{"id":3016,"date":"2026-01-28T08:00:56","date_gmt":"2026-01-28T08:00:56","guid":{"rendered":"https:\/\/research.reading.ac.uk\/met-darc\/?p=3016"},"modified":"2026-01-28T11:28:29","modified_gmt":"2026-01-28T11:28:29","slug":"gaussian-anamorphosis-for-log-normal-distributions","status":"publish","type":"post","link":"https:\/\/research.reading.ac.uk\/met-darc\/2026\/01\/28\/gaussian-anamorphosis-for-log-normal-distributions\/","title":{"rendered":"Gaussian anamorphosis for log-normal distributions"},"content":{"rendered":"
By Yumeng Chen, January 2026.<\/p>\n
Limitations of the normal distribution in data assimilation<\/h4>\n
Operational data assimilation methods typically perform best when both background (forecast) and observation errors can be approximated by a normal distribution. The normal distribution has many attractive mathematical properties. For example:<\/p>\n
\n
It is fully characterised by its mean and (co)-variance.<\/li>\n
Its mean, mode, and median all coincide.<\/li>\n<\/ul>\n
However, a limitation of the normal distribution is that it allows both positive and negative values. This assumption is inappropriate for many strictly non-negative weather and climate variables. For example, we do not expect negative rainfall or cloud cover.<\/p>\n
How could we handle log-normal variables?<\/h4>\n
To address this issue, a common approach is to transform the original variable into a new variable that follows a normal distribution. Data assimilation is then performed for the transformed variables, and the resulting analysis is converted back to the original variable. This approach is known as Gaussian anamorphosis<\/em>.<\/p>\n
Operational marine ecosystem forecasting systems routinely assimilate phytoplankton. These are often considered to be log-normally distributed. This can be transformed to a normal distribution.<\/p>\n
In the one-dimensional case, the mean (\u03bc<\/em>) and variance (\u03c3<\/em>2<\/sup>) of the associated normal distribution fully determine the log-normal distribution. For a normal distribution, \u03bc<\/em>\u00a0defines the location of the distribution, while \u03c3<\/em>\u00a0controls the spread.<\/p>\n
The shape of a log-normal distribution behaves differently. While \u03bc<\/em>\u00a0still controls the median of the distribution, both \u03bc<\/em>\u00a0and \u03c3<\/em>\u00a0influence its variance, mean, and mode. With fixed \u03c3<\/em>, increasing \u03bc<\/em>\u00a0not only shifts the distribution but also broadens it. Likewise, with fixed \u03bc<\/em>, \u03c3<\/em> controls the variance of the log-normal distribution (see Figure 1).<\/p>\n
Are analyses always less uncertain than the background?<\/h4>\n
What does this imply for data assimilation? When variables are transformed to be normally distributed, if the observation has a larger \u03bc<\/em>\u00a0than the background, the analysis will typically increase \u03bc <\/em>toward the observation. At the same time, \u03c3<\/em>\u00a0is reduced relative to the background, reflecting reduced uncertainty after assimilating observations due to additional information from observation in a normal distribution.<\/p>\n
However, when the analysis is transformed back into the original variable, this reduction in \u03c3<\/em>\u00a0does not necessarily lead to reduced uncertainty. Because the shape of the log-normal distribution depends on both \u03bc<\/em> and \u03c3<\/em>, an increase in \u03bc<\/em> can outweigh the decrease in \u03c3<\/em>, leading to a broader distribution than the background. The increased variance comes from large variance from observations with large \u03bc<\/em>. This could be particularly evident when biases exist for \u03bc<\/em>\u00a0in either observations or background.<\/p>\n
The log-normal distribution is an interesting example of how distribution parameters can behave very differently from those of a normal distribution. Nevertheless, in marine ecosystem experiments, Gaussian anamorphosis routinely brings the analyses closer to the observations. This implies some improvements even though the ensemble spread is not necessarily reduced for some observations.<\/p>\nFigure 1: An illustration of a log-normal distribution. Upper left: the probability density function of a log-normal distribution with the same \u03c3<\/em> and different \u00b5<\/em>. Upper right: the same as upper left but with the same \u00b5<\/em> and different \u03c3<\/em>. Bottom: the difference of standard deviation between background and analysis from a univariate data assimilation experiment where background and observations are sampled from a normal distribution of a truth of 0 and \u03c3<\/em> = 1 and \u03c3<\/em> = 0.9, respectively. Here, for demonstration purposes, we only show the case where the observations are larger than the truth (i.e., biased observations). A negative difference means that the analysis variance is larger than the background.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"
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