Theory
of GPS Remote Sensing
·
In
1993, the first GPS remote sensing experiment for probing the earth's
atmosphere was created and called GPS/MET
·
GPS/MET was an active satellite-to-satellite remote sensing experiment that
used the occultation technique
·
Occultation technique - measurement of phase delay (slowing down) and
therefore bending of an electromagnetic wave as it passes through a medium of
different density, i.e., the atmosphere
·
Note
that this is similar to white light passing through a prism
·
This
technique provides vertical profiles of:
1.
N(z) - refractive index
2.
T(z) - temperature
3.
P(z) - pressure
·
In
earth's atmosphere, the bending angle, a can be as large as 1.5 degrees near the
ground (large r) and as small as 10-4 degrees at upper levels (60 km or so)
·
These
small angles can not be measured directly
·
Instead,
phase delays of L1 and L1 are measured - converted into a Doppler-shifted
frequency
·
So,
from the phase delays, one can determine the bending angle, a
·
Once
a is
known, then the refractive index, m(z) can be found
·
Now,
the refractive index depends upon:
1.
Pd
- pressure of dry air (mb)
2.
T
- temperature of the air (K)
3.
e - vapor pressure (mb)
through:
(1)
By
defining N = atmospheric refractivity
as:
(2)
Then (1) can be rewritten as (verify on your own):
(3)
Even though you know N, still have one equation with three unknowns (P, T, and e)
So,
what's one to do????
Use
the equation of state and the hydrostatic equations!!!
First,
let’s use the hydrostatic equation and integrate from some height, z, to ztop, where ztop
is the highest altitude where an occultation takes place.
(4)
(4) becomes:
or (5)
Now, if e = 0 (OK assumption above 5-7 km
AGL), then (3) becomes
(6)
Using the ideal gas law, P(z)=r(z)RT(z), we can eliminate P(z)/T(z)
in (6), then (6) becomes:
or solving for r(z):
(7)
Now, substituting (7) into (5) gives:
(8)
Note that in (8), ztop is usually
around 90 km where Ptop is about 0.002 hPa and so Ptop
can be dropped in (8). So, if N(z)
is known, then so is P(z).
Using (8), once P(z) is known, then from (6):
(9)
Therefore, from (9), T(z) is known.
The temperature retrieved from (9) is the “dry temperature”, recall that we set e =
0.
Hence, a temperature retrieval using the above
methodology is OK in a dry atmosphere (higher up) and breaks down in the
lower-mid troposphere. The above
methodology will produce a cold bias in T(z) when e is non
negligible
Q: What
about the vertical profile of water vapor?????
Consider ground-based GPS receivers as shown in the
figure to the right ŕ
The phase delay along the zenith direction is called
the “zenith delay” and is related to the
atmospheric refractivity, N(z) by:
where
the integral is in the zenith direction.
Recall
that the atmospheric refractivity is expressed as:
(11)
dry term wet term
The dry term contributes up to 240 cm to the total
zenith delay while the wet term contributes up to 40 cm.
In addition to the zenith delay contributed by the
dry and wet atmosphere, free electrons in the ionosphere also contribute to the
zenith delay. This delay, which can be
calculated independently, gives information about the free electron density in
the ionosphere.
Substituting (11) into (10 gives:
zenith delay (12)
zenith dry delay zenith wet delay
REALITY CHECK – the GPS receivers measure the zenith
delay that is the sum of the zenith dry delay
(due to dry atmosphere) and the zenith wet delay
(due to water vapor)
The zenith wet delay is related to the total amount
of water vapor along the zenith direction, this is what we want to calculate.
Well, if we use the hydrostatic equation and convert the zenith dry delay integral from z to p, then:
or (13)
substituting (13) into the zenith dry delay integral in (12) gives:
zenith dry delay (14)
we can simplify (14) if we use the equation of state:
P = rRT or
(15)
Substituting (15) into (14) gives:
zenith dry delay or after doing the
integration:
zenith dry delay
(16)
So, if you measure the pressure at the location of
the GPS receiver, then from (16), the zenith dry delay can be calculated.
Then, from (12):
zenith delay (17)
zenith dry delay zenith wet delay
we know the zenith wet delay since the GPS receiver
measures the zenith delay and the zenith dry delay is known from (16).
FINALLY…….
The precipitable water vapor (PWV) is directly
related to the zenith wet delay through:
PWV = P×zenith wet delay
(18)
where P = 0.15
There it is!
This methodology is done for each ray (GPS satellite):