WAIT A MINUTE..... MARTIAN WIND AUTOCORRELATIONS
R. D. Lorenz, Space Exploration Sector, Johns Hopkins Applied Physics Laboratory, Laurel, MD
20723, USA (ralph.lorenz@jhuapl.edu)
Introduction:
A number of operations on the surface of
Mars benefit from knowledge of the wind : regolith manipulation for sampling or solar array cleaning may have more favorable results under certain
wind conditions, and flight of e.g. a helicopter
may be risky in strong gusts. While it is impossible to reliably predict the instantaneous wind during the turbulent early afternoon period purely as
a function of time, a useful predictive degree of
skill can be brought via time series analysis. Specifically, the turbulent afternoon is characterized
by an alternation between stronger winds and
weaker ones, presumably associated with the updraft walls and downwelling cores of boundary
layer convection cells. The scale of these circulations introduces a characteristic timescale of the
order of 5~20 minutes which defines correlations
in the wind speed time series. The autocorrelation
function (ACF) is above 0.5 for timescales less
than about a minute , whereas the ACF is actually
negative for 200-500s. These statistical features
can be reproduced with simple Markov models,
and can be exploited to improve the odds of operational success via logic of the form “attempt 2
regolith manipulations separated by 8 minutes to
maximize the likelihood that one is during low
winds”. The existence of correlations in the wind
time series may suggest some circumspection
regarding autocorrelation results from seismic
data.

Figure 1 : Wind speed time series on Sol 782.

Analysis :
The autocorrelation function (ACF) of a time
series evaluates the relationship of values separated by an interval termed the ‘lag’ and provides
insights that are distinct from frequency analysis
such as Fourier decomposition. A white noise
signal has an ACF that is unity for zero lag (0,1),
and approximately zero for all other values, since
white noise is uncorrelated – the signal at time
(t+lag) has no dependence on what the signal
happened to be at time (t). A sinusoidal signal
has a sinusoidal ACF, since the signal is perfectly
equal to itself one cycle later, but perfectly anticorrelated with itself half a cycle later (or one-anda-half etc.). Most signals in dynamic systems
have a fall-off from (0,1) that indicates some filtering or ‘memory’ in the system and tails off to zero
at large lag.

Data:
We examine wind speed data from the dualboom TWINS (Temperature & Wind for InSight)
instrument, part of InSight’s Auxiliary Payload
Sensor Suite (APSS). Wind measurements are
acquired at a height of 1.2 m above the surface, at
a cadence that may be typically 0.1 or 1 Hz.
A quasiperiodic character can be discerned in
the time series (Fig.1) – such periodicities have
been noticed previously in Viking wind and seismic data (Lorenz et al., 2017), and in InSight optical depth information (Lorenz et al., 2020) and
wind and temperature data (Lorenz et al., 2021).

Figure 2 :Autocorrelation of the time series in figure 1.
Note the rapid fall-off, indicating the atmosphere loses
~80% of its memory of the wind in about 150s. T he negative region shows that if it is windier than average at
time zero, it is actually more likely than not to be less
windy than average 500-800s later.

The ACF of the noontime wind signal for a representative Sol (782) is shown in figure 2. First, the
correlation falls off to 0.5 at a lag of about 40s,
and 0.2 at around 150s. Thus after 2-3 minutes,
the wind has largely ‘forgotten’ its initial value.
What is most striking about the ACF, however, is
that it does not just decay to zero, but overshoots
to become negative – the signature of a quasiperiodic signal. The negative section reaches
only ~-0.1, and is around 200s wide.
Markov Models:
Crudely, the time series in figure 1 shows a bimodal distribution of values – sequences of low
values interspersed by sequences of high values,
or lulls and gusts respectively. Considering these,
then, as two states, we can represent the time series by a Markov model. This is a discrete system
that at each timestep transitions randomly from
one state into another (or remains in the same
state) according to a set of probabilities called a
transition matrix. In this simple construct (a firstorder Markov model), the present system state is
the only information retained by the system.
The time series that results reflects the transition matrix. If the probability of Lull --> Gust is 0.1
then the probability of Lull --> 0.9, since the two
must sum to unity (the state must either change to
the other state, or remain the same). It follows
then, that once the system falls into the Lull state,
it will remain there for ~10 timesteps before the
next gust occurs. Similarly, if the Gust --> Lull
probability is 0.5, then on average a gust lasts for
2 timesteps, and the ‘duty cycle’ of gusts is
Not coincidentally with the Mars gust application, a two-state Markov model has been shown
to be a reasonable representation of the presence
or absence of dust devils on a terrestrial desert
playa (Lorenz et al., 2018). Since both Mars gusts,
and the presence of dust devils, are physically
associated with the upwelling sheets of warm air
in the convecting planetary boundary layer, it is
unsurprising that the same mathematical model
has some success.
Markov models can be constructed with a
much finer discretization, e.g. states being defined
as wind between 3.0 and 4.0 m/s, 4.0 to 5.0 m/s etc.
Such models have been proposed for over two
decades for wind energy forecasting (e.g. Sahin
and Sen, 2001). Much more elaborate variants can

be developed, with multiple orders to provide
longer memory in the system, or with ‘hidden’
variables. However, the clarity and simplicity of
the first-order two-state is useful in the present
application.
Applications:
The periodicity can be exploited for operations
as described in the introduction. In particular, the
temporal spacing of repeated operations can be
optimized to maximize the chances of success.
Similarly, conditional decision rules can be constructed, e.g. “Attempt operation once winds
have dropped below 4 m/s if winds were >4m/s for
4 minutes previously”.
Some seismic analyses employ autocorrelation
methods to look for echos at characteristic times,
and to interpret these as subsurface reflectors. If
the surface forcing by the atmosphere is itself
quasiperiodic at the relevant periods, interpretation as internal structure may merit careful scrutiny.
Acknowledgement:
RL acknowledges the support of the NASA
InSight Participating Scientist Program via Grant
80NSSC18K1626.
References:

Lorenz, R. D., B. K. Jackson and P. D. Lanagan,
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