

1 hour5 hours1 day7 days

Hydrological forward problem• Approx. Jacob 1950 solution• T = 0.0027 m2s-1

• Sy = 0.1• Q = 300 m3h-1

Geophysical forward problem• Numerical integration with

MATLAB’s dblquad routine• Instrument 5m above initial

water table

Synthetic gravity

100 102 104 106-40

-30

-20

-10

0

10

time (s)

∆ g

( µ g

al)

observations at 10 m from the well, with an instrument error of 5 µgal (expected Scintrex CG5 accuracy)

Hydro-Geophysical inverse problem• Solved with Nelder-Mead

simplex search algorithm in MATLAB

• Error of recovered transmissivity: 0.0003 m2s-1

or 11%100 102 104 106-40

-30

-20

-10

0

10

time (s)

∆ g

( µ g

al)

Real-world Example: Hydrology of Northern Sealand

Hydrological forward problem:• Finite difference

groundwater model implemented in MODFLOW (Jan Jeppesen et al.)

• Total simulation period of 150 years

• Horizontal grid discretization of 125-250 m

• Vertical discretization into 7 layers, partly convertible

x (UTM 32N)

y (U

TM 3

2N)

6.9 7 7.1 7.2 7.3x 105

6.165

6.17

6.175

6.18

6.185

6.19

6.195

x 106

∆ g

(µ G

al)

-20

-15

-10

-5

0

5

10

15

20

25

30Gravity Change, micro-Gal

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Nov-98 Mar-00 Jul-01 Dec-02 Apr-04

Temporal ground-level gravity variation at the Vestvolden site(compares well with in-situ observations)

Ground-level gravity variation in micro-gal in the model domain Oct 2001-Mar 2002

Geophysical forward problem• Grid-based forward routine using prismatic mass storage

elements. Methods adapted from terrain correction of gravity measurements (Forsberg et al., 1989, Leiriao, 2007, He, 2007)

Conclusions• Space borne and ground-based gravity

observations can be used as calibration targets in hydrological modeling

• HYDROGRAV proposes a hydrogeophysicalinversion approach to the problem, consisting of the hydrological forward model, the geophysical forward model and the hydrogeophysicalinversion routine.

• Utility of the approach is demonstrated with a synthetic pump test example and with a real-world groundwater modeling application from Northern Sealand, Denmark.