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.. currentmodule:: oggm



## Mass-balance

The mass-balance (MB) model implemented in OGGM is an extended version of the temperature index model presented by Marzeion et al., (2012). While the equation governing the mass-balance is that of a traditional temperature index model, our special approach to calibration requires that we spend some time describing it.

.. ipython:: python
:suppress:

fpath = "_code/prepare_climate.py"
with open(fpath) as f:
exec(code)



### Climate data

The MB model implemented in OGGM needs monthly time series of temperature and precipitation. The current default is to download and use the CRU TS data provided by the Climatic Research Unit of the University of East Anglia.

#### CRU (default)

If not specified otherwise, OGGM will automatically download and unpack the latest dataset from the CRU servers.

Warning

While the downloaded zip files are ~370mb in size, they are ~5.6Gb large after decompression!

The raw, coarse (0.5°) dataset is then downscaled to a higher resolution grid (CRU CL v2.0 at 10' resolution) following the anomaly mapping approach described by Tim Mitchell in his CRU faq (Q25). Note that we don't expect this downscaling to add any new information than already available at the original resolution, but this allows us to have an elevation-dependent dataset based on a presumably better climatology. The monthly anomalies are computed following [Harris_etal_2010] : we use standard anomalies for temperature and scaled (fractional) anomalies for precipitation.

#### HISTALP

If required by the user, OGGM can also automatically download and use the data from the HISTALP dataset.

The data is available at 5' resolution (about 0.0833°) from 1801 to 2014. However, the data is considered spurious before 1850. Therefore, we recommend to use data from 1850 onwards. This can be done by setting cfg.PARAMS['baseline_y0'] = 1850.

.. ipython:: python

@savefig plot_temp_ts.png width=100%
example_plot_temp_ts()  # the code for these examples is posted below



#### User-provided dataset

You can provide any other dataset to OGGM. See the HISTALP_oetztal.nc data file in the OGGM sample-data folder for an example format.

#### GCM data

OGGM can also use climate model output to drive the mass-balance model. In this case we still rely on gridded observations (CRU) for the baseline climatology and apply the GCM anomalies computed from a preselected reference period (currently: 1961-1990). This method is often called the delta method.

Currently we can process data from the CESM Last Millenium Ensemble project (see :py:func:tasks.process_cesm_data) only, but adding other models will be available soon.

#### Elevation dependency

OGGM needs to compute the temperature and precipitation at the altitude of the glacier grid points. The default is to use a fixed gradient of -6.5K km ^{-1} and no gradient for precipitation. However, OGGM also implements an optional algorithm which computes the local gradient by linear regression of the 9 surrounding grid points. This method requires that the near-surface temperature lapse-rates provided by the climate dataset are good (i.e.: in most of the cases, you should probably use the simple fixed gradient instead).

### Temperature index model

The monthly mass-balance B_i at elevation z is computed as:

B_i(z) = P_i^{Solid}(z) - \mu ^{*} \, max \left( T_i(z) - T_{Melt}, 0 \right)


where P_i^{Solid} is the monthly solid precipitation, T_i the monthly temperature and T_{Melt} is the monthly mean air temperature above which ice melt is assumed to occur (0°C per default). Solid precipitation is computed out of the total precipitation. The fraction of solid precipitation is based on the monthly mean temperature: all solid below temp_all_solid (default: 0°C) all liquid above temp_all_liq (default: 2°C), linear in between.

The parameter \mu ^{*} indicates the temperature sensitivity of the glacier, and it needs to be calibrated.

### Calibration

We will start by making two observations:

• the sensitivity parameter \mu ^{*} is depending on many parameters, most of them being glacier-specific (e.g. avalanches, topographical shading, cloudiness...).
• the sensitivity parameter \mu ^{*} will be affected by uncertainties and systematic biases in the input climate data.

As a result, \mu ^{*} can vary greatly between neighboring glaciers. The calibration procedure introduced by Marzeion et al., (2012) and implemented in OGGM makes full use of these apparent handicaps by turning them into assets.

The calibration procedure starts with glaciers for which we have direct observations of the annual specific mass-balance SMB. We use the WGMS FoG (shipped with OGGM) for this purpose.

For each of these glaciers, time-dependent "candidate" temperature sensitivities \mu (t) are estimated by requiring that the average specific mass-balance B_{31} is equal to zero. B_{31} is computed for a 31 yr period centered around the year t and for a constant glacier geometry fixed at the RGI date (e.g. 2003 for most glaciers in the European Alps).

