Sundermeyer

MAR 550

Spring 2020 1

Laboratory in Oceanography:

Data and Methods

MAR550, Spring 2020

Miles A. Sundermeyer

Methods for

Non-Stationary Means

Sundermeyer

MAR 550

Spring 2020 2

Recall, for OA

• Assumed field is homogeneous and isotropic.

• Assumed errors do not co-vary with themselves or with observations, and

that errors have zero mean.

• Estimated field based on observations and correlation matrix (assumes the

observations are correlated with each other).

• Computed expected error variances (Note, as long as stations don’t change

w/ time, errors also don’t change with time. Can use this to explore possible

station schemes to minimize error in maps.)

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 3

Types of kriging

• Simple kriging (OA, OI) – known constant mean, μ(x) = 0.

• Ordinary kriging - unknown but constant mean, μ(x) = μ, and enough

observations to estimate the variogram/correlation function

• Universal kriging - assumes mean is unknown but linear combination of

known functions,

Extensions

• Lognormal kriging

• Vector fields - incorporate non-divergence (velocity potential, 𝑢 = 𝛻Φ), or

geostrophy (streamfunction, 𝑢 =𝜕𝜓

𝜕𝑦, 𝑣 = −

𝜕𝜓

𝜕𝑥)

• Non-isotropic (challenge for coastal OA – see OAX from Bedford Institute of

Oceanography)

• Multivariate

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 4

Extensions of simple kriging (OI,OA)

Consider problem of a localized tracer, such as dye-release experiment, river

plume, or other localized field.

• Suppose non-zero mean – can always subtract the mean

• Suppose non-isotropic – can scale different directions (assuming correlation

function is still the same)

• Suppose spatially varying mean ... need universal kriging for this

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 5

Example: Dye mapping during Coastal Mixing & Optics Experiment (CMO)

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 6

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• Dye concentration

varies spatially –

approx. Gaussian

in x and y at large

scales.

• Wish to map small-

scale variability –

capture variability

within patch

Sundermeyer

MAR 550

Spring 2020 7

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

Sundermeyer

MAR 550

Spring 2020 8

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• Start with large-

scale interpolation

Sundermeyer

MAR 550

Spring 2020 9

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• Start with large-

scale interpolation

(b=6 km, a=2)

• “interpolate”

smoothed map

onto observation

points as spatially

varying mean.

Sundermeyer

MAR 550

Spring 2020 10

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• compute

covariance

function of

“residual” from first

pass kriging (data

minus spatially

varying mean).

Sundermeyer

MAR 550

Spring 2020 11

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• Do 2nd pass kriging

on “residual”

• Obtain kriging

estimate and error

map

Sundermeyer

MAR 550

Spring 2020 12

Methods for Non-Stationary Means OA (cont’d) and Kriging

Example: CMO

• Do 2nd pass kriging

on “residual”

• Obtain kriging

estimate and error

map

Sundermeyer

MAR 550

Spring 2020 13

Nugget Effect

Though correlation at zero lag is theoretically = 1, sampling error and small

scale variability may cause observations separated by small distances to be

dissimilar. This causes a discontinuity at the origin of the correlation function

called the “nugget” effect.

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 14

Anisotropy

• Kriging/OA can handle different correlation length scales in different

coordinate directions.

• Example: OAX (developed by Bedford Institute of Oceanography)

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 15

Block Kriging

• Use only data within certain range to estimate value at particular location.

Minimizes size of inversion required for OA.

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 16

Time Correlations

• Can also handle time correlations for spatio-temporal data

• Example: AVISO, CNES, France

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 17

http://people.seas.harvard.edu/~leslie/MBST98/ll_analysis.html

“Subjective” Objective analysis …

Need to be mindful of decisions made during OA / kriging analysis

Methods for Non-Stationary Means OA (cont’d) and Kriging

Sundermeyer

MAR 550

Spring 2020 18

References

• A. G. Journel and CH. J. Huijbregts " Mining Geostatistics", Academic Press

1981

Methods for Non-Stationary Means OA (cont’d) and Kriging