# Regularization and trade off parameter

**URL:** <https://simpeg.discourse.group/t/regularization-and-trade-off-parameter/48>\
**Category:** Getting Started\
**Created:** [January 16, 2020, 7:05pm UTC](https://simpeg.discourse.group/t/regularization-and-trade-off-parameter/48 "2020-01-16T19:05:06Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![jcbarreto](https://yyz2.discourse-cdn.com/free1/user_avatar/simpeg.discourse.group/jcbarreto/32/108_2.png) [@jcbarreto](https://simpeg.discourse.group/u/jcbarreto)\
**Post date:** [January 16, 2020, 7:05pm UTC](https://simpeg.discourse.group/t/regularization-and-trade-off-parameter/48/1 "2020-01-16T19:05:06Z")

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How many classes of regularization there are? Which the difference between them? in this file, Regularization.py seems there are a lot: Simple Small Regularization, Simple Smooth Regularization, Tikhonov Regularization, Sparse Regularization. How to choose the beta ratio, cooling factor, and cooling rate?

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**Author:** ![thibaut.astic](https://yyz2.discourse-cdn.com/free1/user_avatar/simpeg.discourse.group/thibaut.astic/32/109_2.png) [@thibaut.astic](https://simpeg.discourse.group/u/thibaut.astic)\
**Post date:** [January 17, 2020, 2:38am UTC](https://simpeg.discourse.group/t/regularization-and-trade-off-parameter/48/2 "2020-01-17T02:38:53Z")

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Hi jcbarreto,

you are right, there are quite a few. They are important because they will strongly influence what our final model will look like.

The most used in Geophysics are the least-squares regularization on the model values (smallness) and model gradients (smoothness). Oldenburg, D. W., and Y. Li, 2005, Inversion for applied geophysics: A tutorial, in D. K. Butler, ed., Near-Surface Geophysics: Society of Exploration Geophysicists, 5, 89-150 is a good reference for those.

Those are registered under two regularizations. `Simple` is the simplest, straight-forward unweighted regularization. This is the one to use if you call any distance or sensitivity-weighting `directives`. The second is `Tikhonov`, which include the cell’s volume as weights, in an attempt at approximating their sensitivity.  
Those two are the ones to start with as a user. When you see `Small` or `Smooth` in the name, it means they refer to those specific parts. So for example, `Simple` has both a `Simple Small` and `Simple Smooth` term in it. You should not have to call any `Small` or `Smooth` on its own.

There are no hard rules for the choice of beta ratio, cooling factor or rate… the rules of thumbs are: “start relatively high, and cool moderately”… starting too high or too low or cooling too fast or too slow this will, of course, affect your convergence.

In “Exploring nonlinear inversions: A 1D magnetotelluric example, Seogi Kang, Lindsey J. Heagy, Rowan Cockett, and Douglas W. Oldenburg, The Leading Edge 2017 36:8, 696-699”, the authors published notebooks you can play with to get a sense of the effect of beta and cooling rate.

> **[simpeg/tle-magnetotelluric\_inversion](https://github.com/simpeg/tle-magnetotelluric_inversion)**
>
> tle - Exploring non-linear inversions: a 1D magnetotelluric example - simpeg/tle-magnetotelluric\_inversion

There are useful directives for beta. `Directives.BetaEstimate_ByEig` estimate the importance of the regularization versus the data misfit. `beta0_ratio` between 1 and 10 is a good first try. Cooling factors such as below are also a good first try.

```auto
beta = Directives.BetaEstimate_ByEig(beta0_ratio=1.)
betaSched = Directives.BetaSchedule(coolingFactor=5., coolingRate=3)

```

The `Sparse` regularization is more involved. It promotes the model to only change at limited locations. See the examples in the documentation.:

1. [http://docs.simpeg.xyz/content/examples/01-basic/plot\_inversion\_linear\_irls.html#sphx-glr-content-examples-01-basic-plot-inversion-linear-irls-py](http://docs.simpeg.xyz/content/examples/01-basic/plot_inversion_linear_irls.html#sphx-glr-content-examples-01-basic-plot-inversion-linear-irls-py)
2. [http://docs.simpeg.xyz/content/examples/05-mag/plot\_nonLinear\_Amplitude.html#sphx-glr-content-examples-05-mag-plot-nonlinear-amplitude-py](http://docs.simpeg.xyz/content/examples/05-mag/plot_nonLinear_Amplitude.html#sphx-glr-content-examples-05-mag-plot-nonlinear-amplitude-py)

finally, not a SimPEG resource, but here is a very basic section about the various parts of an objective function and their respective influence:

[https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html)

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**Author:** ![jcbarreto](https://yyz2.discourse-cdn.com/free1/user_avatar/simpeg.discourse.group/jcbarreto/32/108_2.png) [@jcbarreto](https://simpeg.discourse.group/u/jcbarreto)\
**Post date:** [January 17, 2020, 2:42pm UTC](https://simpeg.discourse.group/t/regularization-and-trade-off-parameter/48/3 "2020-01-17T14:42:57Z")

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Thank you, it is much clear now, and the last link you post is great!