.. ipython:: python

@savefig plot_mu_ts.png width=100%
example_plot_mu_ts()  # the code for these examples is posted below



Around 1900, the climate was cold and wet. As a consequence, the temperature sensitivity required to maintain the 2003 glacier geometry is high. Inversely, the recent climate is warm and the glacier must have a small temperature sensitivity in order to preserve its geometry.

Note that these \mu (t) are just hypothetical sensitivities necessary to maintain the glacier in equilibrium in an average climate at the year t. We call them "candidates", since one (or more) of them is likely to be close to the "real" sensitivity of the glacier.

This is when the mass-balance observations come into play: each of these candidates can be used to compute the mass-balance during the period were we have observations. We then compare the model output with the expected mass-balance and compute the model bias:

.. ipython:: python

@savefig plot_bias_ts.png width=100%
example_plot_bias_ts()  # the code for these examples is posted below



The bias is positive when \mu is too low, and negative when \mu is too high. Here, the bias crosses the zero line twice. All dates correspond to approximately the same \mu (but not exactly, as precipitation and temperature both have an influence on it). These dates at which the \mu candidates are close to the real \mu are called t^* (the associated sensitivities \mu (t^*) are called \mu^*). For the next step, one t^* is sufficient: we pick the one which corresponds to the smallest absolute bias.

At the glaciers where observations are available, this detour via the \mu candidates is not necessary to find the correct \mu^*. Indeed, the goal of these computations are in fact to find t^*, which is the actual value to be interpolated to glaciers where no observations are available.

The benefit of this approach is best shown with the results of a cross-validation study realized by Marzeion et al., (2012) (and confirmed by OGGM):

Benefit of spatially interpolating t^{*} instead of \mu ^{*} as shown by leave-one-glacier-out cross-validation (N = 255). Left: error distribution of the computed mass-balance if determined by the interpolated t^{*}. Right: error distribution of the mass-balance if determined by interpolation of \mu ^{*}.

This substantial improvement in model performance is due to several factors:

• the equilibrium constraint applied on \mu implies that the sensitivity cannot vary much during the last century. In fact, \mu at one glacier varies far less in one century than between neighboring glaciers, because of all the factors mentioned above. In particular, it will vary comparatively little around a given year t : errors in t^* (even large) will result in small errors in \mu^*.
• the equilibrium constraint will also imply that systematic biases in temperature and precipitation (no matter how large) will automatically be compensated by all \mu (t), and therefore also by \mu^*. In that sense, the calibration procedure can be seen as a empirically driven downscaling strategy: if a glacier is here, then the local climate (or the glacier temperature sensitivity) must allow a glacier to be there. For example, the effect of avalanches or a negative bias in precipitation input will have the same impact on calibration: \mu^* should be reduced to take these effects into account, even though they are not resolved by the mass-balance model.

The most important drawback of this calibration method is that it assumes that two neighboring glaciers should have a similar t^*. This is not necessarily the case, as other factors than climate (such as the glacier size) will influence t^* too. Our results (and the arguments listed above) show however that this is an approximation we can cope with.

In a final note, it is important to mention that the \mu^* and t^* should not be over-interpreted in terms of "real" temperature sensitivities or "real" response time of the glacier. This procedure is primarily a calibration method, and as such it can be statistically scrutinized (for example with cross-validation). It can also be noted that the MB observations play a relatively minor role in the calibration: they could be entirely avoided by fixing a t^* for all glaciers in a region (or even worldwide). The resulting changes in calibrated \mu^* will be comparatively small (again, because of the local constraints on \mu). The MB observations, however, play a major role for the assessment of model uncertainty.

### References

 [Harris_etal_2010] Harris, I., Jones, P. D., Osborn, T. J., & Lister, D. H. (2014). Updated high-resolution grids of monthly climatic observations - the CRU TS3.10 Dataset. International Journal of Climatology, 34(3), 623–642. https://doi.org/10.1002/joc.3711

### Implementation details

If you had the courage to read until here, it means that you have concrete questions about the implementation of the mass-balance model in OGGM. Here are some more details:

• the mass-balance in OGGM is computed from the altitudes and widths of the flowlines grid points (see :ref:flowlines). The easiest way to let OGGM compute the mass-balance for you is to use the :py:class:core.massbalance.PastMassBalance.
• the interpolation of t^* is done with an inverse distance weighting algorithm (see :py:func:tasks.local_t_star)
• if more than one t^* is found for some reference glaciers, than the glaciers with only one t^* will determine the most likely t^* for the other glaciers (see :py:func:tasks.compute_ref_t_stars)
• yes, the temperature gradients and the precipitation scaling factor will have an influence on the results, but it is small since any change will automatically be compensated by \mu^*. We are currently quantifying these effects more precisely.

Code used to generate these examples:

.. literalinclude:: _code/prepare_climate.py

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