Inferring Transients inIce Flow, Ice-Sheet Thickness,

and Accumulation Ratefrom Internal Layers

(near the WAIS Divide ice-core site)

Michelle Koutnik, Ed Waddington, Howard Conway University of Washington

Tom Neumann NASA

Steve Price Los Alamos National Laboratory

Radar profile from WAIS Divide

(e.g. Neumann et al. 2008)

Radar profile from WAIS Divide

(e.g. Neumann et al. 2008)

Radar profile from WAIS Divide

ice core

Conway and Rasmussen (2009) Dixon et al. (2004)

Modern surface velocity Modern accumulation rate

How do we infer histories of accumulation and ice dynamics from internal layers?

How well can we infer histories of accumulation and ice dynamics from internal layers?

How do we infer histories of accumulation and ice flow from internal layers?

Estimate unknowns (e.g. accumulation-rate history)

How do we infer histories of accumulation and ice flow from internal layers?

Estimate unknowns (e.g. accumulation-rate history)

Track particlesthrough transient

velocity field

Generateinternallayers

How do we infer histories of accumulation and ice flow from internal layers?

Estimate unknowns (e.g. accumulation-rate history)

Track particlesthrough transient

velocity field

Generateinternallayers

Compare modeled observables to measured quantities; update parameters

iterate.

DATA SET (known)

• Internal layers

• Layer ages

• Modern ice velocity (from GPS)

• Geometry

• Accumulation rates at any point and time

DATA SET (known)

• Internal layers

• Layer ages

• Modern ice velocity (from GPS)

• Geometry

• Accumulation rates at any point and time

PARAMETER SET (unknown)

• Accumulation rate (x,t)

• External-flux forcing (xbounds,t)

• Ice thickness (x0,t0)

• Layer ages

• Ice flux into solution domain (x0,t0)

• Temperature-independent ice softness

• Geothermal flux

FORWARD ALGORITHM

2.5-D thermomechanical ice-flow model.

• Ice-surface evolution

• Ice-temperature evolution

• Ice-velocity field

Track particles to map out an internal layer.

FORWARD ALGORITHM

2.5-D thermomechanical ice-flow model.

• Ice-surface evolution

• Ice-temperature evolution

• Ice-velocity field

Track particles to map out an internal layer.

INVERSE ALGORITHM

Use a gradient inverse method.

• Regularized problem:

• Fit data within a tolerance

• Smooth accumulation profile

• Linearized problem

Find updates to parameter estimates.

“There may be no model that exactly fits the data.”

“If exact solutions exist, they may not be unique…”

“The process of computing an inverse solution can be, and often is, extremely unstable in that a small change in measurement can lead to an enormous change in the estimated model.”

(Aster et al. 2005, pg. 12)

Regularized problem

)Tν(+=I p222 dp

Model size Model residuals

pN

kk

k

iii p

p

rrr

1

0

Linearized problem

)Tν(+=I p222 dp

pN

kk

k

iii p

p

ccc

1

0

Model size Model residuals

How well can we infer histories of accumulation and ice flow from internal layers?

• Accumulation rate (x,t)

• External-flux forcing (xbounds,t)

• Ice thickness (x0,t0)

• Layer ages

• Ice flux into solution domain (x0,t0)

• Temperature-independent ice softness

• Geothermal flux

Two histories.

Icedivide

Ice surface

data = “data”

magenta = initial guessgrey = actual (known) solutionblue = inferred solution

surface

bed

magenta = initial guessgrey = actual (known) solutionblue = inferred solution

presentday

What if accumulation rates are known through time?

magenta = initial guessgrey = actual (known) solutionblue = inferred solutionblack = inferred with accumulation rates through time

magenta = initial guessgrey = actual (known) solutionblue = inferred solutionblack = inferred with accumulation rates through time

… and with a different history…

“West Antarctica”

History of external flux?

Preliminary results:

- Internal layers can be used to infer:

accumulation-rate history ice-thickness (ice divide) history

externally forced flux history

- There may be some tradeoff between parameters, but accumulation rates through time may provide rate control

- Requiring a spatially smooth accumulation history can sufficiently regularize this inverse problem

Near WAIS Divide ice-core site:

- Extend spatial and temporal histories beyond the Holocene

- Use layers dated from the ice core

Regularized problem

(see e.g. Eisen 2008)