Extracting Meaningful Data: Distinguishing Signal from Noise in Climate Change Q. Steven Hu School of Natural Resources University of Nebraska-Lincoln

In general, noise is the part of information that we want to tease out and signal is the part that we want to keep. Noise is “stubborn” and always present and interferes with signal, forcing us to identify it and find ways to separate it in the data. We don’t want to throw away everything (data) we have collected because of presence of noise, but we cannot keep everything either. We want to keep the signal (the baby) after clearing the noise (the bath water)!

Except for some “absolute noise,” noise and signal are relative and they are determined by the interest of studies. Example 1: The atmosphere contains variations at rather wide ranges of frequencies and spatial scales. All those variations are signals to the climate. Yet, if we are interested in studying a particular frequency variation, e.g., interannual variation – changes of rainfall from one summer to the next, or variation at a specific spatial scale, e.g., the synoptic scale – in the order of 1000km, all the other signals become “noises.” We must identify and remove them before we can examine variations at the interested frequency and scale and understand their behavior and change.

Example 2: In phonological research, an overwhelming number of studies have examined long-term data on phonological patterns and life- events of animals and plants, and found earlier migration to breeding sites, birds laying eggs on earlier dates, and plants flowering earlier. These rather diverse yet consistent changes of phenology are, as believed, responses to a warming climate.

This phenological signal of change is an independent source of information of climate change. To a great extent, this signal is free of errors and noises resulting from gathering and manipulating instrumentation records, thus providing a independent check of this following result from instrumentation data. This phenological signal of change is an independent source of information of climate change. To a great extent, this signal is free of errors and noises resulting from gathering and manipulating instrumentation records, thus providing a independent check of this following result from instrumentation data.

However, the phonological change can only tell us the direction of climate change and cannot tell the rate or magnitude of the change. Can we know how may degrees the temperature may have increased from how many days earlier a bird or a butterfly has migrated north to certain latitude?

Not yet, because the correlations, which have been exclusively used in connecting changes in evolution of life-events of animals and plants with environmental conditions, “do not allow us to discern whether the earlier reproduction is a direct response to warmer temperatures, or to other factors that may also vary with climate, such as reproductive resources and inter- and intra-specific competition,” as elaborated in Post et al. (2001, in Proc. R. Soc. London, B).

When we try to tease the information and single out those responses, the one organism that biologically connects a species reproduction behavior with warmer temperature and dominates the other organisms would be considered the signal, and the rest would be noises. This signal-noise relationship can change in different analyses of varying aspects of the problem. There are, of course, other ways to treat several major organisms simultaneously, and even include their nonlinear interactions (e.g., data mining).

To summarize the previous slides: There are “absolute” noises and erroneous information in data, but more often the noises are relative to the signal we want to examine.

Now, let’s examine what are the “absolute noises” in the meteorological and climatic data ► Instrumentation drift induced noise to data (ground sensor and satellite drift) ► Instrumentation upgrading induced changes in data (sensor differences) ► Station’s geographical location change induced shift to the data (terrain and surface differences) ► Local surroundings change induced noise (a tree grows into full canopy and cools the surroundings of the station, inducing a cooling noise. Similarly the urban expansion may warm a previously rural area, inducing a warming noise to the rural station temperature data) ► Different ways observers read the instrument (low vs. high angle) ► Observation time differences add noise in the data (for precipitation in some frequencies)

How well have we extracted climate signal? (How have we identified and treated the noises?) Many of the noises have been identified and their effects on signals minimized. ► Instrumentation drift induced noise to data ( satellite position drift have been calculated and included in retrieval schemes for surface temperature and precipitation ) ► Instrumentation upgrading induced changes in data (may have considered in developing temperature data series) ► Observation time differences add noise in the data ( Observation time effects were also estimated and included in finalizing the station observed precipitation ) Others remain to be specified and their effects removed in developing climate data (lacking station history has made these following noises very difficult to clarify). ► Station’s geographical location change induced shift to the data ► Local surroundings change induced noise ► Different ways observers read the instrument (can never be certain)

Those noises and biases in the climate data have left uncertainties in the results derived from the data. They have made it particularly difficult to examine local climate change – because for large regions the noise and biases may cancel each other and reduce their effects on the results.

A very brief discussion on Noise in climate models and their outputs

Let me use an example to show the noise resulting from numerical treatment of the governing equations of the atmospheric and oceanic motions. The generalized linear system of governing equations, describing a number of types of wave motions in the atmosphere and ocean, can be written as:

In finite difference, this equation is written (at time = n×Δt) So, the true solution to the equation should have an invariant amplitude, U(0) = the initial amplitude.

In numerical models, various “finite differencing schemes” are used to calculate the values of U at time t and locations. These values are estimates of true U’s and contain noises intrinsic to those schemes. To evaluate the effect of those “numerical noises” on the solutions we use the von Neumann method. By defining a variable λ (“distortion” of the solution from the true one) we can get

Let’s focus on the amplitude of the amplitude of the modeled solution, |λ| n ×U(0). It is different from the analytic (true) solution, and the coefficient |λ| n measures how different the amplitude, U(nΔt), at time step n is from the true solution. We have these possibilities: 1. |λ|>1  the numerical noise grows every time step and quickly overwhelms the signal (the solution of the equation); 2. |λ|=1  neutral solution, noise is minimal (good); and 3. |λ| 1  the numerical noise grows every time step and quickly overwhelms the signal (the solution of the equation); 2. |λ|=1  neutral solution, noise is minimal (good); and 3. |λ|<1  damping solution, noise “erodes” the signal and it will be gone during the model integration.

The value of |λ| for some popular time differencing schemes is shown below.

To summarize: Although researchers have strived to minimize the numerical noises resulting from various numerical schemes used in models, those noises remain and make numerical models and their predictions of climate suffer uncertainties.

Concluding remarks: ► Except for the “absolute noise,” noise and signal are relative and are determined by the nature of a problem. ► The sources of noise can be determined after the research question is well defined. ► Various methods can be used to filter out or attenuate the noises and minimize their effect on signal, although such effect may always exist to varying magnitudes. ► Conventional climate data have many types of noises. ► Phenology data of life-events of animals and plants provide an independent source of information to detect climate and environmental change. A challenge for us to use the data effectively is that the signal are biologically and chemically intertwined with “noises.”

(“Sandhills Cranes in Flight” – photo by Michael Forsberg)