Correcting Negatively-biased Radio Occultation (RO)

Refractivity below Ducts using an Optimal Estimation Approach

Kuo-Nung Wang1 (Eric), Chi O Ao1, Manuel de La Torre Juarez1 and Feiqin Xie2

(1)NASA Jet Propulsion Laboratory, Pasadena, CA, United States

(2)Texas A&M University - Corpus Christi, Corpus Christi, TX, United States

© 2017 California Institute of Technology. Government sponsorship acknowledged.

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Ducting

Motivation

• Negative N-bias (2007~2008)

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[Xie et al, Geophysical Research Letters, Vol. 37, 2010]

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Ducting

Introduction

• Ducting occurs when dN / dh < -157

(N-units/km) at the top of PBL

• The curvature of the signal path will

equal or larger than the curvature of

the Earth

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Earth Ducting layer

[A. T. Young, http://aty.sdsu.edu/explain/simulations/ducting/duct_intro.html]

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Ducting

N-bias

• No tangent point lies within the ducting layer

• x=rn(r) becomes a multi-value function w.r.t. height (r)

• Bending angle w.r.t. x will remain seamless

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h

x

hb

ht

h

α

h

N

True Abel

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Parametrization

Reconstruction

• The h(x) below ducting can be derived analytically based

on bi-linear assumption

• Using other assumptions a h(x) profile can be defined by

a pair of parameter [xb, xm]

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ℎ1 𝑥 = ෨ℎ 𝑥 +2

𝜋ℎ𝑡 − ℎ𝑏 𝑧 − 1 + 𝑧2 𝑡𝑎𝑛−1 Τ1 𝑧

where 𝑧 = ൗ𝑥𝑏−𝑥

𝑥𝑚−𝑥𝑏

xmxb

ht

hb

hm

h1(x)

h2(x)

h3(x)

h4(x)

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Ill-posed problem

Reconstruction

• Infinite number of refractivity solutions corresponds to

the same bending angle profile (Xie et al. 2006)

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Which [xb, xm] pair is correct?

Reconstruction

• Precipitable water (PW): different profiles (different [xb,

xm]) contains different PW

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[Wang et al, AMT discussion, 2017]

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Which [xb, xm] pair is correct?

Reconstruction

• Optimal Estimation

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xb

xm

Apriori

Measurement

Apriori

Apriori

𝐶0

Estimated [xb, xm]

𝐶𝑦

PW

Optimal

Estimation

Ƹ𝑠𝑛+1 = 𝑠0 + (𝐶0−1 + 𝐾𝑛

𝑇𝐶𝑦−1𝐾𝑛)

−1𝐾𝑛𝑇𝐶𝑦

−1 𝑦 − 𝑦𝑛 − 𝐾𝑛 𝑠0 − Ƹ𝑠𝑛

y

𝑠 Ƹ𝑠

𝐾𝑛 = ቤ𝜕𝑦

𝜕𝑠𝑠=𝑠𝑛

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Refractivity Reconstruction (Simulation)

Results

• End-to-end validation

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PW

Radiosonde

Refractivity

A-priori:

[xb, xm]

Reconstructed

Refractivity

Forward

Abel

Optimal

Estimation

Bending

Angle

Inverse

Abel

Abel

Refractivity

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Refractivity Reconstruction (Simulation)

Results

• VOCALS (VAMOS Ocean-Cloud-Atmosphere-Land

Study) campaign collocated with AMSR-E

• Location: Southeast pacific ocean

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Refractivity Reconstruction (Simulation)

Results

• X-h curves and N-h curves

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Refractivity Reconstruction (Simulation)

Results

• 19 cases (reconst. - RAOB)

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Refractivity Reconstruction (Simulation)

Results

• 6 cases (reconst. - RAOB)

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Refractivity Reconstruction (Actual RO signals)

Results

• 8 cases

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Refractivity Reconstruction (Actual RO signals)

Results

• 8 cases (Reconst. - RAOB)

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Summary

• Optimal estimation constrained by PW measurements reduces the

RO N-bias to < 1% in simulation and < 5% in actual data compared

to 15% using inverse Abel

• Critical layer height, thickness and refractivity increment can also be

better estimated with PW-constrained optimization than with Abel

integrals

• In addition to PW constraint, the flexibility of the optimal estimation

framework allows to use other physical constraints to correct the N-

bias in the presence of ducting

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jp l .nasa.gov

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Planetary Boundary Layer (PBL)

Introduction

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Earth’s surface

~1km PBL

~10km Free Troposphere

• Temperature inversion:

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GPS Radio Occultation (RO)

Introduction

• Forward Model:

𝛼 𝑟𝑜 = −2𝑛 𝑟𝑜 𝑟𝑜න𝑟0

∞ 1

𝑛 𝑟

𝑑𝑛 𝑟

𝑑𝑟

1

𝑛 𝑟 𝑟 2 − 𝑛 𝑟𝑜 𝑟𝑜2𝑑𝑟

• Changing variable: x = rn(r)

• Assume the value of x has 1-to-1 relationship with r.

• Forward Abel Transform:

𝛼 𝑥𝑜 = −2𝑥𝑜න𝑥0

∞ 1

𝑛 𝑥

𝑑𝑛 𝑥

𝑑𝑥

1

𝑥2 − 𝑥02

𝑑𝑥

where 𝑥 = 𝑟𝑛 𝑟

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Refractive index

Bending Angle

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GPS Radio Occultation (RO)

Introduction

• Inverse Abel Transform:

𝑛 𝑥𝑜 = 𝑒𝑥𝑝1

𝜋න𝑥0

∞ 𝛼 𝑥

𝑥2 − 𝑥𝑜2𝑑𝑥

• The retrieved refractivity profile has 1-to-1 relationship

with the bending angle observation.

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Bending Angle

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Parametrization (Updated)

Progress

• Steps: (for each [x1, x2])

– h3 can be calculated based on x1

– ෨ℎ 𝑥 is a square root function of x at x1: linear regression is used

to fit ෨ℎ 𝑥 and calculate C (to get h1)

• ෨ℎ 𝑥 − ℎ3 = − 𝐶 𝑥1 − 𝑥

• 𝐶 = −16

𝜋2ℎ3−ℎ1

2

𝑥2−𝑥1

– Get the profile below h1:

– The slope of h-x close to x1 should remain the same (get h2)

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ℎ𝐴 𝑥 = ෨ℎ 𝑥 +2

𝜋ℎ3 − ℎ1 𝑧 − 1 + 𝑧2 𝑡𝑎𝑛−1 ൗ1 𝑧

x2x1

h3

h1

h2

a

b

c

d

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Which [xb, xm] pair is correct?

Reconstruction

• Apriori for xb:

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Which [xb, xm] pair is correct?

Reconstruction

• PW calculation for certain [xb, xm] :

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𝑃𝑊 = න𝑞(𝑁, 𝑇)𝑑𝑝

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Refractivity Reconstruction (Simulation)

Results

• 6 cases

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Refractivity Reconstruction (Actual RO signals)

Results

• 8 cases

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Refractivity Reconstruction (Actual RO signals)

Results

• 8 cases

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Refractivity Reconstruction (Actual RO signals)

Results

• Reconstruction on RO signals

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PW

GPS-RO

Refractivity

A-priori:

[xb, xm]

Reconstructed

Refractivity

Optimal

Estimation

Radiosonde

Refractivity